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Introduction to Cartesian Geometry

Welcome to Introduction to Cartesian Geometry

In Cartesian or Analytic Geometry we will learn how to represent points, lines, and planes using the Cartesian Coordinate System, also called Rectangular Coordinate System. This can be applied to solve a broad range of problems from geometry to algebra and it will be very useful later on Calculus.

=Cartesian Coordinate System=

The foundations of Analytic Geometry lie in the search for describing geometric shapes by using algebraic equations. One of the most important mathematicians that helped to accomplish this task was René Descartes for whom the name is given to this exciting subject of mathematics.

The Coordinate System
For a coordinate system to be useful we want to give to each point an attribute that helps to distinguish and relate different points. In the Cartesian system we do that by describing a point using the intersection of two(2D Coordinates) or more(Higher Dimensional Coordinates) lines. Therefore a point is represented as P(x1,x2,x3,...,xn) in "n" dimensions.

2D Coordinates
You can look at a 2D Coordinate system as a system in which a point can be defined using two values. We do this by using two perpendicular and directed lines called the abscissa(x-axis) and the ordinate(y-axis). The point of intersection of these two lines is called the origin denoted O(0,0). Any point can be determined as P(x,y), where x is the value in the x-axis and y is the value in the y-axis.

=3d Coordinate Geometry=

The coordinate system in two dimensions uses values containing width and depth expressed as values of $$X$$ and/or $$Y.$$

Coordinate geometry in three dimensions adds the dimension of height, usually expressed as a value of $$Z.$$ The axes of $$X$$ and $$Y$$ usually represent the horizontal plane. The $$Z$$ axis is thus vertical or normal to the plane of $$X$$ and $$Y.$$

In three dimensional space a point is shown as $$(x_1, y_1, z_1)$$ where the 3 values locate the point relative to the origin along each of the $$X, Y, Z$$ axes respectively. The value $$(0, 0, 0)$$ is valid, usually the origin.

Within this page the values $$x_1, y_1, z_1$$ are all real numbers.

Examples of points in 3 dimensional space.
Points of 1 dimension

Each of the points in the diagram has exactly one dimension.

The point $$(10,\ 0,\ 0)$$ has only the dimension of width. It is on the $$X$$ axis.

The point $$(0,\ 12,\ 0)$$ has only the dimension of depth. It is on the $$Y$$ axis.

The point $$(0,\ 0,\ 13)$$ has only the dimension of height. It is on the $$Z$$ axis.

$$$$$$$$$$$$$$$$

Points of 2 dimensions

Each of the points in the diagram has exactly two dimensions.

The point $$(8,\ 6,\ 0)$$ has the dimensions of width and depth, no height.

The point $$(0,\ 12,\ 6)$$ has the dimensions of depth and height, no width.

The point $$11,\ 0,\ -7)$$ has the dimensions of width and height, no depth.

Points of 3 dimensions

Each of the points in the diagram has three dimensions.

Distance between 2 points

Consider the point $$Q\ (4, 3, 12).$$ What is the distance from point $$Q$$ to the origin?

Point $$P$$ is the projection of $$Q$$ on to the plane of $$X,Y.$$

Calculate length $$OP = \sqrt{4^2 + 3^2 }.$$

Then length $$OQ = \sqrt{(OP)^2 + 12^2 } = \sqrt{4^2 + 3^2 + 12^2 } = \sqrt{169} = 13.$$

Length $$OQ = \sqrt{x_1^2 + y_1^2 + z_1^2}.$$ In this case $$(x_1, y_1, z_1) = (4, 3, 12).$$

The general case:

For two points $$M\ (x_M, y_M, z_M)$$ and $$N\ (x_N, y_N, z_N)$$

the distance $$MN = \sqrt{(x_M - x_N)^2 + (y_M - y_N)^2 + (z_M - z_N)^2 }.$$

Length $$OQ$$ is the special case in which $$(x_N, y_N, z_N) = (0, 0, 0).$$

The distance $$MN$$ may be $$0$$ in which case $$M$$ and $$N$$ are the same point.

as 2 points
In the example above line $$OP$$ passes through the points $$O\ (0,0,0)$$ and $$P\ (4,3,0).$$

The line $$OP$$ may be defined by the two points $$O$$ and $$P$$ provided that the length $$OP$$ is non-zero.

Line $$OP = (O, P) = ((0,0,0),\ (4,3,0))$$

There are other ways to define a line. Add a descriptor to avoid confusion:

Line $$OP = (O, P) = ((0,0,0),\ (4,3,0),$$ '$$2$$points'$$)$$ where:

$$(x_O, y_O, z_O) = (0,0,0)$$ and $$(x_P, y_P, z_P) = (4,3,0).$$

Line $$PO = (P, O) = ((4,3,0),\ (0,0,0),$$ '$$2$$points'$$).$$

Similarly:

line $$PQ = (P, Q) = ((4,3,0),\ (4,3,12),$$ '$$2$$points'$$)$$

line $$QP = (Q, P) = ((4,3,12),\ (4,3,0),$$ '$$2$$points'$$)$$

line $$OQ = (O, Q) = ((0,0,0),\ (4,3,12),$$ '$$2$$points'$$)$$ and

line $$QO = (Q, O) = ((4,3,12),\ (0,0,0),$$ '$$2$$points'$$).$$

as point and 3 direction numbers
For information about direction numbers see Line_(Geometry).

Line $$OQ$$ above is defined by 2 points $$O\ (x_1,y_1,z_1)=(0,0,0)$$ and $$Q\ (x_2,y_2,z_2)=(4,3,12).$$

The direction numbers of line $$OQ = [A,B,C] = [x_2-x_1,y_2-y_1,z_2-z_1] = [4,3,12].$$

A line may be defined as $$((x_1,y_1,z_1),\ [A_1,B_1,C_1])$$ where:

$$(x_1,y_1,z_1)$$ is a known point on the line, and

$$[A_1,B_1,C_1]$$ are the direction numbers of the line.

In practice:

The programming language python, for example, recognizes  as a list and code can interpret these values as direction numbers.

Python recognizes  as a tuple and code can interpret these values as coordinates of a point.

In python a line may be defined as:

 * or

Line $$OQ$$ may be defined as $$((0,0,0), (4,3,12),$$ 'p&dn'$$).$$

Given a point on the line and the direction numbers of the line, any other point on the line may be established by introducing constant $$K.$$

On line $$((x_1,y_1,z_1),\ (A_1,B_1,C_1),$$ 'p&dn'$$)$$ any second point $$(x_2,y_2,z_2) = (x_1+KA_1, y_1+KB_1, z_1+KC_1)$$ and distance from $$(x_1,y_1,z_1)$$ to $$(x_2,y_2,z_2)$$

$$= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$

$$= \sqrt{(x_1+KA_1-x_1)^2 + (x_2+KB_1-y_1)^2 + (z_1+KC_1-z_1)^2}$$

$$= \sqrt{(KA_1)^2 + (KB_1)^2 + (KC_1)^2}$$

$$= K\sqrt{A_1^2 + B_1^2 + C_1^2}$$

It would be convenient if the value $$A_1^2 + B_1^2 + C_1^2$$ would $$= 1.$$

If $$A_1^2 + B_1^2 + C_1^2 == 1,$$ the values $$A_1, B_1, C_1$$ are the direction cosines of the line.

To convert direction numbers to direction cosines divide each direction number by $$\sqrt{A_1^2 + B_1^2 + C_1^2}.$$

$$(\frac{A_1}{\sqrt{A_1^2 + B_1^2 + C_1^2} })^2 + ( \frac{B_1}{\sqrt{A_1^2 + B_1^2 + C_1^2} }     )^2 + (  \frac{C_1}{\sqrt{A_1^2 + B_1^2 + C_1^2} }   )^2$$

$$= \frac{A_1^2}{A_1^2 + B_1^2 + C_1^2} + \frac{B_1^2}{A_1^2 + B_1^2 + C_1^2} + \frac{C_1^2}{A_1^2 + B_1^2 + C_1^2}$$

$$= \frac{A_1^2 + B_1^2 + C_1^2}{A_1^2 + B_1^2 + C_1^2} = 1.$$

If $$[A,B,C]$$ are the direction cosines of the line, any second point on the line exactly $$K$$ units from $$(x_1, y_1, z_1)$$ has coordinates $$(x_1 + KA, y_1+KB, z_1+KC).$$

While point $$(0,0,0)$$ is valid, direction numbers $$[A,B,C]=[0,0,0]$$ are not valid. For the group of direction numbers to be valid, at least one of $$A,B,C$$ must be non-zero.

Angle between two lines
In the diagram line $$OM$$ has direction numbers $$[A_1,B_1,C_1].$$ Point $$M$$ has coordinates $$(A_1,B_1,C_1).$$

Line $$ON$$ has direction numbers $$[A_2,B_2,C_2]$$ and point $$N$$ has coordinates $$(A_2,B_2,C_2).$$

Length $$OM = \sqrt{A_1^2 + B_1^2 + C_1^2}.$$

Length $$ON = \sqrt{A_2^2 + B_2^2 + C_2^2}.$$

Length $$MN = \sqrt{(A_1-A_2)^2 + (B_1-B_2)^2 + (C_1-C_2)^2}.$$

By the cosine rule $$MN^2 = ON^2 + OM^2 - 2\cdot OM\cdot ON\cdot\cos(\theta).$$

$$\cos(\theta) = \frac{MN^2 - ON^2 - OM^2}{- 2\cdot OM\cdot ON} = \frac{ON^2 + OM^2 - MN^2}{2\cdot OM\cdot ON}$$ $$= \frac{A_1\cdot A_2 + B_1\cdot B_2 + C_1\cdot C_2}{ \sqrt{A_1^2 + B_1^2 + C_1^2} \cdot \sqrt{A_2^2 + B_2^2 + C_2^2} }$$

If all values are direction cosines, $$\cos(\theta) = A_1\cdot A_2 + B_1\cdot B_2 + C_1\cdot C_2.$$

If $$\cos(\theta) == 0,$$ the lines are normal.

If $$\cos(\theta) == 1$$ or $$\cos(\theta) == -1,$$ the lines are parallel.

Angle between line and each axis:

Let a line have direction cosines $$[A_1,B_1,C_1].$$

Let $$X$$ axis have direction cosines $$[A_2,B_2,C_2] = [1,0,0].$$

Let $$\theta_X$$ be angle between line and $$X$$ axis.

$$\cos(\theta_X) = A_1\cdot 1 + B_1\cdot 0 + C_1\cdot 0 = A_1.$$

Hence the expression direction cosine. $$A_1$$ is the cosine of the angle between line and $$X$$ axis.

Similarly $$B_1$$ is cosine of angle between line and $$Y$$ axis and $$C_1$$ is cosine of angle between line and $$Z$$ axis.

Parallel lines:

Let one line have direction numbers $$[A,B,C]$$ and a second line have $$[KA,KB,KC].$$

$$\cos(\theta)$$ $$= \frac{A_1\cdot A_2 + B_1\cdot B_2 + C_1\cdot C_2}{ \sqrt{A_1^2 + B_1^2 + C_1^2} \cdot \sqrt{A_2^2 + B_2^2 + C_2^2} }$$ $$= \frac{A\cdot KA + B\cdot KB + C\cdot KC}{ \sqrt{A^2 + B^2 + C^2} \cdot K\sqrt{A^2 + B^2 + C^2} }$$ $$= \frac{K(A^2 + B^2 + C^2)}{ K(A^2 + B^2 + C^2 ) } = 1.$$

$$\theta = 0^\circ$$ and the lines are parallel.

Similarly, if the second line has direction numbers $$[-KA,-KB,-KC],\ \cos(\theta) = -1,\ \theta = 180^\circ$$ and the lines are parallel.

