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Catenary

In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.

The catenary curve has a U-like shape, superficially similar in appearance to a parabola, which it is not.

The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings.

The catenary is also called the alysoid, chainette, or, particularly in the materials sciences, an example of a funicular. Rope statics describes catenaries in a classic statics problem involving a hanging rope.

Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid, is a minimal surface, specifically a minimal surface of revolution. A hanging chain will assume a shape of least potential energy which is a catenary. Galileo Galilei in 1638 discussed the catenary in the book Two New Sciences recognizing that it was different from a parabola. The mathematical properties of the catenary curve were studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691.

Catenaries and related curves are used in architecture and engineering (e.g., in the design of bridges and arches so that forces do not result in bending moments). In the offshore oil and gas industry, "catenary" refers to a steel catenary riser, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. In the rail industry it refers to the overhead wiring that transfers power to trains. (This often supports a contact wire, in which case it does not follow a true catenary curve.)

In optics and electromagnetics, the hyperbolic cosine and sine functions are basic solutions to Maxwell's equations. The symmetric modes consisting of two evanescent waves would form a catenary shape.

History

The word "catenary" is derived from the Latin word catēna, which means "chain". The English word "catenary" is usually attributed to Thomas Jefferson, who wrote in a letter to Thomas Paine on the construction of an arch for a bridge:

"I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium."

It is often said that Galileo thought the curve of a hanging chain was parabolic. However, in his Two New Sciences (1638), Galileo wrote that a hanging cord is only an approximate parabola, correctly observing that this approximation improves in accuracy as the curvature gets smaller and is almost exact when the elevation is less than 45°. The fact that the curve followed by a chain is not a parabola was proven by Joachim Jungius (1587–1657); this result was published posthumously in 1669.

The application of the catenary to the construction of arches is attributed to Robert Hooke, whose "true mathematical and mechanical form" in the context of the rebuilding of St Paul's Cathedral alluded to a catenary. Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon.

In 1671, Hooke announced to the Royal Society that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin anagram in an appendix to his Description of Helioscopes, where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building." He did not publish the solution to this anagram in his lifetime, but in 1705 his executor provided it as ut pendet continuum flexile, sic stabit contiguum rigidum inversum, meaning "As hangs a flexible cable so, inverted, stand the touching pieces of an arch."

In 1691, Gottfried Leibniz, Christiaan Huygens, and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli; their solutions were published in the Acta Eruditorum for June 1691. David Gregory wrote a treatise on the catenary in 1697 in which he provided an incorrect derivation of the correct differential equation.

Euler proved in 1744 that the catenary is the curve which, when rotated about the $x$-axis, gives the surface of minimum surface area (the catenoid) for the given bounding circles. Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796.

Inverted catenary arch
Catenary arches are often used in the construction of kilns. To create the desired curve, the shape of a hanging chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material.

The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is incorrect. It is close to a more general curve called a flattened catenary, with equation $y = A cosh(Bx)$, which is a catenary if $AB = 1$. While a catenary is the ideal shape for a freestanding arch of constant thickness, the Gateway Arch is narrower near the top. According to the U.S. National Historic Landmark nomination for the arch, it is a "weighted catenary" instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form.

Catenary bridges

In free-hanging chains, the force exerted is uniform with respect to length of the chain, and so the chain follows the catenary curve. The same is true of a simple suspension bridge or "catenary bridge," where the roadway follows the cable.

A stressed ribbon bridge is a more sophisticated structure with the same catenary shape.

However, in a suspension bridge with a suspended roadway, the chains or cables support the weight of the bridge, and so do not hang freely. In most cases the roadway is flat, so when the weight of the cable is negligible compared with the weight being supported, the force exerted is uniform with respect to horizontal distance, and the result is a parabola, as discussed below (although the term "catenary" is often still used, in an informal sense). If the cable is heavy then the resulting curve is between a catenary and a parabola.

Anchoring of marine objects
The catenary produced by gravity provides an advantage to heavy anchor rodes. An anchor rode (or anchor line) usually consists of chain or cable or both. Anchor rodes are used by ships, oil rigs, docks, floating wind turbines, and other marine equipment which must be anchored to the seabed.

When the rope is slack, the catenary curve presents a lower angle of pull on the anchor or mooring device than would be the case if it were nearly straight. This enhances the performance of the anchor and raises the level of force it will resist before dragging. To maintain the catenary shape in the presence of wind, a heavy chain is needed, so that only larger ships in deeper water can rely on this effect. Smaller boats also rely on catenary to maintain maximum holding power.

Cable ferries and chain boats present a special case of marine vehicles moving although moored by the two catenaries each of one or more cables (wire ropes or chains) passing through the vehicle and moved along by motorized sheaves. The catenaries can be evaluated graphically.