To verify that two lines are not parallel, check that $$|\cos(\theta)|$$ does not equal $$1.$$

If one line has direction numbers $$[A_1,B_1,C_1]$$ and the second line has direction numbers $$[A_2,B_2,C_2] = [KA_1,KB_1,KC_1]$$ then:

$$\frac{A_2}{A_1} = \frac{B_2}{B_1} = \frac{C_2}{C_1} = K$$

$$A_2B_1 = A_1B_2,\ B_2C_1 = B_1C_2,\ A_2C_1 = A_1C_2$$

The values $$B_1C_2 - C_1B_2,$$ $$C_1A_2 - A_1C_2,$$ $$A_1B_2 - B_1A_2$$ are the direction numbers of the normal to both lines. If the normal to both lines has direction numbers  the lines are parallel.

Normal to 2 lines
A line normal or perpendicular to two lines is normal to each line.

Before attempting to calculate the direction numbers of the normal, first verify that the two lines are not parallel.

Let the normal have direction numbers $$[A_1,B_1,C_1].$$

Let the 2 lines have direction numbers $$[A_2,B_2,C_2],\ [A_3,B_3,C_3].$$

Then:

$$A_1A_2 + B_1B_2 + C_1C_2 = 0\ \dots\ (1)$$ and

$$A_1A_3 + B_1B_3 + C_1C_3 = 0\ \dots\ (2)$$

$$(1)*C_3,\ A_1A_2C_3 + B_1B_2C_3 + C_1C_2C_3 = 0\ \dots\ (3)$$

$$(2)*C_2,\ A_1A_3C_2 + B_1B_3C_2 + C_1C_3C_2 = 0\ \dots\ (4)$$

$$(3)-(4),\ A_1(A_2C_3-A_3C_2) + B_1(B_2C_3 - B_3C_2) = 0\ \dots\ (5)$$

$$\frac{A_1}{B_1} = \frac{B_3C_2-B_2C_3}{A_2C_3-A_3C_2}\ \dots\ (6)$$

If divisor $$A_2C_3-A_3C_2 == 0,$$ $$\frac{B_1}{C_1} = \frac{C_3A_2-C_2A_3}{B_2A_3-B_3A_2}\ \dots\ (7)$$

If divisor $$B_2A_3-B_3A_2 == 0,$$ $$\frac{C_1}{A_1} = \frac{A_3B_2-A_2B_3}{C_2B_3-C_3B_2}\ \dots\ (8)$$

Provided that all inputs are valid, then at least one of $$[A_1,B_1,C_1]$$ must be non-zero and at least one of $$(6)$$ or $$(7)$$ or $$(8)$$ must be valid.

From $$(6), (7), (8)$$ above:

$$\frac{A_1}{B_3C_2 - B_2C_3} = \frac{B_1}{A_2C_3-A_3C_2} = \frac{C_1}{B_2A_3-B_3A_2}$$

Values of $$[A_1,B_1,C_1]$$ that satisfy this condition are:

$$[B_3C_2 - B_2C_3,\ A_2C_3-A_3C_2,\ B_2A_3-B_3A_2].$$

Proof:

Angle between $$[A_1,B_1,C_1]$$ and $$[A_2,B_2,C_2]:$$

$$\cos (\theta) = (B_3C_2 - B_2C_3)A_2 + (A_2C_3-A_3C_2)B_2 + (B_2A_3-B_3A_2)C_2$$ $$= B_3C_2A_2 - B_2C_3A_2 + A_2C_3B_2-A_3C_2B_2 + B_2A_3C_2-B_3A_2C_2$$ $$= B_3C_2A_2 - B_3A_2C_2 + A_2C_3B_2 - B_2C_3A_2 + B_2A_3C_2 - A_3C_2B_2$$ $$= 0$$

Therefore the two groups of direction numbers are normal.

Similarly it can be shown that the two groups of direction numbers $$(A_1,B_1,C_1)$$ and $$(A_3,B_3,C_3)$$ are normal.

The line normal to $$[A_2,B_2,C_2]$$ and $$[A_3,B_3,C_3]$$ has direction numbers:

$$A_1 = B_3C_2 - C_3B_2$$

$$B_1 = C_3A_2 - A_3C_2$$

$$C_1 = A_3B_2 - B_3A_2$$

Parallel input:

Let $$[A_3,B_3,C_3] = [KA_2,KB_2,KC_2]$$

Then $$A_1 = B_3C_2 - C_3B_2 = (KB_2)(C_2) - (KC_2)(B_2) = 0$$

If the two groups of direction numbers $$[A_2,B_2,C_2], [A_3,B_3,C_3]$$ are parallel, $$[A_1,B_1,C_1] = [0,0,0].$$

Normal to 1 line
If a given line has direction numbers then any of the following groups of direction numbers is normal to the given line:

For example:

If the given line has direction numbers arbitrary normals have direction numbers In this case only the groups  are valid.

If the given line has direction numbers arbitrary normals have direction numbers In this case all three groups of direction numbers are valid.

If the given line has direction numbers arbitrary normals have direction numbers In this case all three groups of direction numbers are valid. However, the groups are parallel.

Point and line
In the diagram line $$PQ$$ and point $$M$$ are well defined and $$\theta = \angle MPQ.$$$$$$$$$$

Point $$N$$ is on line $$PQ.$$

Line $$MN$$ is perpendicular to line $$PQ.$$

Calculate the coordinates of point $$N$$ and length $$MN.$$

Initial considerations:

 * Points $$M, P$$ may be the same point in which case points $$M,N,P$$ are the same point and lengths $$MN=NP=PM=0.$$

 * Point $$M$$ may be on line $$PQ$$ in which case lengths $$PM=PN$$ and length $$MN=0.$$

 * line $$MP$$ may be perpendicular to line $$PQ$$ in which case lengths $$MP=MN$$ and length $$PN=0.$$

Calculate direction cosines of line $$PQ$$: $$[A1,B1,C1].$$

Calculate direction cosines of line $$PM$$: $$[A2,B2,C2].$$

Calculate $$\cos(\theta).$$

Calculate length $$PM.$$

Calculate length $$PN = PM\cdot \cos(\theta).$$

Let point $$P$$ have coordinates $$(x_1,y_1,z_1)$$

Then point $$N = ( x_1+$$length$$PN*A1,\ y_1+$$length$$PN*B1,\ z_1+$$length$$PN*C1 ).$$

Calculate length $$MN.$$

By extending line PQ with constant K
Let line $$PN$$ be defined as

Let point $$N$$ have coordinates

Let point $$M$$ have coordinates

Then line $$MN$$ has direction numbers

Lines $$MN, PN$$ are normal. Therefore:

$$A(p+KA-s) + B(q+KB-t) + C(r+KC-u) = 0$$ and

$$K = \frac{As - Ap + Bt - Bq + Cu - Cr}{A^2 + B^2 + C^2}.$$

Use this value of $$K$$ to calculate the coordinates of point $$N$$ and the direction numbers of line $$MN.$$

Example:

is defined as

Point  has coordinates

Calculate coordinates of point  on   so that line is normal to

Distance between two parallel lines:

Let $$M$$ be a point on a line parallel to line $$PQ.$$

Distance between the two lines is length $$MN.$$

Points of closest approach
Points of closest approach are the 2 points at which the distance between 2 skew lines is minimum.

Let 2 lines be  and. Verify that the lines are not parallel.

Calculate point $$P_1$$ on    and  point $$P_2$$ on    so that line $$P_1P_2$$ is normal to   and line $$P_2P_1$$ is normal to

It's possible that length $$P_1P_2$$ may be zero in which case points $$P_1, P_2$$ are the same point and the lines intersect.

Let  be defined as point $$(p,q,r)$$ with direction numbers $$[A_1,B_1,C_1].$$

Let  be defined as point $$(s,t,u)$$ with direction numbers $$[A_2,B_2,C_2].$$

Calculate the direction numbers of the normal: $$[A_3,B_3,C_3].$$

Relative to  point $$P_1 = ( p+K_1\cdot A_1,\ q+K_1\cdot B_1,\ r+K_1\cdot C_1)$$

and point $$P_2 = ( p+K_1\cdot A_1+K_3\cdot A_3,\ q+K_1\cdot B_1+K_3\cdot B_3,\ r+K_1\cdot C_1+K_3\cdot C_3).$$

Relative to  point $$P_2 = ( s+K_2\cdot A_2,\ t+K_2\cdot B_2,\ u+K_2\cdot C_2)$$

Therefore:

$$p+K_1\cdot A_1+K_3\cdot A_3 = s+K_2\cdot A_2$$

$$q+K_1\cdot B_1+K_3\cdot B_3 = t+K_2\cdot B_2$$

$$r+K_1\cdot C_1+K_3\cdot C_3 = u+K_2\cdot C_2$$

or:

$$A_1\cdot K_1 - A_2\cdot K_2 + A_3\cdot K_3 + p - s = 0\ \dots\ (1)$$

$$B_1\cdot K_1 - B_2\cdot K_2 + B_3\cdot K_3 + q - t = 0\ \dots\ (2)$$

$$C_1\cdot K_1 - C_2\cdot K_2 + C_3\cdot K_3 + r - u = 0\ \dots\ (3)$$

This is a system of three equations with three unknowns: $$K_1, K_2, K_3.$$

If $$[A_3,B_3,C_3]$$ are direction cosines, $$K_3 =$$ length $$P_1P_2.$$

Example 1:

For function  see  "Solving_3_by_4"

Example 2:

Two lines are defined as:

and

Calculate the points of closest approach.

In this case, $$K_3 = 0$$ and the lines intersect at point $$(3,-5,11).$$

=The plane in 3 dimensions=

In two dimensions the point that is always equidistant from two fixed points is the line.

In three dimensions the point that is always equidistant from two fixed points is the plane.

Let point $$P_1$$ have coordinates $$(a,b,c).$$

Let point $$P_2$$ have coordinates $$(d,e,f).$$

Distance $$P_1P_2$$ must be non-zero.

Let point $$P_{xyz}$$ have coordinates $$(x,y,z).$$

Then length $$P_1 P_{xyz}$$ = length $$P_2 P_{xyz}.$$

$$(x-a)^2 + (y-b)^2 + (z-c)^2 = (x-d)^2 + (y-e)^2 + (z-f)^2$$

$$x^2 -2ax + a^2 + y^2 - 2by + b^2 + z^2 - 2cz + c^2 $$$$= x^2 -2dx + d^2 + y^2 - 2ey + e^2 + z^2 - 2fz + f^2$$

$$ -2ax + a^2 - 2by + b^2 - 2cz + c^2 =  -2dx + d^2 - 2ey + e^2  - 2fz + f^2$$

$$ -2ax - 2by - 2cz + a^2 + b^2 + c^2 =  -2dx - 2ey - 2fz + d^2  + e^2  + f^2$$

$$ 2(d-a)x + 2(e-b)y + 2(f-c)z + a^2 + b^2 + c^2 - (d^2  + e^2  + f^2)= 0 $$

This equation has the form $$Ax + By +Cz + D = 0$$ where:

$$A= 2(d-a);\ d-a = \frac{A}{2}$$

$$B= 2(e-b);\ e-b = \frac{B}{2}$$

$$C= 2(f-c);\ f-c = \frac{C}{2}$$

$$D = a^2 + b^2 + c^2 - (d^2 + e^2  + f^2)$$

Normal to the plane:
Line $$P_1P_2$$ is normal to the plane and it has direction numbers $$[d-a, e-b, f-c]$$ or $$[\frac{A}{2},\frac{B}{2},\frac{C}{2}].$$

When a plane is defined as $$Ax+By+Cz+D = 0,$$ the line normal to the plane has direction numbers $$[A,B,C].$$

Distances:
Distance $$P_1P_2 = \sqrt{(d-a)^2 +(e-b)^2 +(f-c)^2}$$ $$=\sqrt{    (\frac{A}{2})^2 +   (\frac{B}{2})^2 +  (\frac{C}{2})^2   }$$ $$= \frac{\sqrt{A^2 + B^2 + C^2}}{2}  $$

Distance from each point $$P_1,P_2$$ to plane $$= \frac{\sqrt{A^2 + B^2 + C^2}}{4} .$$

Consider the expression $$Ax + By + Cz + D.$$ Substitute $$(d,e,f)$$ or $$(a,b,c)$$ for $$(x,y,z).$$

$$Ad + Be + Cf + D = \frac{A^2 + B^2 + C^2}{4};$$ $$\frac{Ad+Be+Cf+D}{  \sqrt{A^2 + B^2 + C^2}     }$$$$   =\frac{\sqrt{A^2 + B^2 + C^2}}{4} $$

Let $$A_1 = \frac{A}{\sqrt{A^2 + B^2 + C^2} },$$ $$\ B_1 = \frac{B}{\sqrt{A^2 + B^2 + C^2} },$$ $$\ C_1 = \frac{C}{\sqrt{A^2 + B^2 + C^2} }.$$ $$\ D_1 = \frac{D}{\sqrt{A^2 + B^2 + C^2} }.$$

Then $$A_1d + B_1e + C_1f + D_1$$= distance from point $$(d,e,f)$$ to plane.

$$Aa + Bb + Cc + D = \frac{-(A^2 + B^2 + C^2)}{4}.$$

Similarly, $$A_1a + B_1b + C_1c + D_1$$= distance from point $$(a,b,c)$$ to plane with opposite sign.

When the values $$(A,B,C)$$ are all direction cosines, the distance from point $$(a,b,c)$$ to plane $$= Aa+Bb+Cc+D$$ with the sign of the result determined by the position of the point relative to the origin.

If on the same side of the plane as the origin, the sign of the result is the sign of D.

If on the side of the plane opposite to that of the origin, the sign of the result is the sign of -D.

To visualize the plane:
Think of the normal to the plane. The normal has direction numbers $$[A,B,C].$$ The plane is perpendicular to the normal.

Note which direction number or numbers are missing. The plane is parallel to the missing axes.

The plane $$x=4$$ is parallel to both $$Y,Z$$ axes.

The plane $$y=3$$ is parallel to both $$X,Z$$ axes.

The plane $$z=-2$$ is parallel to both $$X,Y$$ axes.

The plane $$3x - 4y + 7 = 0$$ is parallel to the $$Z$$ axis.

The plane $$3x - 4z = 7$$ is parallel to the $$Y$$ axis.

The plane $$3y - 4z = 7$$ is parallel to the $$X$$ axis.

The plane $$9x - 8y + 12z + 9 = 0$$ is not parallel to any axis.

By definition above
Given two points as above, the equation $$Ax + By + Cz + D = 0$$ may be calculated.

If this is too complicated, calculate the three direction numbers and the point midway between the two given points. Proceed as under "Point and direction numbers" below.

By point and direction numbers
A point and direction numbers usually define a line. In this case it is a special line, the line normal to the plane with the given point being a point in the plane.

Given $$((x_1,y_1,z_1),[A_1,B_1,C_1]):$$

The equation of the plane is $$A_1x + B_1y + C_1z + D = 0$$ where $$D = -(A_1x_1 + B_1y_1 + C_1z_1).$$

By three points
Verify that the three points form a real triangle with all sides non-zero.

From the three points produce two groups of direction numbers. Calculate the direction numbers of the normal to the two groups of direction numbers. Using any one of the points and the direction numbers of the normal proceed as under "Point and direction numbers" above.

Plane and line
Given a plane and a line in 3 dimensions some relevant questions are:

 * Is the line parallel to the plane? if so, what is distance from line to plane?

 * If not parallel, what is the point within the plane at which the line enters the plane?

Let the plane be defined as: $$Ax + By + Cz + D = 0.$$

Let the line be defined as point $$(x_1, y_1, z_1)$$ and direction numbers $$[A_1, B_1, C_1].$$

The point at which the line enters the plane is: $$(x_1 + KA_1, y_1+KB_1, z_1+KC_1).$$

These points satisfy the equation of the plane:

$$A(x_1 + KA_1) + B(y_1+KB_1) + C(z_1+KC_1) + D = 0$$

$$K = \frac{-(+ Ax_1 + By_1 + Cz_1 + D )}{AA1 + BB1 + CC1}.$$

The divisor $$AA1 + BB1 + CC1$$ is the top line of the equation for $$\cos \theta,$$ angle between two lines.

If $$\cos \theta == 0,$$ the angle between line and normal to the plane is $$90^\circ$$ and line is parallel to plane.

The point of intersection is: $$(x_1 + KA_1, y_1+KB_1, z_1+KC_1).$$

If the direction numbers $$[A_1, B_1, C_1]$$ are all direction cosines, distance from point to plane along the line is $$K.$$

For example:

In the diagram point $$P$$ has coordinates $$(8,11,-12).$$

Line $$PR$$ through $$P$$ has direction numbers $$[5,25,-7].$$

Point $$R$$ is within the plane $$9x + 12y + 8z +181 = 0.$$

What are the coordinates of point $$R$$?

With $$[ A, B, C ]$$ as direction cosines the coefficients of the plane $$= (A,B,C,D) = ( \frac{9}{17}, \frac{12}{17}, \frac{8}{17}, \frac{181}{17} ).$$

As direction cosines the direction numbers of the line $$= [A_1,B_1,C_1] = [ \frac{5}{\sqrt{699}}, \frac{25}{\sqrt{699}}, \frac{-7}{\sqrt{699}} ].$$

Using $$K = \frac{-(+ Ax_1 + By_1 + Cz_1 + D )}{AA_1 + BB_1 + CC_1} = \frac{-17}{0.643} = -26.4386,$$

The point of intersection $$=(x_1 + KA_1, y_1+KB_1, z_1+KC_1) = (3,-14,-5).$$

Provided that all values $$[A,B,C], [A_1,B_1,C_1]$$ are direction cosines,

$$-(+ Ax_1 + By_1 + Cz_1 + D ) =$$ length $$PQ$$ and $$AA_1 + BB_1 + CC_1 = \cos(\theta).$$

If the value of $$\angle \theta_1$$ is desired, $$\sin (\theta_1) = \cos(\theta).$$

Plane and point
Given a plane and a point in 3 dimensions some relevant questions are:

 * What is distance from point to plane?

 * What is the point within the plane at which the line through the point and normal to plane enters the plane?

Let the plane be defined as: $$Ax + By + Cz + D = 0.$$

Any line normal to the plane has direction numbers $$(A, B, C).$$

Let the point be defined as point $$(x_1, y_1, z_1).$$

Let  be defined as the line through point $$(x_1, y_1, z_1)$$ with direction numbers $$(A, B, C).$$

Ensure that the values $$(A, B, C)$$ are direction cosines.

Use the method above to calculate the point at which  enters the plane.

Distance from point to plane is K. Alternatively, distance from point to plane is $$Ax_1 + By_1 + Cz_1 + D.$$

For example:

is defined as $$9x+12y+8z-108 = 0.$$

Point $$P$$ has coordinates $$(13,20,5).$$

 * At what point $$P_1$$ does the normal through $$P$$ enter ?

 * What is the distance from $$P$$ to $$P_1$$?

Convert the equation of  to direction cosines: $$\frac{9}{17}x+\frac{12}{17}y+\frac{8}{17}z-\frac{108}{17} = 0.$$

Using $$K = \frac{-(+ Ax_1 + By_1 + Cz_1 + D )}{AA1 + BB1 + CC1}.$$

$$K = \frac{-(+ \frac{9}{17}x_1 + \frac{12}{17}y_1 + \frac{8}{17}z_1 -\frac{108}{17} )}{A^2 + B^2 + C^2}$$ $$= -(+ \frac{9}{17}13 + \frac{12}{17}20 + \frac{8}{17}5 -\frac{108}{17} ) = -17.$$

Point $$P_1 = (13-17\frac{9}{17},\ 20-17\frac{12}{17},\ 5-17\frac{8}{17}) = (4,8,-3).$$

Distance from $$P$$ to $$P_1 = -17.$$

Plane and plane
If two planes are parallel:

 * they do not intersect.

 * their normals are parallel.

The distance between two parallel planes $$A_1x + B_1y + C_1z + D_1 = 0$$ and $$A_2x + B_2y + C_2z + D_2 = 0$$ is $$D_1 - D_2$$ provided that both planes are defined with direction cosines.

Intersection of 2 planes:

The intersection of plane and plane is a line. The word "intersection" implies that the two planes are not parallel.

If two planes intersect, the line normal to both normals is parallel to the line of intersection.

As the intersection of two planes, the line cannot be defined by a neat algebraic equation like that of the plane.

However the concept is useful because the intersection of line and plane or line and sphere can be solved with algebra.

The line defined as the intersection of two planes can be defined with the familiar point and direction numbers:

As point and direction numbers:

Line $$PQ$$ is defined as the intersection of two planes  and

Using any point (the origin $$O$$ is convenient,) calculate the coordinates of point $$R$$ within  and on the normal from point $$O$$ to.

Calculate the direction numbers of line $$PQ.$$

Calculate the direction numbers of line $$RS,$$ normal to both lines $$OR, PQ.$$

Point $$S$$ is the point at which line $$RS$$ enters.

Line PQ may be defined as point $$S$$ with direction numbers of line $$PQ.$$

Any point on line $$PQ$$ has distance $$0$$ from both planes  and

Algebraic method:

Let a line be defined by two planes:

$$\pi_1:\ x+y+z+1 = 0$$

$$\pi_2:\ x+2y+3z+4 = 0$$

From $$\pi_1,\pi_2$$ derive other equations:

$$\pi_3:\ y + 2z + 3 = 0\ \dots\ \pi_2 - \pi_1$$

$$\pi_4:\ x - z - 2 = 0\ \dots\ 2\pi_1-\pi_2$$

$$\pi_5:\ 2x + y - 1 = 0\ \dots\ 3\pi_1-\pi_2$$ $$$$

The line may be defined by any 2 of $$\pi_1,\ \pi_2,\ \pi_3,\ \pi_4,\ \pi_5.$$

To find a point on the line, let $$x = 0.$$

From $$\pi_4:\ z = -2$$

From $$\pi_5:\ y = 1$$

Therefore $$(0,1,-2)$$ is a point on the line and the line has direction numbers $$(A,B,C)=(-1,2,-1).$$

Any point on the line has coordinates $$(x+KA,\ y+KB,\ z+KC) = (0+K(-1),\ 1+K(2),\ -2+K(-1)).$$

For proof:

Substitute these values into $$\pi_1:$$ $$x+y+z+1 = (-K) + (1+2K) + (-2-K) + 1 = 0.$$

Substitute these values into $$\pi_2:$$ $$x+2y+3z+4 = (-K) + 2(1+2K) + 3(-2-K) + 4 = -K + 2 + 4K -6-3K + 4 = 0.$$

Line  defined as intersection of 2 planes:

Reviewing:
Now that information about the plane is available to the reader, some of the concepts above can be reviewed and simplified.

Point and line
Line $$PQ$$ and point $$M$$ are given.

Calculate the coordinates of point $$N$$ on line $$PQ$$ so that line $$MN$$ is normal to line $$PQ.$$

Calculate the direction numbers of line $$PQ:\ (A,B,C).$$

Using $$(A,B,C)$$ and point $$M$$ calculate the plane normal to line $$PQ$$ and containing point $$M,$$ plane $$MNR.$$

Point $$N$$ is the point at which line $$PQ$$ enters plane $$MNR.$$

Points of closest approach
Lines $$PQ,RS$$ are skew.

Calculate point $$R$$ so that distance from point $$R$$ to line $$PQ$$ is minimum.

Calculate direction numbers of line $$RT$$ normal to both $$PQ, RS.$$

Calculate direction numbers of line $$UV$$ normal to both $$PQ, RT.$$

Using direction numbers of line $$UV$$ and point $$P$$ or point $$Q,$$ calculate plane $$PQT.$$

Point $$R$$ is the point at which line $$SR$$ enters plane $$PQT.$$

Three planes
The intersection of 3 planes is a point. The word "intersection" implies that the three planes actually intersect.

Given three random planes, there are two situations that do not produce a point of intersection:

 * If two of the planes are parallel.