Equation

The equation of a catenary in Cartesian coordinates has the form

$$y = a \cosh \left(\frac{x}{a}\right) = \frac{a}{2}\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right),$$ where $cosh$ is the hyperbolic cosine function, and where $a$ is the distance of the lowest point above the x axis. All catenary curves are similar to each other, since changing the parameter $a$ is equivalent to a uniform scaling of the curve.

The Whewell equation for the catenary is $$\tan \varphi = \frac{s}{a},$$ where $$\varphi$$ is the tangential angle and $a$ the arc length.

Differentiating gives $$\frac{d\varphi}{ds} = \frac{\cos^2\varphi}{a},$$ and eliminating $$\varphi$$ gives the Cesàro equation $$\kappa=\frac{a}{s^2+a^2},$$ where $$\kappa$$ is the curvature.

The radius of curvature is then $$\rho = a \sec^2 \varphi,$$ which is the length of the normal between the curve and the $s$-axis.

Relation to other curves
When a parabola is rolled along a straight line, the roulette curve traced by its focus is a catenary. The envelope of the directrix of the parabola is also a catenary. The involute from the vertex, that is the roulette traced by a point starting at the vertex when a line is rolled on a catenary, is the tractrix.

Another roulette, formed by rolling a line on a catenary, is another line. This implies that square wheels can roll perfectly smoothly on a road made of a series of bumps in the shape of an inverted catenary curve. The wheels can be any regular polygon except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.

Geometrical properties
Over any horizontal interval, the ratio of the area under the catenary to its length equals $x$, independent of the interval selected. The catenary is the only plane curve other than a horizontal line with this property. Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the perpendicular segment connecting the centroid of the curve itself and the $a$-axis.

Science
A moving charge in a uniform electric field travels along a catenary (which tends to a parabola if the charge velocity is much less than the speed of light $x$).

The surface of revolution with fixed radii at either end that has minimum surface area is a catenary revolved about the $c$-axis.

Model of chains and arches
In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the chain is parallel to the chain. The analysis of the curve for an optimal arch is similar except that the forces of tension become forces of compression and everything is inverted. An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium. Equations which define the shape of the curve and the tension of the chain at each point may be derived by a careful inspection of the various forces acting on a segment using the fact that these forces must be in balance if the chain is in static equilibrium.

Let the path followed by the chain be given parametrically by $r = (x, y) = (x(s), y(s))$ where $x$ represents arc length and $r$ is the position vector. This is the natural parameterization and has the property that

$$\frac{d\mathbf{r}}{ds}=\mathbf{u}$$

where $u$ is a unit tangent vector.

A differential equation for the curve may be derived as follows. Let $c$ be the lowest point on the chain, called the vertex of the catenary. The slope $r$ of the curve is zero at $T_{0}$ since it is a minimum point. Assume $c$ is to the right of $T$ since the other case is implied by symmetry. The forces acting on the section of the chain from $r$ to $(0, −ws)$ are the tension of the chain at $c$, the tension of the chain at $dy⁄dx$, and the weight of the chain. The tension at $c$ is tangent to the curve at $r$ and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written $c$ where $c$ is the magnitude of the force. The tension at $r$ is parallel to the curve at $c$ and pulls the section to the right. The tension at $r$ can be split into two components so it may be written $c$, where $s$ is the magnitude of the force and $T$ is the angle between the curve at $c$ and the $φ$-axis (see tangential angle). Finally, the weight of the chain is represented by $(−T_{0}, 0)$ where $x$ is the weight per unit length and $w$ is the length of the segment of chain between $T_{0}$ and $r$.

The chain is in equilibrium so the sum of three forces is $r$, therefore

$$T \cos \varphi = T_0$$ and $$T \sin \varphi = ws\,,$$

and dividing these gives

$$\frac{dy}{dx}=\tan \varphi = \frac{ws}{T_0}\,.$$

It is convenient to write

$$a = \frac{T_0}{w}$$

which is the length of chain whose weight is equal in magnitude to the tension at $r$. Then

$$\frac{dy}{dx}=\frac{s}{a}$$

is an equation defining the curve.

The horizontal component of the tension, $Tu = (T cos φ, T sin φ)$ is constant and the vertical component of the tension, $r$ is proportional to the length of chain between $(0, −ws)$ and the vertex.

Derivation of equations for the curve
The differential equation $$dy/dx = s/a$$, given above, can be solved to produce equations for the curve. We will solve the equation using the boundary condition that the vertex is positioned at $$s_0=0$$ and $$(x,y)=(x_0,y_0)$$.