 * If the three planes form a "tent" or "awning."

In diagram to the right:

 * No two planes are parallel.

 * The line of intersection of any 2 planes is parallel to the third plane.

For example, line $$CD$$ is the line of intersection of two planes

Line $$CD$$ is parallel to  and there is no point of intersection.

Point of intersection:

In the diagram line $$CD$$ is line of intersection of two planes

Point $$E$$, the point of intersection of the 3 planes, is the point at which line $$CD$$ enters

Similarly:

 * Line $$EF$$ is common to  Point $$E$$ is the point at which line $$FE$$ enters

 * Line $$EG$$ is common to  Point $$E$$ is the point at which line $$GE$$ enters

=The sphere=

The sphere is the locus of a point that is always a fixed non-zero distance from a given fixed point.

Let the point be  and the distance be

Let  be any point on the surface of the sphere.

Then $$(x-d)^2 + (y-e)^2 + (z-f)^2 = r^2$$

$$x^2 - 2dx + d^2 + y^2 - 2ey + e^2 + z^2 - 2fz + f^2 = r^2$$

$$x^2 + y^2 + z^2 - 2dx - 2ey - 2fz + d^2 + e^2 + f^2 - r^2 = 0$$

$$x^2 + y^2 + z^2 + Ax + By + Cz + D = 0$$

where:

$$A = -2d,$$ $$B = -2e,$$ $$C = -2f,$$ $$D = d^2 + e^2 + f^2 - r^2.$$

Given a sphere defined as  the sphere may be defined as  where it is understood that the coefficients of $$(x^2,\ y^2,\ z^2)$$ are

reversing the conversion
Given a sphere defined as  calculate and

$$d = \frac{-A}{2},$$ $$e = \frac{-B}{2},$$ $$f = \frac{-C}{2},$$ $$r = \sqrt{ d^2 + e^2 + f^2 - D }.$$

and

Two definitions of the sphere:

 * , the algebraic definition of the sphere.

 * , the trigonometric definition of the sphere.

examples
Errors in definition:

The tuple  does not define a valid sphere because radius=0.

The tuple  does not define a valid sphere because the value $$( d^2 + e^2 + f^2 - D )$$ is negative. Consequently the value $$r = \sqrt{ d^2 + e^2 + f^2 - D }$$ is a complex number.

Sphere at origin:

If center of sphere is at origin, $$(d, e, f) = (0,0,0).$$

For example, a sphere at origin with radius $$7$$ may be defined as:

or

Point inside, on or outside sphere:

If distance from point to center $$< r,$$ point is inside sphere.

If distance from point to center $$== r,$$ point is on surface of sphere.

If distance from point to center $$> r,$$ point is outside sphere.

A sphere is defined as

3 points are given:

For each point, determine position of point relative to surface of sphere.

 * for point (3,4,3), distance from point to center is  Point is inside sphere.

 * for point (-3,19,0), distance from point to center  Point is on surface of sphere.

 * for point (9,-5,6), distance from point to center is  Point is outside sphere.

Sphere and line
Given a sphere and a line:

 * The line may pierce the sphere in 2 places,

 * The line may touch the sphere in 1 place, or

from the center of the sphere is always greater than the radius of the sphere.
 * The line may not touch or pierce the sphere. In other words, the distance to the line

The possible point/s of intersection may be calculated by trigonometry or by algebra.

Given a sphere defined as  and a line defined as   calculate the point/s of intersection, if any.

By trigonometry:
Let center of sphere $$P = (3,5,-2).$$ and point $$M = (4,-5,10).$$

Calculate coordinates of point $$N$$ on the line so that line $$PN$$ is normal to the given line.

$$N = (5.5, 1, -3.5).$$

Calculate length $$PN = \sqrt{24.5} = 4.949747468305833.$$ This length is less than the radius of the sphere. The line intersects the sphere in 2 points. Let these points be $$S,T.$$

Length $$NS =$$ length $$NT = \sqrt{ (7)^2 - 24.5 } = \sqrt{24.5}.$$$$$$

Redefine the line so that direction numbers are direction cosines: $$\frac{1}{\sqrt{98}},\frac{4}{\sqrt{98}},\frac{-9}{\sqrt{98}}$$

Point $$S = ( 5.5 + \sqrt{24.5}\cdot\frac{1}{\sqrt{98}}, 1 + \sqrt{24.5}\cdot\frac{4}{\sqrt{98}}, -3.5 + \sqrt{24.5}\cdot\frac{-9}{\sqrt{98}} )$$

Point $$T = ( 5.5 - \sqrt{24.5}\cdot\frac{1}{\sqrt{98}}, 1 - \sqrt{24.5}\cdot\frac{4}{\sqrt{98}}, -3.5 - \sqrt{24.5}\cdot\frac{-9}{\sqrt{98}} )$$

Proof:

Distance from point  to center $$= \sqrt{ (6-3)^2 + (3-5)^2 + (-8-(-2))^2 } = 7 =$$ radius.

Distance from point  to center $$= \sqrt{ (5-3)^2 + (-1-5)^2 + (1-(-2))^2 } = 7 =$$ radius.

Direction numbers of line

Direction numbers of line

Lines  and original line are all parallel and all pass through point

Points  are on the original line.

By extending line with constant K
Let the line be defined as point and direction numbers:

Let the point at which the line enters the sphere be:

These values of $$(x,y,z)$$ satisfy the equation of the sphere:

$$x^2 + y^2 + z^2 + Ax + By + Cz + D = 0.$$

Therefore: $$(s+KE)^2 + (t+KF)^2 + (u+KG)^2 + A(s+KE) + B(t+KF) + C(u+KG) + D = 0$$

Expand and the result is:

$$(E^2 + F^2 + G^2)K^2 + (AE + BF + CG + 2Es + 2Ft + 2Gu)K + As + Bt + Cu + D + s^2 + t^2 + u^2 = 0.$$

This is a quadratic equation in $$K:$$

$$A_0K^2 + B_0K + C_0 = 0$$ where:

$$A_0 = E^2 + F^2 + G^2$$

$$B_0 = AE + BF + CG + 2Es + 2Ft + 2Gu$$ and

$$C_0 = As + Bt + Cu + D + s^2 + t^2 + u^2.$$

By algebra:
Let the sphere be defined as $$A_1, B_1, C_1, D_1$$

Let the line be defined as the intersection of 2 planes $$A_2, B_2, C_2, D_2$$ and $$A_3, B_3, C_3, D_3$$

The relevant equations are :

$$x^2 + y^2 + z^2 + A_1x + B_1y + C_1z + D_1 = 0\ \dots\ (1)$$

$$A_2x + B_2y + C_2z + D_2 = 0\ \dots\ (2)$$

$$A_3x + B_3y + C_3z + D_3 = 0\ \dots\ (3)$$

The point/s of intersection are the solution/s of these 3 equations for $$x, y, z.$$

The value/s of $$x$$ are the solution/s of the quadratic equation $$c_2x^2 + c_1x + c_0 = 0.$$

Exchange rows
Rows 2 and 3 may be exchanged, one for the other:

Exchange columns
If data supplied to  produces an error (as above), try exchanging two columns: Exchange appropriate columns in output to produce correct values:

Defined as 4 points
If 4 unique points on surface of sphere are known, the center and radius may be calculated.

From above, equation of sphere is: $$x^2 + y^2 + z^2 + Ax + By + Cz + D = 0$$

When $$x, y, z$$ are known and $$A, B, C, D$$ are unknown the above equation becomes: $$xA + yB + Cz + (1)D + x^2 + y^2 + z^2 = 0\ \dots\ (1)$$

With four points provided, equation $$(1)$$ may be used to construct a system of 4 simultaneous equations that, when solved, provide the values $$(A, B, C, D).$$

For example:

The above matrix of 4 rows by 5 columns is input to function See Solving_M_by_(M+1)

Center as intersection of 3 planes
Let center of sphere be $$(d, e, f).$$

Let 4 points on surface of sphere be $$(a,b,c),\ (g,h,j),\ (k,l,m),\ (n,p,q).$$

$$(d-a)^2 + (e-b)^2 + (f-c)^2 = r^2$$

$$(d-g)^2 + (e-h)^2 + (f-j)^2 = r^2$$

$$(d-k)^2 + (e-l)^2 + (f-m)^2 = r^2$$

$$(d-n)^2 + (e-p)^2 + (f-q)^2 = r^2$$

$$(d-a)^2 + (e-b)^2 + (f-c)^2 = (d-g)^2 + (e-h)^2 + (f-j)^2$$ or

$$(d-a)^2 + (e-b)^2 + (f-c)^2 - ( (d-g)^2 + (e-h)^2 + (f-j)^2 ) = 0\ \dots\ (1)$$

$$(d-a)^2 + (e-b)^2 + (f-c)^2 - ( (d-k)^2 + (e-l)^2 + (f-m)^2 ) = 0\ \dots\ (2)$$

$$(d-a)^2 + (e-b)^2 + (f-c)^2 - ( (d-n)^2 + (e-p)^2 + (f-q)^2 ) = 0\ \dots\ (3)$$

When expanded, equation $$(1)$$ above has the form $$Ad + Be + Cf + D = 0.$$ This is the equation of the plane normal to line $$((a,b,c),\ (g,h,j),$$ 'line'$$)$$ and containing the point midway between points $$(a,b,c),\ (g,h,j).$$ For function  see  "Solving_3_by_4"

=Third line=

Given:

 * has direction numbers

 * has direction numbers

 * is cosine of angle between  and

 * is cosine of angle between  and

Calculate direction numbers of line3.

In the diagram:

has direction cosines $$[a_1, b_1, c_1].$$

has direction cosines $$[a_2, b_2, c_2].$$

$$\theta_1$$ is angle between  and

$$\theta_2$$ is angle between  and

$$= \cos(\theta_1).$$

$$= \cos(\theta_2).$$

Length

Length

Point  has coordinates

Point  has coordinates

is normal to  and contains point

is normal to  and contains point

Point  is on line of intersection of   and and point  is distance   from point

Point  may have no value. $$(\theta_1 + \theta_2 < \angle MON)$$

Point  may have 1 value. $$(\theta_1 + \theta_2 = \angle MON)$$

Point  may have 2 values. $$(\theta_1 + \theta_2 > \angle MON)$$

For function  see  "Solving_3_by_4"

An example:

Rotate line through given angle
Given:

 * defined as

 * wholly within  and with direction numbers

 * angle $$\theta$$

Calculate:

 * direction numbers of  wholly within   and making angle $$\theta$$ with

This is a special case of function  above in which   makes the angle of $$90^\circ$$ with the normal to the plane, and angle $$\theta$$ with A simple example:

Spokes of a wheel
Given:

 * defined as

 * wholly within  and with direction numbers

Rotate  5 times through $$72^\circ,$$ wholly within

5 times $$72^\circ = 360^\circ.$$ After 5 rotations the direction cosines $$[a_1, b_1, c_1]$$ match their initial values.

By algebra
Ensure that all values $$[a_1,b_1,c_1],\ [a_2,b_2,c_2]$$ are direction cosines.

The relevant equations are:

$$a_1 \cdot a_3 + b_1 \cdot b_3 + c_1 \cdot c_3$$  $$\dots\ (1)$$

$$a_2 \cdot a_3 + b_2 \cdot b_3 + c_2 \cdot c_3$$  $$\dots\ (2)$$

$$a_3^2 + b_3^2 + c_3^2 = 1\ \dots\ (3)$$

Because $$a_3$$ may have exactly $$0$$ or $$1$$ or $$2$$ values, $$a_3$$ is the solution of the quadratic equation $$C_2\cdot a_3^2 + C_1\cdot a_3 + C_0 = 0$$ where $$C_2, C_1, C_0$$ are as defined in the python code below.

Then values $$b_3, c_3$$ may be calculated as shown below.

The function above simplifies to:

Normal to 2 lines
This is the special case in which  and the calculation of $$a_3$$ becomes:

$$C_2\cdot a_3^2 = -C_0$$

$$a_3^2 = \frac{A_3^2}{A_3^2 + B_3^2 + C_3^2}$$

$$a_3 = \frac{A_3}{\sqrt{A_3^2 + B_3^2 + C_3^2}}$$

whence: $$b_3 = \frac{a_3\cdot B_3}{A_3} = \frac{B_3}{\sqrt{A_3^2 + B_3^2 + C_3^2}}$$

and $$c_3 = \frac{-(a_3\cdot a_1 + b_3\cdot b_1)}{c_1}$$

Normal to X,Y axes
In this case the result below was derived from "# output line 3" in the Python code above. 3 -0.0, 0.0, 1.0], [-0.0, 0.0, -1.0

=Links to Related Topics=

Coordinate_systems

Functions_(mathematics)/Graphs/Coordinations

Functions_(mathematics)/Graphs

Investigating_3D_geometric_algebra

Line_(Geometry)

Functions_(mathematics)/Graphs/Cartesian_Coordinate

title: Introduction to Cartesian Geometry


text:
Web Design/Embedded CSS

Example 1

As you can see the style information is placed between the  &   tags, inside the   &   tags

The CSS code below should make the:
 * Background color = yellow
 * Text inside a  tag = green & a font size of 10 pixels
 * Text inside a  tag = blue

Type the code below into your favorite code editor and save the file as cssexample.html

Exercise 2 – Changing the font family and font style using CSS.

The CSS code below should make the:
 * Text within the  tag, appear as helvetica, impact, sans-serif font.
 * Text within the  tag, appear as italic.
 * Text within the  tag, appear as gill sans, new baskerville, sans-serif font.

Paste the code below into your code editor, create a new page called cssexample2.html

Exercise 3 – Changing the border and aligning the text using CSS.

The CSS code below should make the:
 * Background within the  tag, appear as pink.
 * Border around the  tag, appear as solid green.
 * Text within the  tag, aligned in the center.
 * Border around the  tag, to appear as dotted red.

Paste the code below into your code editor, create a new page called cssexample3.html

Exercise 4 – Changing the margin padding using CSS.

The CSS code below should make the:
 * The text within the  tag will have a left margin of 2cm.

Paste the code below into your code editor, create a new page called cssexample4.html

Exercise 5 – Changing the color of a Horizontal Rule using CSS.

The CSS code below should make the:
 * The hortizontal rule  tag will appear as blue (instead of grey).

Paste the code below into your code editor, create a new page called cssexample5.html

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Web Design/Embedded CSS


text:
User:Nrose

About Nathan

 * Web Design Teacher in Richmond, NSW, Australia.

title: User:Nrose


text:
Web Design/External CSS

If you want to use the same CSS for several Web pages it is worth considering using a separate style sheet which you then link from each page. You can do this as follows:

The link tag should be placed within the & element. Here is an HTML file with a link to an external style sheet:

The link element's rel attribute must be set to the value stylesheet to allow the browser to recognize that the href attribute gives the Web address (URL) for your style sheet.

Exercise 1 - Creating a HTML page with an external CSS

In this example we will create:
 * a file called externalcss.html (which is the HTML web page); and
 * a file called external.css (which holds all the style information);

Both files should be located in the same directory.

externalcss.html will obtain style information from external.css

Part 1 - Creating externalcss.html

Type the code below into your code editor and save the file as externalcss.html (this page is a HTML webpage)

My CSS Webpage

Some heading This is a webpage that uses an external stylesheet. The style is controlled from the file style.css. This is an advantage because you can call this same CSS Style on many pages. When you want to edit the style for the whole website you just have to change code within the external.css file.

Part 2 - Create a page called external.css

The CSS code below should make the:
 * Left and right margin indent by 10%
 * Text color appear as white
 * Background color appear as Black

Type the code below into your code editor and save the file as external.css

/* style.css - a simple style sheet */ body { margin-left: 10%; margin-right: 10%; color: white; background: black; }

View externalcss.html in your internet browser and you will notice that the css code from external.css is formatting the webpage.

title: Web Design/External CSS


text:
Web Design/Inline CSS

 * The simplest form of applying styles to a web page is by using inline styles.
 * The style is applied in a line of your XHTML, so it is called an inline style.
 * It is the least powerful style application because it only applies to the line you put it in and not to the whole page or many pages.
 * Inline styles override any other style applied to a web page.
 * Inline Styles are placed within the particular tag you wish to format.
 * E.g. Some text

Example 1

Type the code below into your code editor and save the file as inlinecss.html

Inline Styles

This text does not have any style applied to it.

This text has the font-size style applied to it, making it 20pt.

 This text has the font-size and color styles applied to it, making it 20pt. and blue.

 * The first inline style that appears just after <h1, this will make the color of anything text within the tag to appear as red.
 * The second inline style coding appears. It starts straight after the <p or paragraph element in this example.
 * Attribute style specifies the style for an element. For example in the one above the style then is followed by a property here being font-size.
 * Each property is followed by a colon and a value. In this case the property is font-size and the value is 20 point.
 * There is a third example of inline styles, but in this case there are two properties, font-size and color. The two properties are separated by a semi-colon and the colour name can be specified by a hexadecimal code or by a colour name e.g. blue.

Above text adapted from: Garrett, J, (2007), “Develop Cascading Style Sheets (CSS): Week One”.

title: Web Design/Inline CSS


text:
Web Design/CSS Classes

What are Classes in CSS?
 * Let's say you want to give an HTML element multiple looks with CSS.
 * Say for example, sometimes you want the font to be small and white with a black background, while other times you would prefer the font to be big and blue with a light blue background. CSS allows you to do this with the use of classes.
 * The name of the actual class can be called whatever you wish!

To make a class, in between & type the name of the class following the tag name. p.first { color: blue; } Then in the HTML tag add the class This is some text

The last code will set the text color to blue for all of the 'p' elements with a class value of 'first'. It's also possible to apply the desired style to all of the elements with an specific class value no matter which element the class attribute is at. To do that, the '*' character is used.
 * .first { color: blue; }

Example 1

The code below should make the:
 * Paragraph with the first class, appear as blue color.
 * Paragraph with the Second class, appear as red color.
 * Paragraph with the no class, appear as normal.

Type the code below into your code editor and save the file as cssclass.html

 p.first{ color: blue; } p.second{ color: red; }

This is a normal paragraph.

This is a paragraph that uses the p.first CSS code!

This is a paragraph that uses the p.second CSS code!

Example 2

The code below should make the:
 * Heading2 appear red with font size of 20px.
 * Paragraph with the firstpara class, appear as gray color.
 * Paragraph with the Secondpara class, appear as red color.
 * Paragraph with the Thirdpara class, appear as white text with purple background.
 * Paragraph with the no class, appear as normal.

Type the code below into your code editor and save the file as cssclass2.html

h2 { color: red; font-size: 20px; } p.firstpara { background-color: gray; } p.secondpara { background-color: red; } p.thirdpara { background: purple; color: white; }

CSS Classes This is the p.firstpara paragraph

This is the p.secondpara paragraph

This is the p.thirdpara paragraph

This has no CSS formatting

References

http://www.tizag.com/cssT/class.php

title: Web Design/CSS Classes


text: File:Sample Matte Painting 2 cropped.png


text:
User talk:BlueZEN

Hello everybody !!! I am a new-comer. I'm study in a university.

title: User talk:BlueZEN


text:
File:EmotiBase-looking down evil 000.png

Example for the Storyboard Artwork Project


text:
File:EmotiBase-point forward worry 000 shoes tele.png

Example for the Storyboard Artwork Project


text:
File:EmotiBase-point forward worry 000 shoes.png

Example for the Storyboard Artwork Project


text:
User:Areil123

I am now to be Areil314 since someone used Areil123 on wikipedia, and I needed to be able to login to both.

To talk to me click here.

I want to create a course in introductory computers with my students that I teach in a college.Areil123 16:29, 16 March 2007 (UTC)

Controversies in Science Coursework

 * water fluoridation
 * big bang
 * panspermia
 * LHC black hole risk?
 * recycling
 * nuclear power greener?
 * are humans causing global warming?
 * was there a mitochondrial Eve?
 * what killed the dinosaurs?
 * RNA world vs metabolism first
 * autism from vaccines?

Computer Science Coursework

 * Here's where I have my draft
 * Introduction to Computer Science
 * Introduction to Computers
 * level tests

useful links

 * Table of page editing
 * Course titles and numbers
 * Help:101 things to do with your class on Wikiversity
 * School:Advanced general studies, User:Dionysios

Discussions about course

 * Colloquium
 * Colloquium Flashcard dicussion
 * Help_talk:Quiz
 * proposal
 * proposal list

AV tests
[[Media:ipod_test1.m4v|m4v video podcast]]

title: User:Areil123


text:
User talk:AReil123

test

title: User talk:AReil123


text:
User:Areil123/draft comp course

Here's where I'll post my draft comp course

 * Introduction
 * Hardware
 * Software
 * Personal technology
 * Networks
 * Challenges (bugs and crime)
 * Web
 * Programming
 * Databases
 * Artificial Intelligence

title: User:Areil123/draft comp course


text: School:Aviation/box-header

title: School:Aviation/box-header


text:
User:Areil123/draft comp course/Introduction

=Introduction=


text: User:Areil123/draft comp course/Hardware


text: User:Areil123/draft comp course/Software


text: User:Areil123/draft comp course/Personal technology


text: User:Areil123/draft comp course/Artificial Intelligence


text: User:Areil123/draft comp course/Networks


text: User:Areil123/draft comp course/Internet and WWW


text: User:Areil123/draft comp course/Challenges (bugs and crime)


text: User:Areil123/draft comp course/Databases


text:
Draft:Operating system/Operating Systems/Memory Management

Memory Management Algorithms
There are four memory management algorithms.

 * First-Fit: A chunk of data is put in the first available block of memory that is big enough for the data to fit.
 * Next-Fit: A modification of First-Fit, for which the search space starts from where the last process was placed instead of the beginning of the memory. When the end of the memory is reached while searching, searching starts from the beginning of the memory all over again.
 * Best-Fit: A chunk of data is put into the absolutely smallest available block of memory that can still fit the data.
 * Worst-Fit: A chunk of data is put into the largest available block of memory. This is the opposite of the Best-Fit algorithm.

Memory Paging
Paging, put simply is when data is moved between the disk and ram. In some cases this may be done if a piece of data has not been touched in a while, so, to free up ram the operating system takes the unused data, and puts it in a special page file, essentially "unloading it". This is called paging out memory. How ever, if a program asks for the memory back, it has to be "paged in". Paging in is when you take requested data and move it back it to ram.


text:
Latin stream/Conjugations

In Latin, verbs are said to be conjugated. Conjugation is the method Latin uses to add the subject of simple sentences to the verb itself. This lesson will show how conjugations work.

Conjugations
In English sentences can be split into subject, verb, and object. An example of this is the simple sentence 'They love you'. 'They' is the subject of the sentence, the person, group or thing performing the action. 'Love' is the verb, the thing being done. 'You' is the object, the person, group, or thing that is being acted on. English normally follows the predictable pattern of subject, verb, object. As an exercise pick up any English book and try to identify the subject, verb, and object of some sentences.

A Roman faced with the sentence 'They love you' would be very confused as to who was giving the love, and who was receiving it. The reason for this is that Latin has a much more flexible word order than English. The information that is encoded in word order in English is encoded by changing the ending of words in Latin. This means the personal pronouns used in English ( I, we, you, they ) are merged into the end of the verb in Latin. The Latin word for love is conjugated below.

The part of the word to the left of the hyphen ( ama ) is called the stem of the word. The part to the right is called the end. All verbs are conjugated with more or less the same pattern so you don't have to worry about having to remember 6 times the number of words! The reason the first person singular does not look like it contains the stem is because the original word was "amao", but was shortened to "amo" after time. This has happened in all verb conjugations.