First, invoke the formula for arc length to get $$\frac{ds}{dx} = \sqrt{1+\left(\frac{dy}{dx}\right)^2} = \sqrt{1+\left(\frac{s}{a}\right)^2}\,,$$ then separate variables to obtain $$\frac{ds}{\sqrt{1+(s/a)^2}} = dx\,.$$

A reasonably straightforward approach to integrate this is to use hyperbolic substitution, which gives $$a \sinh^{-1}\frac{s}{a} + x_0 = x$$ (where $$x_0$$ is a constant of integration), and hence $$\frac{s}{a} = \sinh\frac{x-x_0}{a}\,.$$

But $s/a = dy/dx$, so $$\frac{dy}{dx} = \sinh\frac{x-x_0}{a}\,,$$ which integrates as $$y = a \cosh\frac{x-x_0}{a} + \delta$$ (with $$\delta=y_0-a$$ being the constant of integration satisfying the boundary condition).

Since the primary interest here is simply the shape of the curve, the placement of the coordinate axes are arbitrary; so make the convenient choice of $x_0=0=\delta$ to simplify the result to $$y = a \cosh\frac{x}{a} \quad\square$$

For completeness, the $$y \leftrightarrow s$$ relation can be derived by solving each of the $$x \leftrightarrow y$$ and $$x \leftrightarrow s$$ relations for $$x/a$$, giving: $$\cosh^{-1}\frac{y-\delta}{a} = \frac{x-x_0}{a} = \sinh^{-1}\frac{s}{a}\,,$$ so $$y-\delta = a\cosh\left(\sinh^{-1}\frac{s}{a}\right)\,,$$ which can be rewritten as $$y-\delta = a\sqrt{1+\left(\frac{s}{a}\right)^2} = \sqrt{a^2 + s^2}\,.$$

Alternative derivation
The differential equation can be solved using a different approach. From

$$s = a \tan \varphi$$

it follows that

$$\frac{dx}{d\varphi} = \frac{dx}{ds}\frac{ds}{d\varphi}=\cos \varphi \cdot a \sec^2 \varphi= a \sec \varphi$$ and $$\frac{dy}{d\varphi} = \frac{dy}{ds}\frac{ds}{d\varphi}=\sin \varphi \cdot a \sec^2 \varphi= a \tan \varphi \sec \varphi\,.$$

Integrating gives,

$$x = a \ln(\sec \varphi + \tan \varphi) + \alpha$$ and $$y = a \sec \varphi + \beta\,.$$

As before, the $s$ and $x$-axes can be shifted so $y$ and $α$ can be taken to be 0. Then

$$\sec \varphi + \tan \varphi = e^\frac{x}{a}\,,$$ and taking the reciprocal of both sides $$\sec \varphi - \tan \varphi = e^{-\frac{x}{a}}\,.$$

Adding and subtracting the last two equations then gives the solution $$y = a \sec \varphi = a \cosh\left(\frac{x}{a}\right)\,,$$ and $$s = a \tan \varphi = a \sinh\left(\frac{x}{a}\right)\,.$$

Determining parameters
In general the parameter $β$ is the position of the axis. The equation can be determined in this case as follows:

Relabel if necessary so that $c$ is to the left of $r$ and let $T_{H}$ be the horizontal and $a$ be the vertical distance from $0$ to $c$. Translate the axes so that the vertex of the catenary lies on the $H$-axis and its height $v$ is adjusted so the catenary satisfies the standard equation of the curve $$y = a \cosh\left(\frac{x}{a}\right)$$

and let the coordinates of $T cos φ = T_{0}$ and $T sin φ = ws$ be $r$ and $P_{1}$ respectively. The curve passes through these points, so the difference of height is

$$v = a \cosh\left(\frac{x_2}{a}\right) - a \cosh\left(\frac{x_1}{a}\right)\,.$$

and the length of the curve from $P_{2}$ to $P_{1}$ is

$$L = a \sinh\left(\frac{x_2}{a}\right) - a \sinh\left(\frac{x_1}{a}\right)\,.$$

When $P_{2}$ is expanded using these expressions the result is

$$L^2-v^2=2a^2\left(\cosh\left(\frac{x_2-x_1}{a}\right)-1\right)=4a^2\sinh^2\left(\frac{H}{2a}\right)\,,$$ so $$\frac 1H \sqrt{L^2-v^2}=\frac{2a}H \sinh\left(\frac{H}{2a}\right)\,.$$

This is a transcendental equation in $y$ and must be solved numerically. Since $$\sinh(x)/x$$ is strictly monotonic on $$x > 0$$, there is at most one solution with $P_{1}$ and so there is at most one position of equilibrium.

However, if both ends of the curve ($P_{2}$ and $(x_{1}, y_{1})$) are at the same level ($(x_{2}, y_{2})$), it can be shown that $$a = \frac {\frac14 L^2-h^2} {2h}\,$$ where L is the total length of the curve between $P_{1}$ and $P_{2}$ and $a$ is the sag (vertical distance between $L^{2} − v^{2}$, $a > 0$ and the vertex of the curve).