In our example sentence 'They love you' a third person plural is performing the action to the word we want is 'amant'. The word for you is 'tu,' and is declined to 'te' in the accusative (see Declensions), so the sentence 'They love you' could be written in Latin as 'amant te'. The following table demonstrates every conjugation of the word 'amo'.

title: Latin stream/Conjugations


text:
Talk:Latin stream/Conjugations

This is a simple draft introduction to Latin conjugations. -Unixtastic

"You love you" is nonsensical
Why not change the object to a "real" noun, rather than a pronoun? Ifnkovhg 04:19, 23 April 2008 (UTC)

title: Talk:Latin stream/Conjugations


text: User talk:Vermishis

title: User talk:Vermishis


text:
User:RedSkunk

I have a user page on wikipedia at http://en.wikipedia.org/wiki/User:Redskunk. If you want me to respond quickly to queries message me at my User talk on wikipedia. Cheers! RedSkunktalk 00:46, 17 March 2007 (UTC)

title: User:RedSkunk


text:
File:Hangulcharacterposition.PNG

Summary
{{Information }}
 * Description=Chart of Character Positioning in Korean Hangul
 * Source=Own Work
 * Date=2007-3-16
 * Author=Brian Cox (User:RedSkunk}
 * Permission=This can be used by anyone for any non-commercial reasons.
 * other_versions=None

title: File:Hangulcharacterposition.PNG


text: User talk:RedSkunk

title: User talk:RedSkunk


text: User talk:Gardengirl

title: User talk:Gardengirl


text:
User talk:Unixtastic~enwikiversity

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Unixtastic. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Unixtastic~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 00:09, 18 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:24, 19 April 2015 (UTC)

title: User talk:Unixtastic~enwikiversity


text:
User:Rdubrayj

I am a graduate student in Indiana University's Instructional Systems Technology program.

title: User:Rdubrayj


text:
Portal talk:Participants

This and the new list either need to be combined, or one should be eliminated. Or something. The Jade Knight 03:07, 17 March 2007 (UTC)

title: Portal talk:Participants


text:
User:Blueminneapolis~enwikiversity

Languages I'm Interested In Learning
spanish arabic

Languages I Have Some Familiarity With
japanese german I will be brushing up on these later.

Personal Navigation
Personal Menu

title: User:Blueminneapolis~enwikiversity


text:
Wikiversity:Developing Wikiversity through action research/List of uses of Wikiversity

This is a page for listing the ways that people have used Wikiversity. If context is not obvious, please give links to pages, or individual comments (through diffs).

Uses of Wikiversity

 * Developing and discussing educational materials
 * Courses (eg. Filmmaking)
 * Collaborative learning projects (eg. Portal:Learning projects)
 * Collaborative working space for extra-Wikiversity classes (eg. Media literacy, Instructional Technology)
 * Collaborative research (eg. Developing Wikiversity through action research)
 * Collaborative data collection (eg. Bloom clock project)
 * Other resources for extra-Wikiversity class (eg. Trentcosta)
 * Brainstorming and writing space for proposals (eg. Sandbox Server)
 * Discussions and Debates (eg. Wikiversity as narrative, War and Iran)
 * Bulletin board (I am not convinced of this one - any examples? Cormaggio talk 16:14, 22 August 2008 (UTC))
 * Exam preparation (eg. Category:OSCE)

Wikiversity activity
This section might not be for uses to which Wikiversity can be put - but is to capture the activity of its participants.

 * Surveying and discussing pedagogical methods (eg Learning models
 * Welcoming and helping newcomers
 * Discussing and writing policy pages
 * Writing personal reflections (not only in personal blogs - the Colloquium has worked as a collective reflective space)
 * Designing key pages - particularly main page
 * Designing templates
 * Vandalism and dealing with vandalism
 * Discussing and forming a 'structure' for Wikiversity (topics, schools etc)
 * Asking "what's this all about?" or "why is this the way it is?" (often by newcomers - and it has often been a very productive question to have asked.)


text:
User:Joseph Grey

Hello. I am Joseph Grey. I think Wikiversity is a great idea.

title: User:Joseph Grey


text:
User:JWSchmidt/Blog/17 March 2007

Stable versions  Anyone who has spent time dealing with vandalism has probably thought about ways to limit vandalism and ways to help detect vandalism. One recent helpful innovation is the change to the recent changes tool that shows the number of characters added or removed by each edit. Last summer Jimmy Wales wrote about plans for "version flagging", a proposed new wiki editing process in which there would be a distinction between wiki page versions shown to the public and the newest page versions produced by some editors. The idea being to let everyone edit the wiki pages, but allow the community to review some edits before the edits go public. "Some editors" in this case means wiki editors who do not register a user name or editors who only recently created their user account.

Rumor had it that the German Wikipedia would be the first to test some kind of system for allowing edits to protected articles, review of those edits, and approval of the edits by long-term registered users (here "long-term" means about four days). This system was to apply to the relatively few articles that are protected or semi-protected. This blog post marks the start of my attempt to find out if anything has or will come of this proposal.

The Wikipedia Signpost of 20 March 2007 reported that a contractor was hired to work on "stable versions". The project seems to be moving towards a system where people browsing Wikipedia will be able to request that they see only only reviewed versions of pages.

April 28: With a bit of luck, it can undergo the review process and perhaps be tested on a Wikipedia soon - Rob Church (blog)

title: User:JWSchmidt/Blog/17 March 2007


text:
User:Blueminneapolis~enwikiversity/Personal Menu

Links

 * Topic:Foreign_Language_Learning
 * Topic:Spanish
 * Spanish

Wikibooks

 * Spanish/Contents


text:
User talk:Donnaidh sidhe~enwikiversity

Greece?
So, you know Greek... Are you Greek yourself, you live in any ather country apart from Greece but you have Greek relatives there or are you just a student? Banshee 18:39, 17 March 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Donnaidh sidhe. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Donnaidh sidhe~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:31, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:06, 19 April 2015 (UTC)

title: User talk:Donnaidh sidhe~enwikiversity


text:
User:Sorinelb

SORINEL BALAN Profesional experience:

2016-prezent: ZENITH Consulting: Intranet Consulting, Sharepoint, eColaborare, Digital Workplace for  https://zenify.net/  http://zen-it.ro/   & Achizitii Online, e=Procurement for https://achizio.eu/ http://achizitii-online.ro/

BCR 2000-2013: I defined, created and administrated BCR Intranet (content management) from a single HTML page to a SPS 2003-2007 for the whole BCR organisation 10K employes. In my evaluation with Intranet Review Toolkit, at that moment was probably the best and the biggest Intranet in the country. Management and development BCR web poratls.

Bankcoop s.a. IT division. I work with CIP, Central of Paying Incidents. I created all the pages for Intranet site, I worked at Y2K project. I made web promotion with submitting programs.

Programmer at Bancorex Telecommunications division. I collaborated to the designing and exploitation of banking applications, installations of different applications, opening new branches, training. At my own initiative first Internet connection of Bancorex, first web page, and first Internet presence were achieved. I designed web pages for our Intranet and I proposed to our top management to introduce a Quality Management in Bancorex. For this reason I created a Quality section on our Intranet (TQM, ISO, QM articles and information)

Programmer and secretary at International  Press Service, a  private press agency authorized dealer for New York Times Syndication  Sales and Representative Press Office of Council of Europe in Romania.

Programmer at the Institute for Computers Technique - ITCI

Articles published after 1990

"Online Banking via Internet become reality" - Computer World (83) - 16-31 July, 1997

"Intranet for all" – Computer World - (16) 16-30 Sept 1997

"Webmaster mission" - Computer World

"Banks interested in Quality Management" - Adevarul Economic (11) 13-19, March, 1998

"Information technology and brick and mortar branches" – Adevarul Economic - (18) din 1-7 May 1998

"Unified Field in Computer Science" - COMPUTERWORLD 5 March 1995

"A testimonial on Transcendental Meditation experience" - Magazin 12.05.1990

Scientific activity

Learning Through Double Interface - a possibly revolution in education, Sorinel Balan, Florin Vasilita, The XIII Symposium on Information Technology and Management, CONDINF'89, Cluj-Napoca, 7-9 June 1989

Computer-Aided preschool Education - an interdisciplinary model, Sorinel Balan, Mariana Broscareanu, Conference Progress in Computer Science, PROCOMP'89, Bucuresti

Consideration on Double Interface Terminal - Sorinel Balan, Comsa Radu, Conference - Progress in Computer Science, PROCOMP'89, Bucuresti, 28-29 Sept 1989

Double Interface as a Model of Interdisciplinary Connection Approach, Sorinel Balan, Florin Vasilita, Symposium Computers in Manufacturing Process Modernizing, Miercurea Ciuc, 16-17 November 1989

Courses followed:

https://www.scribd.com/lists/4503665/Certificates     |  https://www.scribd.com/lists/13806753/CURSURI-AUDIATE   |  https://www.scribd.com/lists/16395186/INTRANET

PSR: VP & webmaster AMTR - TM Association from Romania   http://ro.tm.org

My Amazon author page: https://www.amazon.com/author/sorinelbalan/  | Wikiversity:  https://en.wikiversity.org/wiki/User:Sorinelb   | Academia.edu: https://independent.academia.edu/SorinelBalan   |

PSR: VP & webmaster AMTR - TM Association from Romania http://ro.tm.org

title: User:Sorinelb


text:
File:3duStaciLoRes 045 down.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 045 down.png


text:
File:3duStaciLoRes 135 down.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 135 down.png


text:
File:3duStaciLoRes 080 down.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 080 down.png


text:
File:3duStaciLoRes 010 down.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 010 down.png


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File:3duStaciLoRes 080.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 080.png


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File:3duStaciLoRes 010 up.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 010 up.png


text:
File:3duStaciLoRes 010.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 010.png


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File:3duStaciLoRes 135.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 135.png


text:
Greek Language/Vowels

Greek vowels have similar pronunciations as those of Spanish. If you know Spanish, then you will have an easy time pronouncing these. Do not automatically apply standard English vowel pronunciations to Greek words! You will probably be pronouncing them wrong. Please refer to the page linked at the end of this article for sound clips of the correct prounciations.

You will probably have noticed that some of the letters are redundant; Greek has three letters representing the sound /i/ ('ee') and two representing the sound /o/ ('oo'). This is because in Classical Greek the different letters did actually have different sounds associated with them. However, over the centuries, the pronunciations shifted to what they are today.

title: Greek Language/Vowels


text:
File:3duStaciLoRes 045.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 045.png


text:
File:3duStaciLoRes 135 up.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 135 up.png


text:
File:3duStaciLoRes 080 up.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 080 up.png


text:
File:3duStaciLoRes 045 up.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 045 up.png


text:
File:3duStaciLoRes NoHair 000.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes NoHair 000.png


text:
File:3duStaciLoRes NoHair 010.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes NoHair 010.png


text:
File:3duStaciLoRes NoHair 045.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes NoHair 045.png


text:
File:3duStaciLoRes NoHair 080.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes NoHair 080.png


text:
File:3duStaciLoRes NoHair 135.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes NoHair 135.png


text:
File:3duStaciLoRes head 000.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes head 000.png


text:
File:3duStaciLoRes 000 up.png

Example for the Storyboard Artwork Project

title: File:3duStaciLoRes 000 up.png


text:
Greek Language/Consonants

As with the vowels, many Greek consonants are pronounced the same as in Romance languages, especially Italian and Spanish. The main differences between Greek pronunciations and English pronunciations of consonants is that the Greek pronunciations sound lighter, involving palatalization.

Some sounds that are common in other languages simply do not exist in Greek. These sounds are the postalveolar fricatives and affricates: [ʃ] as in "fish," [ʒ] as in "mirage", [tʃ] as in "cheat" and [dʒ] as in "jade". Other common sounds, such as [b], [d] and [ɡ], do exist in Greek, but do not have single letters to represent them. Instead, two letters are used. The sound [b] is represented by μπ (e. g. ball – μπάλα), [d] is written as ντ (e. g. doughnut – ντόνατ) and [ɡ] as γκ (e. g. golf – γκολφ).