It can also be shown that $$L = 2a \sinh \frac {H} {2a}\,$$ and $$H = 2a \operatorname {arcosh} \frac {h+a} {a}\,$$ where H is the horizontal distance between $P_{1}$ and $P_{2}$ which are located at the same level ($y_{1} = y_{2}$).

The horizontal traction force at $P_{1}$ and $P_{2}$ is $P_{1}$, where $a$ is the weight per unit length of the chain or cable.

Tension relations
There is a simple relationship between the tension in the cable at a point and its $h$- and/or $w$- coordinate. Begin by combining the squares of the vector components of the tension: $$(T\cos\varphi)^2 + (T\sin\varphi)^2 = T_0^2 + (ws)^2$$ which (recalling that $$T_0=wa$$) can be rewritten as $$\begin{align} T^2(\cos^2\varphi + \sin^2\varphi) &= (wa)^2 + (ws)^2 \\[6pt] T^2 &= w^2 (a^2 + s^2) \\[6pt] T &= w\sqrt{a^2+s^2} \,. \end{align}$$ But, as shown above, $$y = \sqrt{a^2 + s^2}$$ (assuming that $$y_0=a$$), so we get the simple relations $$T = wy = wa \cosh\frac{x}{a}\,.$$

Variational formulation
Consider a chain of length $$L$$ suspended from two points of equal height and at distance $$D$$. The curve has to minimize its potential energy $$ U = \int_0^D w y\sqrt{1+y'^2} dx $$ (where $x$ is the weight per unit length) and is subject to the constraint $$ \int_0^D \sqrt{1+y'^2} dx = L\,.$$

The modified Lagrangian is therefore $$ \mathcal{L} = (w y - \lambda )\sqrt{1+y'^2}$$ where $$\lambda $$ is the Lagrange multiplier to be determined. As the independent variable $$x$$ does not appear in the Lagrangian, we can use the Beltrami identity $$ \mathcal{L}-y' \frac{\partial \mathcal{L} }{\partial y'} = C $$ where $$C$$ is an integration constant, in order to obtain a first integral $$\frac{(w y - \lambda )}{\sqrt{1+y'^2}} = -C$$

This is an ordinary first order differential equation that can be solved by the method of separation of variables. Its solution is the usual hyperbolic cosine where the parameters are obtained from the constraints.

Nonuniform chains
If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.

Let $y$ denote the weight per unit length of the chain, then the weight of the chain has magnitude

$$\int_\mathbf{c}^\mathbf{r} w\, ds\,,$$

where the limits of integration are $P_{2}$ and $P_{1}$. Balancing forces as in the uniform chain produces

$$T \cos \varphi = T_0$$ and $$T \sin \varphi = \int_\mathbf{c}^\mathbf{r} w\, ds\,,$$ and therefore $$\frac{dy}{dx}=\tan \varphi = \frac{1}{T_0} \int_\mathbf{c}^\mathbf{r} w\, ds\,.$$

Differentiation then gives

$$w=T_0 \frac{d}{ds}\frac{dy}{dx} = \frac{T_0 \dfrac{d^2y}{dx^2}}{\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}}\,.$$

In terms of $w$ and the radius of curvature $w$ this becomes

$$w= \frac{T_0}{\rho \cos^2 \varphi}\,.$$

Suspension bridge curve

A similar analysis can be done to find the curve followed by the cable supporting a suspension bridge with a horizontal roadway. If the weight of the roadway per unit length is $φ$ and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable (see the figure in Catenary) from $P_{2}$ to $H = x_{2} − x_{1}$ is $ρ$ where $w$ is the horizontal distance between $P_{1}$ and $P_{2}$. Proceeding as before gives the differential equation

$$\frac{dy}{dx}=\tan \varphi = \frac{w}{T_0}x\,. $$

This is solved by simple integration to get

$$y=\frac{w}{2T_0}x^2 + \beta$$

and so the cable follows a parabola. If the weight of the cable and supporting wires is not negligible then the analysis is more complex.

Catenary of equal strength
In a catenary of equal strength, the cable is strengthened according to the magnitude of the tension at each point, so its resistance to breaking is constant along its length. Assuming that the strength of the cable is proportional to its density per unit length, the weight, $wx$, per unit length of the chain can be written $x$, where $w$ is constant, and the analysis for nonuniform chains can be applied.

In this case the equations for tension are

$$\begin{align} T \cos \varphi &= T_0\,,\\ T \sin \varphi &= \frac{1}{c}\int T\, ds\,. \end{align}$$

Combining gives

$$c \tan \varphi = \int \sec \varphi\, ds$$

and by differentiation

$$c = \rho \cos \varphi$$

where $T⁄c$ is the radius of curvature.

The solution to this is

$$y = c \ln\left(\sec\left(\frac{x}{c}\right)\right)\,.$$

In this case, the curve has vertical asymptotes and this limits the span to $T_{0} = wa$. Other relations are

$$x = c\varphi\,,\quad s = \ln\left(\tan\left(\frac{\pi+2\varphi}{4}\right)\right)\,.$$

The curve was studied 1826 by Davies Gilbert and, apparently independently, by Gaspard-Gustave Coriolis in 1836.

Recently, it was shown that this type of catenary could act as a building block of electromagnetic metasurface and was known as "catenary of equal phase gradient".

Elastic catenary
In an elastic catenary, the chain is replaced by a spring which can stretch in response to tension. The spring is assumed to stretch in accordance with Hooke's Law. Specifically, if $c$ is the natural length of a section of spring, then the length of the spring with tension $c$ applied has length

$$s=\left(1+\frac{T}{E}\right)p\,,$$

where $ρ$ is a constant equal to $T$, where $E$ is the stiffness of the spring. In the catenary the value of $kp$ is variable, but ratio remains valid at a local level, so $$\frac{ds}{dp}=1+\frac{T}{E}\,.$$ The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.

The equations for tension of the spring are

$$T \cos \varphi = T_0\,,$$ and $$T \sin \varphi = w_0 p\,,$$

from which

$$\frac{dy}{dx}=\tan \varphi = \frac{w_0 p}{T_0}\,,\quad T=\sqrt{T_0^2+w_0^2 p^2}\,,$$

where $k$ is the natural length of the segment from $r$ to $c$ and $r$ is the weight per unit length of the spring with no tension. Write $$a = \frac{T_0}{w_0}$$ so $$\frac{dy}{dx}=\tan \varphi = \frac{p}{a} \quad\text{and}\quad T=\frac{T_0}{a}\sqrt{a^2+p^2}\,.$$

Then $$\begin{align} \frac{dx}{ds} &= \cos \varphi = \frac{T_0}{T} \\[6pt] \frac{dy}{ds} &= \sin \varphi = \frac{w_0 p}{T}\,, \end{align}$$ from which $$\begin{alignat}{3} \frac{dx}{dp} &= \frac{T_0}{T}\frac{ds}{dp} &&= T_0\left(\frac{1}{T}+\frac{1}{E}\right) &&= \frac{a}{\sqrt{a^2+p^2}}+\frac{T_0}{E} \\[6pt] \frac{dy}{dp} &= \frac{w_0 p}{T}\frac{ds}{dp} &&= \frac{T_0p}{a}\left(\frac{1}{T}+\frac{1}{E}\right) &&= \frac{p}{\sqrt{a^2+p^2}}+\frac{T_0p}{Ea}\,. \end{alignat}$$

Integrating gives the parametric equations

$$\begin{align} x&=a\operatorname{arsinh}\left(\frac{p}{a}\right)+\frac{T_0}{E}p + \alpha\,, \\[6pt] y&=\sqrt{a^2+p^2}+\frac{T_0}{2Ea}p^2+\beta\,. \end{align}$$

Again, the $T$ and $p$-axes can be shifted so $x$ and $y$ can be taken to be 0. So

$$\begin{align} x&=a\operatorname{arsinh}\left(\frac{p}{a}\right)+\frac{T_0}{E}p\,, \\[6pt] y&=\sqrt{a^2+p^2}+\frac{T_0}{2Ea}p^2 \end{align}$$

are parametric equations for the curve. At the rigid limit where $α$ is large, the shape of the curve reduces to that of a non-elastic chain.

Chain under a general force
With no assumptions being made regarding the force $c$ acting on the chain, the following analysis can be made.

First, let $r$ be the force of tension as a function of $β$. The chain is flexible so it can only exert a force parallel to itself. Since tension is defined as the force that the chain exerts on itself, $πc$ must be parallel to the chain. In other words,

$$\mathbf{T} = T \mathbf{u}\,,$$

where $E$ is the magnitude of $p$ and $c$ is the unit tangent vector.

Second, let $r$ be the external force per unit length acting on a small segment of a chain as a function of $s$. The forces acting on the segment of the chain between $T$ and $w_{0}$ are the force of tension $G$ at one end of the segment, the nearly opposite force $T = T(s)$ at the other end, and the external force acting on the segment which is approximately $T$. These forces must balance so

$$\mathbf{T}(s+\Delta s)-\mathbf{T}(s)+\mathbf{G}\Delta s \approx \mathbf{0}\,.$$

Divide by $T$ and take the limit as $u$ to obtain

$$\frac{d\mathbf{T}}{ds} + \mathbf{G} = \mathbf{0}\,.$$

These equations can be used as the starting point in the analysis of a flexible chain acting under any external force. In the case of the standard catenary, $G = G(s)$ where the chain has weight $s$ per unit length.

title: Catenary


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Color temperature

Color temperature is a parameter describing the color of a visible light source by comparing it to the color of light emitted by an idealized opaque, non-reflective body. The temperature of the ideal emitter that matches the color most closely is defined as the color temperature of the original visible light source. The color temperature scale describes only the color of light emitted by a light source, which may actually be at a different (and often much lower) temperature.