There is another sound that exists in English and Greek that does not have its own letter in either language: [ŋ] (ingma), which is the sound ending the word "ring". This sound is much rarer in Greek than in English, but when it shows up, it appears as the combination γχ. Examples: άγχος (anxiety), έλεγχος (checking).

Categories
Greek consonants are divided into 3 categories according to specific criteria:

 * According to their sound
 * unvoiced (Greek: άηχα): κ, π, τ, χ, φ, θ, σ, τσ*
 * voiced (Greek: ηχηρά): γ, β, δ, ζ, λ, μ, ν, ρ, τζ*, μπ*, ντ*, γκ*

 * According to their duration
 * instantaneous (Greek: στιγμιαία): κ, π, τ, τσ*, τζ*, μπ*, ντ*, γκ*
 * continuous (Greek: εξακολουθητικά): γ, β, δ, ζ, λ, μ, ν, ρ, χ, φ, θ, σ

 * According to the part of mouth where they are formed
 * labial (Greek: χειλικά): π, β, φ, μπ*, μ
 * dental (Greek: οδοντικά): τ, δ, θ, ντ*
 * alveolar: σ, ζ, τσ*, τζ*, λ, ρ, ν
 * velar: κ, γ, χ, γκ*

* Two-letter consonants

title: Greek Language/Consonants


text:
User:David Don

About David W Don
David is a Clinical Hypnotherapist and Psychotherapist who has also spent many years researching human motivation and behaviour. He is a skilled user of a number of specialised approaches to personal therapy and has dealt with a wide range of issues over the last six years.

For example:

 * Anorexia nervosa and Bulimia
 * Anxiety, esp. Phobias, Panic attacks
 * Counselling about significant life decisions
 * Depression and mood disorders
 * Drug and alcohol addictions
 * Erectile dysfunction/ impotence
 * Gambling and other compulsive behaviours
 * Low self esteem, Stress
 * Major physical/ organic health issues
 * PTSD and Early childhood trauma
 * Sleep disorders (incl. sleep eating etc)
 * Smoking cessation
 * Weight control.

From 1996 to 2001, David ran a company called Medistrat Pty Ltd. Medistrat specialises in medical market research. Prior to this he was responsible for new product development at ICI Pharmaceuticals and held several research and planning roles.

title: User:David Don


text: Portal:Film/External links

title: Portal:Film/External links


text:
Template:Cite encyclopedia

 .

title: Template:Cite encyclopedia


text:
User:Glen

w:User talk:Glen

title: User:Glen


text:
User talk:Glen

w:User talk:Glen

title: User talk:Glen


text:
User talk:Yeshy~enwikiversity

Image licence
An image you uploaded didn't include any information on licencing. Wikiversity policy is to delete such images, but if you could update the image page with the proper info we can avoid that. Thanks. Historybuff 22:56, 9 April 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Yeshy. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Yeshy~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:48, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:25, 19 April 2015 (UTC)

title: User talk:Yeshy~enwikiversity


text:
Biblical psychology

A BIBLICAL PSYCHOLOGY
Lesson 1: Positive Disintegration <> “Unless a grain of wheat falls into the earth and dies …” (John 12:24)

INTRODUCTION: This Lesson will present you with a theory of soul / psyche which matches up more closely with the Story of Job* and the psychological growth of Biblical Exemplars** than any other theory of Modern Psychology.
 * Chronologically, “Job” was the first written book of the Bible.
 * i.e. Jacob, David, Moses, Paul

PURPOSE: This Lesson will... 1) Introduce Kazmierz Dabrowski, author of the Theory of Positive Disintegration; Note: The Editor will, from Lesson 2 on, attempt a virtually seamless transition to [an overtly biblical] Psychology entitled “Creative Disintegration”;2) Introduce, as Psychology, “Unless a grain of wheat  falls into the earth and dies …” (John 12:24);3) Introduce the role of conflict in the developmental process;4) Offer both a cryptic contrast of “Biblical Psychology” to Freudand a comparison  with Jung.

LEARNING HINTS: As you read the Sources, look for the following: 1) A sense of both the person and the achievement of Professor Dabrowski; 2) An initial grasp of the essence of both "Unless a grain of wheat … ” and "Over-excitability"; 3) A sense of the role of conflict in the developmental process; 4) An initial "handle" on: a) Freud <> A Biblical Psychology contrast;     b) Jung <> A Biblical Psychology similarities. <<>>

To continue on to the Lesson itself, Click:

title: Biblical psychology


text:
User:Julien1311

Wikiversity fr · de · en · es · it · beta | Wikipedia fr · en | Wikibooks fr | Wiktionary fr · en | Wikiquote fr | Commons | Meta | MediaWiki

My page on the French Wikiversity.

WV:IBC

de:Benutzer:Julien1311 es:Usuario:Julien1311 fr:Utilisateur:Julien1311 it:Utente:Julien1311

title: User:Julien1311


text:
User:Iwantfrench

Iwantfrench

Interests
 * 1) Learn French
 * 2) Learn Latin
 * 3) Learn German
 * 4) Learn Italian

title: User:Iwantfrench


text:
Efficiency in computing project

This project will explore efficiency in computing from a variety of perspectives. The project will explore maximizing the efficiency of computers in the following areas:
 * Noise production
 * Energy consumption
 * Cost
 * Weight
 * Environmental impact
 * Life/durability
 * Design/usability
 * Ease of use
 * Pleasantness of experience during use

To participate, collaborate... brainstorm...

Project news

 * March 17, 2007 - Project started

Wikipedia

 * Geode
 * Solid-state fan
 * VIA Eden

title: Efficiency in computing project


text:
User:Polikmus

Welcome to Wikiversity. I'm glad you are taking an interesting in participating. Please don't interpret my comments as biting. I've been told certain things, and something in my gut has felt annoyed, but ultimately I have found the input is generally the best for the project, group and community collectively.

Anyway, please consider if your material would be more appropriate as book material. If you think your contributions would be more appropriate for a book, please consider contributing to Wikibooks.

If you would like to start a department you can use the Department template.

If you want to start a learning project, a course, or feel like doing some research, go a head and be bold. I am under the impression that there are no firm set ways of doing these things. It's a new frontier! =)

If you have any questions please feel free to ask me or post them at the colloquium--Remi 23:08, 17 March 2007 (UTC)

title: User:Polikmus


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User:JWSchmidt/monobook.js

import_script("User:MichaelBillington/ircform.js"); import_script("User:Darklama/monobook.js");

title: User:JWSchmidt/monobook.js


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Template:Note

Not to be confused with.

title: Template:Note


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User:Nimur

Wikipedia User Page

title: User:Nimur


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Electrostatic fields

A static field
We call a field static field when it does not change with time explicitly. Electric fields due to static or stationary charge distributions are electrostatic fields. For example, if we have a charge held fixed somewhere in the space, it creates an electrostatic field. And when it is moving, it creates an electrodynamic field (the charge distribution changes with time explicitly). But when I have a static current of charges in a wire, it creates an electrostatic field (and later it will be seen that moving charges create magnetic field too) because although the charges are moving, the charge distribution in the space is not changing with time as we have a fixed current.

The idea of static field must not be confused as it is the same always. Sometimes in electrostatics we encounter problems when the charge distribution do change but we wish to study the final theme when everything is static (as example, when we place a small charge near a conducting surface, it creates induced charges on the conducting surface and thus changes the charge distribution; but in study of electrostatics we are concerned about what the ultimate charge distribution or field will be there and not about what was happening when the charge distribution was changing).

Electric Field
The electric force acting on a charged object can be interpreted as being produced by some property of the space where that object is located. That property of space that leads to forces over the electrostatic charges is called Electric field denoted as E

In mathematical terms, the electric field can be defined precisely in the following way: at any point in space the electric field is a vector $$\vec{E}$$ obtained by dividing the force $$\vec{F}$$ acting on a point charge $$q_0$$ placed in that point, over the value of the charge:

$$\vec{E} = \frac{\vec{F}}{q_0}$$

With
 * $$\vec{E}$$ Vector Electric field
 * $$F$$ Electric force
 * $$q_0$$ Point charge

Electric Potential
The source of the electric field can be either other electrostatic charges, or a varying magnetic field. If the field is produced by static electric charge distribution, it is called electrostatic field. In that case, it can be shown that is is a conservative field; namely, the path integral of the field between two points is independent of the path. As a consequence, it is possible to define a potential function: the potential in any point of space $$(x,y,z)$$ is equal to the path integral of the electrostatic field from an arbitrary reference point $$P_0$$ until the point $$(x,y,z)$$.

$$V(x,y,z) = -\int_{P_0}^{(x,y,z)} \vec{E}\cdot d\vec{l}$$

Thus, the electrostatic potential (also known as voltage) is a scalar field and it can be proved that the gradient of that scalar field is equal to minus the electrostatic field:

$$\vec{E} = -\vec{\nabla} V$$

title: Electrostatic fields


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School:History/Tasks

Welcome to the School of History's Task page! This page is designed as a simple place where our Departments can list things that they need help on, so that you can dig in and get involved contributing to Wikiversity. If any of the following sound like things you'd be interested in doing, jump right in! Be bold !

The following list is organized by Department and Project.

All Departments
With the new Browse system in place at Wikiversity, categorization is important. If you have some spare time, consider adding and organizing categories for School of History projects.

General learning project

 * History of the World

Timelines
I have started using timelines. Please see Tudor Origins of the British Empire/Timeline. Perhaps the format can be improved. But if we have a shared format, then specific facts can easily be copied from different timelines. i.e. Crown of Ireland Act 1542 could be included in a timeline for Ireland. I have created a space to put links for timelines here.

Departmental Task pages
The format is "pagename/Tasks". It would be helpful if each department had its own task page, as below.

🇨🇴
 * African History
 * Chinese History
 * European History
 * Historiography

 * History of Racism
 * Military History
 * North American History

title: School:History/Tasks


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Wikiversity talk:Open Proxies/List15

List blocked, needs tags--Rayc 05:21, 18 March 2007 (UTC)

title: Wikiversity talk:Open Proxies/List15


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Talk:Beginning composition

i think this is a terrible article. it doesn't provide any information or direction on the subject of beginning composition, and as a whole i think it would be more a detriment than a help to anyone wishing to learn about this subject. also the language of the stub is emotionally charged and does not sound objective and impartial.

I agree. This article does not give constructive information about methods of learning composition. It is true that there is not one hard-and-fast method, but there are several types of compositional strategies that could be discussed, like chord progression and traditional part-writing rules. Traditional part-writing or the "rules of counterpoint" have been taugh from Beethoven's time to the present. It's useful for composers of Western music to learn them, even though they do not always have to follow them.

Perhaps twelve-tone composition could be discussed as well. MusicWeaver 08:11, 10 May 2007 (UTC)

title: Talk:Beginning composition


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Talk:Introduction to Proofs

what happened to this topic

title: Talk:Introduction to Proofs


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File:Seduced By The Dark Side-Seyi Longe.pdf

Used with the WikiU Film School Lesson on how to format a movie script


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HTML/Links

Links are used to connect web pages together. They can do more than link, but for right now, we will focus on the basics.

The tag
A link is denoted by the  tag and requires a   closing tag. For example: The text in between the   tag and the closed   tag is the text that the link will be displayed at. The example above will look like Example. The  attribute, a characteristic with a value inside the opening HTML tag, is the thing that tells  the link where the person is going to go when they click on it. In between the two quotes is the URL of where you want the link to go.

Anchors
Anchors can be used to create links to specific points in an HTML page.

Firstly, place  at the spot you want to link to.

Secondly, place  to create a link on the same page or if its a link from a different page, use   e.g.,.

Tooltips
Tooltips allow revealing details about a link when hovering above it with the mouse pointer:. Sites use it for previewing the meaning of abbreviations and exact time stamps of a relative time.

This can also be done for plain text with  ( demo ). For abbreviations, the dedicated  tag can be used, which browsers usually highlight with a dotted underline, and line breaks can be inserted with   (demo with line break ).