Color temperature has applications in lighting, photography, videography, publishing, manufacturing, astrophysics, and other fields. In practice, color temperature is most meaningful for light sources that correspond somewhat closely to the color of some black body, i.e., light in a range going from red to orange to yellow to white to bluish white. Although the concept of correlated color temperature extends the definition to any visible light, the color temperature of a green or a purple light rarely is useful information. Color temperature is conventionally expressed in kelvins, using the symbol K, a unit for absolute temperature.

Color temperatures over 5000 K are called "cool colors" (bluish), while lower color temperatures (2700–3000 K) are called "warm colors" (yellowish). "Warm" in this context is with respect to a traditional categorization of colors, not a reference to black body temperature. The hue-heat hypothesis states that low color temperatures will feel warmer while higher color temperatures will feel cooler. The spectral peak of warm-colored light is closer to infrared, and most natural warm-colored light sources emit significant infrared radiation. The fact that "warm" lighting in this sense actually has a "cooler" color temperature often leads to confusion.

Categorizing different lighting

The color temperature of the electromagnetic radiation emitted from an ideal black body is defined as its surface temperature in kelvins, or alternatively in micro reciprocal degrees (mired). This permits the definition of a standard by which light sources are compared.

To the extent that a hot surface emits thermal radiation but is not an ideal black-body radiator, the color temperature of the light is not the actual temperature of the surface. An incandescent lamp's light is thermal radiation, and the bulb approximates an ideal black-body radiator, so its color temperature is essentially the temperature of the filament. Thus a relatively low temperature emits a dull red and a high temperature emits the almost white of the traditional incandescent light bulb. Metal workers are able to judge the temperature of hot metals by their color, from dark red to orange-white and then white (see red heat).

Many other light sources, such as fluorescent lamps, or light emitting diodes (LEDs) emit light primarily by processes other than thermal radiation. This means that the emitted radiation does not follow the form of a black-body spectrum. These sources are assigned what is known as a correlated color temperature (CCT). CCT is the color temperature of a black-body radiator which to human color perception most closely matches the light from the lamp. Because such an approximation is not required for incandescent light, the CCT for an incandescent light is simply its unadjusted temperature, derived from comparison to a black-body radiator.

The Sun
The Sun closely approximates a black-body radiator. The effective temperature, defined by the total radiative power per square unit, is 5772 K. The color temperature of sunlight above the atmosphere is about 5900 K.

The Sun may appear red, orange, yellow, or white from Earth, depending on its position in the sky. The changing color of the Sun over the course of the day is mainly a result of the scattering of sunlight and is not due to changes in black-body radiation. Rayleigh scattering of sunlight by Earth's atmosphere causes the blue color of the sky, which tends to scatter blue light more than red light.

Some daylight in the early morning and late afternoon (the golden hours) has a lower ("warmer") color temperature due to increased scattering of shorter-wavelength sunlight by atmospheric particulates – an optical phenomenon called the Tyndall effect.

Daylight has a spectrum similar to that of a black body with a correlated color temperature of 6500 K (D65 viewing standard) or 5500 K (daylight-balanced photographic film standard).

For colors based on black-body theory, blue occurs at higher temperatures, whereas red occurs at lower temperatures. This is the opposite of the cultural associations attributed to colors, in which "red" is "hot", and "blue" is "cold".

Lighting

For lighting building interiors, it is often important to take into account the color temperature of illumination. A warmer (i.e., a lower color temperature) light is often used in public areas to promote relaxation, while a cooler (higher color temperature) light is used to enhance concentration, for example in schools and offices.

CCT dimming for LED technology is regarded as a difficult task, since binning, age and temperature drift effects of LEDs change the actual color value output. Here feedback loop systems are used, for example with color sensors, to actively monitor and control the color output of multiple color mixing LEDs.

Aquaculture
In fishkeeping, color temperature has different functions and foci in the various branches.

 * In freshwater aquaria, color temperature is generally of concern only for producing a more attractive display. Lights tend to be designed to produce an attractive spectrum, sometimes with secondary attention paid to keeping the plants in the aquaria alive.
 * In a saltwater/reef aquarium, color temperature is an essential part of tank health. Within about 400 to 3000 nanometers, light of shorter wavelength can penetrate deeper into water than longer wavelengths,   providing essential energy sources to the algae hosted in (and sustaining) coral. This is equivalent to an increase of color temperature with water depth in this spectral range. Because coral typically live in shallow water and receive intense, direct tropical sunlight, the focus was once on simulating this situation with 6500 K lights.