Note that these are not widely supported on mobile devices as of 2021

title: HTML/Links


text: User talk:Ageru

title: User talk:Ageru


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User:Chrisfow

Chris Fowler
This is the user page of Chrisfow. The person behind this username is Chris Fowler, an undergraduate in the King's College London Department of War Studies.

I am a regular contributor to the English Wikipedia under the same username, and there is more information about me on my Wikipedia userpage. I found Wikiversity through involvement in that project.

Specialisms
I specialise in:
 * Modern era (post-1789) military history
 * Naval history and modern day applications
 * Current military and security issues
 * International relations, theory in general and specific practice of supra-national organisations

Feel free to message me any questions regarding these topics in particular or History and IR in general on my talk page.

My stuff

 * User:Chrisfow/sandbox
 * School:Strategic_Studies
 * Topic:War

title: User:Chrisfow


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User talk:Chrisfow

Welcome to S3
Hello and thanks for getting involved in the School of Strategic Studies ! I've just seen you begun several important courses, that's just great. Do not hesitate to change things in the curriculum: I've made it by improving on Dnjkirk's great job, but the results are far from perfect. A few comments about Clausewitz: First, you were totally right in removing subjective comment: even if they were mine, it wasn't the right place; second, about translations: you're right to say Howard & Paret's translation isn't perfect, but it's still widely recognized among scholars as the best english translation. Once more, thanks for getting in ! --CorentinB 22:34, 29 March 2007 (UTC)

title: User talk:Chrisfow


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Google/SketchUp 6, review

This page is under construction this week {| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:98%" Filmmaking Software Review for Narrative Film Production
 * colspan="2" style="width:100%; vertical-align:top; border:1px solid Sienna; background-color:Cornsilk;" |

Google's Free SketchUp 6

 * This is an extremely unusual program which can be useful for pre-visualization in filmmaking.

 * }

{| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:98%" Review by Robert Elliott
 * style="width:60%; vertical-align:top; border:1px solid Gold; background-color: LightYellow;" rowspan="3"|

Google's Free SketchUp 6

 * Google purchase SketchUp and then released two new versions. One is the free SketchUp version 6 and the other is the professional version (not free) SketchUp Pro 6

 * Unusual
 * This program is unusual. It is very exacting but highly unintuitive.  This takes a bit of getting used to but once you do understand how it works, it is extremely powerful as a 3D sketching tool.

 * Other confused programs
 * Other programs which take a long time to learn are often confusing because the authors of the program are confused and not too bright. (PageMaker is an example of a program which works well but is awkward to learn because the authors of the program were never typesetters.  Apple's Logic is another program which is a hopeless mess because the authors added on to it without fixing the previous parts.)

 * The authors of SketchUp
 * Not so with SketchUp. The authors of SketchUp are extremely bright and very focused.  However, the design that they came up with is not what the average person will call intuitive.  Even people who have used many other sophisticated computer programs will be very confused by this program.

 * Very rewarding challenge
 * Think of SketchUp as a computer puzzle game. You must put on your thinking cap to master this program.  Fortunately, your time and effort will not be wasted.  Every step you learn allows you to be more productive with the program.

 * The challenge
 * The authors of SketchUp realize they have a problem educating people how to use this program. Therefore, the program comes with a huge number of very sophisticated learning aides.  This is by far, the best example of a well designed instructions for any program I have ever seen.

 * Five Stars
 * If you can master this program, you will find it very useful and fun. If you do not master this program, it can be total pain.

 * Highly recommended
 * Since it is free, it is worth downloading immediately and trying. If you like the way it works, you will be extremely happy with this program.  If you don't then you are not out anything.

 * style="padding:0em 0.5em 0em 0.5em;" |

SketchUp's uses for film students
 * style="width:40%; vertical-align:top; border:1px solid #abd5f5; background-color:#f1f5fc;" |

 * Pre-visualization of movie sets
 * 3D Storyboarding
 * Having fun!!!
 * style="padding:0em 0.5em 0em 0.5em;" |
 * style="padding:0em 0.5em 0em 0.5em;" |

Rating
 * style="width:40%; vertical-align:top; border:1px solid #abd5f5; background-color:#f1f5fc;" |

 * Five Stars

 * The program is extremely unique.

 * style="padding:0em 0.5em 0em 0.5em;" |
 * style="padding:0em 0.5em 0em 0.5em;" |

Program Information
 * style="width:40%; vertical-align:top; border:1px solid #abd5f5; background-color:#f1f5fc;" |

 * }

title: Google/SketchUp 6, review


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User talk:Sundance Raphael

Poetry
Hi Sundance - I have edited the poem some more - I hope I got it right. Please let me know if I can help you. Best wishes --Herby talk thyme 13:30, 26 April 2007 (UTC)

I just noticed...
...my birthday is the day after yours. :-) Cormaggio talk 08:21, 28 April 2007 (UTC)

Hello!
Hi there! I wonder if you would be willing to help out at German stream? We could really use your help :-) --HappyCamper 20:51, 5 May 2007 (UTC)

Bit busy
Hi Anne,

I am a bit busy this weekend. I don't know when i will be online. Especially saturday will be difficult starting in the morning. You can discuss Thucydides with Erkan. I really enjoyed reading it.--Daanschr 12:24, 13 December 2007 (UTC)

Meeting next sunday
Hi Sunny,

I will be moving to a new home on sunday and will not have internet yet. I will not be able to be around on sunday evening and i don't know when i will be available again. You will have to continue the reading groups with Mike and Erkan for the moment being.--Daanschr 17:17, 28 December 2007 (UTC)

New meeting for discussing the reading groups and Thucydides
Hi Sunny,

I proposed to Erkan and Mike to meet again next saturday. We could discuss the future of the reading groups on 19:00 hour and discuss Thucydides without a proposed time on saturday. The agenda for discussing the future of the reading groups is a bit of a mess, you can read it here.

Liked the poem of Shakespeare by the way, but can´t listen to your voice, because i don´t have sound at home.--Daanschr 14:57, 13 January 2008 (UTC)

music barnstar
Thank you for your contributions to Wikiversity. JWSchmidt 04:52, 26 August 2008 (UTC)

Music
I saw you sang songs and made them available to wikiversity. I am creating a project to record music performances, made by ourselves. I am good with the piano, so I will start recording some of Beethoven sonatas. Check Music performances page. The goal is to use a channel or something like that in Wiki Campus Radio. Davichito 02:00, 6 November 2008 (UTC)

title: User talk:Sundance Raphael


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Bloom clock project/2007 data

This page will list data from the 2007 clocks. To record a plant in bloom, go to the current clock and follow the directions at the top of the page!

title: Bloom clock project/2007 data


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Wikimania

Welcome to Wikimania at Wikiversity! This is the place to coordinate Wikiversity's contributions to the Wikimania fray and is for "show and tell" about all things Wikimaniacal.

Wikimania 2012
Wikimania 2012 will be the 8th annual Wikimedia Conference. It will be held in Washington, D.C. on July 12–15, 2012. See Wikimania 2012 for details of this event. See Wikiversity at Wikimania 2012 if you would like to be involved in organising Wikiversity input.

Wikimania 2012 will be the 8th annual Wikimedia Conference. It will be held in Washington, D.C. on July 12–15, 2012.

Wikimania 2009, the fifth annual international Wikimedia conference was held August 25-29, 2009 in Buenos Aires, Argentina. It was the first held in Spanish and in the Southern Hemisphere.

Being there
"Wikimania" is an annual face-to-face gathering of enthusiasts of Wikipedia, Wiktionary, Wikisource and other Wikimedia projects. However, you don't have to physically attend Wikimania to participate. Furthermore, at Wikiversity, we are attempting to find ways to sustain this contact throughout the year. This could include having online meetings - perhaps with the rationale of helping combat climate change by reducing the amount of people travelling the world in jet aircraft. (Have your say!)

An adventure
At Wikimania 2006 in Boston, Wikiversity was announced as the newest addition to the Wikimedia family of projects. Several learning projects and content development groups here at Wikiversity plan to participate in Wikimania:
 * The School of Media Studies will build learning resources for and from the Wikimania experience through Internet Audio and Video.

title: Wikimania


text: Template:Wikimainia

title: Template:Wikimainia


text: Talk:Wikimania

title: Talk:Wikimania


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User talk:Devourer09

Welcome
Have you browsed Lunar Boom Town yet? 8) I know you have seen cisLunarFreighter.   Here is my account signature.   Back to IRC now.  Mirwin 00:27, 9 January 2008 (UTC)

Proposal -> Policy
Thanks for your help. About this: I didn't find yet a vote that this was accepted by the community in that version ? Could you point me to there ? If there is no such vote, please do not state it as policy. You could initiate a discussion about it in the Colloquium if you like. Thanks, Erkan Yilmaz uses the Chat (try) 17:09, 25 August 2008 (UTC)

First barnstar
OOOOOOOOOH! My very first barnstar! Shinyyy... I was actually worried at first that I'd get asked to "stop importing all those nasty Wikipedia templates, they won't be used"! But what a great surprise to get a pointy barnstar instead. If someone else tries to tell me the templates aren't needed, do I have permission to use my barnstar as a shuriken against them? :P

Apologies for cluttering up the Recent Changes though hehe. Gaidheal1 22:29, 3 June 2011 (UTC)
 * Heck yes you have my permission! Go to town with your ninja skillz! :D Devourer09  ( t · c ) 16:59, 5 June 2011 (UTC)

PLEs
I just looked at some PLE pages + saw your PLE with: "User:Erkan Yilmaz/PLE - Maybe I can learn something from how he uses his PLE." Your views would help me also, but I think users would also benefit on your experience about your usage of PLEs, since I see you are editing still there, Erkan Yilmaz uses the Chat + Identi.ca 01:11, 22 October 2011 (UTC)
 * just in case this edit caused you sleepless nights or so, feel free to tell (me) about it: e.g. here ;-) Erkan Yilmaz uses the Chat + Identi.ca 05:37, 13 February 2012 (UTC)

Complain!
I know your complain about What is Wikiversity?/En. I have moved What is Wikiversity?/En to What is Wikiversity?! Thanks for complaining, I have to show a custodian why I moved the page! Goodbye now! --Draubb (discuss • contribs) 20:03, 23 May 2013 (UTC)

title: User talk:Devourer09


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Wikiversity talk:Assume good faith

This is a very good policy. It is one of the most important. It makes many good points. It is right when it says that the wiki would not of made it if the majority of the people did not edit in good faith. That is something many people need to remember. If they did I think they would deal with people in a more civil way. Peace:) --Sir James Paul 00:04, 19 March 2007 (UTC)

title: Wikiversity talk:Assume good faith


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Talk:Google/SketchUp 6, review

Robert, you have a really good start on this. It looks very nice:) --Sir James Paul 00:18, 19 March 2007 (UTC)

title: Talk:Google/SketchUp 6, review


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Portal talk:Nuclear engineering

Its pretty good. I took out "Welcome to the department of nuclear engineering" because I feel as if it really is not needed. Peace:) --Sir James Paul 00:33, 19 March 2007 (UTC)

 * I think somewhere very early in the listed topics or coursework should be radiation basics, nuclear medicine, hazards, precautions, etc.  This might be by links to other areas or site.   A few years back a kid in the SouthEast, USA built himself an isotope pile by buying up old watchs with faces and hands painted with thorium isotopes.  Using these he created other unstable isotopes.  When confronted by newpaper reporters after the authorities arrived and confiscated the pile and started decontaminating the neighborhood the child's parents seemed to have no idea that radioactivity was unhealthy and to be avoided.  Only proud their child had attracted national attention.  When such juveniles are using Wikiversity it would be nice to point them at appropriate precautions when studying nuclear aspects of nature.  Most Jimmy Neutron's or surrounding enablers would like to live long enough to enjoy their inventions. Mirwin 20:38, 19 December 2007 (UTC)

title: Portal talk:Nuclear engineering