Digital photography
In digital photography, the term color temperature sometimes refers to remapping of color values to simulate variations in ambient color temperature. Most digital cameras and raw image software provide presets simulating specific ambient values (e.g., sunny, cloudy, tungsten, etc.) while others allow explicit entry of white balance values in kelvins. These settings vary color values along the blue–yellow axis, while some software includes additional controls (sometimes labeled "tint") adding the magenta–green axis, and are to some extent arbitrary and a matter of artistic interpretation.

Photographic film
Photographic emulsion film does not respond to lighting color identically to the human retina or visual perception. An object that appears to the observer to be white may turn out to be very blue or orange in a photograph. The color balance may need to be corrected during printing to achieve a neutral color print. The extent of this correction is limited since color film normally has three layers sensitive to different colors and when used under the "wrong" light source, every layer may not respond proportionally, giving odd color casts in the shadows, although the mid-tones may have been correctly white-balanced under the enlarger. Light sources with discontinuous spectra, such as fluorescent tubes, cannot be fully corrected in printing either, since one of the layers may barely have recorded an image at all.

Photographic film is made for specific light sources (most commonly daylight film and tungsten film), and, used properly, will create a neutral color print. Matching the sensitivity of the film to the color temperature of the light source is one way to balance color. If tungsten film is used indoors with incandescent lamps, the yellowish-orange light of the tungsten incandescent lamps will appear as white (3200 K) in the photograph. Color negative film is almost always daylight-balanced, since it is assumed that color can be adjusted in printing (with limitations, see above). Color transparency film, being the final artefact in the process, has to be matched to the light source or filters must be used to correct color.

Filters on a camera lens, or color gels over the light source(s) may be used to correct color balance. When shooting with a bluish light (high color temperature) source such as on an overcast day, in the shade, in window light, or if using tungsten film with white or blue light, a yellowish-orange filter will correct this. For shooting with daylight film (calibrated to 5600 K) under warmer (low color temperature) light sources such as sunsets, candlelight or tungsten lighting, a bluish (e.g. #80A) filter may be used. More-subtle filters are needed to correct for the difference between, say 3200 K and 3400 K tungsten lamps or to correct for the slightly blue cast of some flash tubes, which may be 6000 K.

If there is more than one light source with varied color temperatures, one way to balance the color is to use daylight film and place color-correcting gel filters over each light source.

Photographers sometimes use color temperature meters. These are usually designed to read only two regions along the visible spectrum (red and blue); more expensive ones read three regions (red, green, and blue). However, they are ineffective with sources such as fluorescent or discharge lamps, whose light varies in color and may be harder to correct for. Because this light is often greenish, a magenta filter may correct it. More sophisticated colorimetry tools can be used if such meters are lacking.

Desktop publishing
In the desktop publishing industry, it is important to know a monitor's color temperature. Color matching software, such as Apple's ColorSync Utility for MacOS, measures a monitor's color temperature and then adjusts its settings accordingly. This enables on-screen color to more closely match printed color. Common monitor color temperatures, along with matching standard illuminants in parentheses, are as follows:

 * 5000 K (CIE D50)
 * 5500 K (CIE D55)
 * 6500 K (D65)
 * 7500 K (CIE D75)
 * 9300 K

D50 is scientific shorthand for a standard illuminant: the daylight spectrum at a correlated color temperature of 5000 K. Similar definitions exist for D55, D65 and D75. Designations such as D50 are used to help classify color temperatures of light tables and viewing booths. When viewing a color slide at a light table, it is important that the light be balanced properly so that the colors are not shifted towards the red or blue.

Digital cameras, web graphics, DVDs, etc., are normally designed for a 6500 K color temperature. The sRGB standard commonly used for images on the Internet stipulates a 6500 K display white point.

TV, video, and digital still cameras
The NTSC and PAL TV norms call for a compliant TV screen to display an electrically black and white signal (minimal color saturation) at a color temperature of 6500 K. On many consumer-grade televisions, there is a very noticeable deviation from this requirement. However, higher-end consumer-grade televisions can have their color temperatures adjusted to 6500 K by using a preprogrammed setting or a custom calibration. Current versions of ATSC explicitly call for the color temperature data to be included in the data stream, but old versions of ATSC allowed this data to be omitted. In this case, current versions of ATSC cite default colorimetry standards depending on the format. Both of the cited standards specify a 6500 K color temperature.

Most video and digital still cameras can adjust for color temperature by zooming into a white or neutral colored object and setting the manual "white balance" (telling the camera that "this object is white"); the camera then shows true white as white and adjusts all the other colors accordingly. White-balancing is necessary especially when indoors under fluorescent lighting and when moving the camera from one lighting situation to another. Most cameras also have an automatic white balance function that attempts to determine the color of the light and correct accordingly. While these settings were once unreliable, they are much improved in today's digital cameras and produce an accurate white balance in a wide variety of lighting situations.

Artistic application via control of color temperature

Video camera operators can white-balance objects that are not white, downplaying the color of the object used for white-balancing. For instance, they can bring more warmth into a picture by white-balancing off something that is light blue, such as faded blue denim; in this way white-balancing can replace a filter or lighting gel when those are not available.

Cinematographers do not "white balance" in the same way as video camera operators; they use techniques such as filters, choice of film stock, pre-flashing, and, after shooting, color grading, both by exposure at the labs and also digitally. Cinematographers also work closely with set designers and lighting crews to achieve the desired color effects.

For artists, most pigments and papers have a cool or warm cast, as the human eye can detect even a minute amount of saturation. Gray mixed with yellow, orange, or red is a "warm gray". Green, blue, or purple create "cool grays". Note that this sense of temperature is the reverse of that of real temperature; bluer is described as "cooler" even though it corresponds to a higher-temperature black body.

Lighting designers sometimes select filters by color temperature, commonly to match light that is theoretically white. Since fixtures using discharge type lamps produce a light of a considerably higher color temperature than do tungsten lamps, using the two in conjunction could potentially produce a stark contrast, so sometimes fixtures with HID lamps, commonly producing light of 6000–7000 K, are fitted with 3200 K filters to emulate tungsten light. Fixtures with color mixing features or with multiple colors (if including 3200 K), are also capable of producing tungsten-like light. Color temperature may also be a factor when selecting lamps, since each is likely to have a different color temperature.

Color rendering index
The CIE color rendering index (CRI) is a method to determine how well a light source's illumination of eight sample patches compares to the illumination provided by a reference source. Cited together, the CRI and CCT give a numerical estimate of what reference (ideal) light source best approximates a particular artificial light, and what the difference is.

Spectral power distribution

Light sources and illuminants may be characterized by their spectral power distribution (SPD). The relative SPD curves provided by many manufacturers may have been produced using 10 nm increments or more on their spectroradiometer. The result is what would seem to be a smoother ("fuller spectrum") power distribution than the lamp actually has. Owing to their spiky distribution, much finer increments are advisable for taking measurements of fluorescent lights, and this requires more expensive equipment.

Color temperature in astronomy
In astronomy, the color temperature is defined by the local slope of the SPD at a given wavelength, or, in practice, a wavelength range. Given, for example, the color magnitudes B and V which are calibrated to be equal for an A0V star (e.g. Vega), the stellar color temperature $$T_C$$ is given by the temperature for which the color index $$B-V$$ of a black-body radiator fits the stellar one. Besides the $$B-V$$, other color indices can be used as well. The color temperature (as well as the correlated color temperature defined above) may differ largely from the effective temperature given by the radiative flux of the stellar surface. For example, the color temperature of an A0V star is about 15000 K compared to an effective temperature of about 9500 K.

For most applications in astronomy (e.g., to place a star on the HR diagram or to determine the temperature of a model flux fitting an observed spectrum) the effective temperature is the quantity of interest. Various color-effective temperature relations exist in the literature. There relations also have smaller dependencies on other stellar parameters, such as the stellar metallicity and surface gravity

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

title: Color temperature


text:
Cartoon

A cartoon is a type of visual art that is typically drawn, frequently animated, in an unrealistic or semi-realistic style. The specific meaning has evolved, but the modern usage usually refers to either: an image or series of images intended for satire, caricature, or humor; or a motion picture that relies on a sequence of illustrations for its animation. Someone who creates cartoons in the first sense is called a cartoonist, and in the second sense they are usually called an animator.

The concept originated in the Middle Ages, and first described a preparatory drawing for a piece of art, such as a painting, fresco, tapestry, or stained glass window. In the 19th century, beginning in Punch magazine in 1843, cartoon came to refer – ironically at first – to humorous artworks in magazines and newspapers. Then it also was used for political cartoons and comic strips. When the medium developed, in the early 20th century, it began to refer to animated films that resembled print cartoons.

Fine art
A cartoon (from cartone and karton—words describing strong, heavy paper or pasteboard) is a full-size drawing made on sturdy paper as a design or modello for a painting, stained glass, or tapestry. Cartoons were typically used in the production of frescoes, to accurately link the component parts of the composition when painted on damp plaster over a series of days (giornate). In media such as stained tapestry or stained glass, the cartoon was handed over by the artist to the skilled craftsmen who produced the final work.

Such cartoons often have pinpricks along the outlines of the design so that a bag of soot patted or "pounced" over a cartoon, held against the wall, would leave black dots on the plaster ("pouncing"). Cartoons by painters, such as the Raphael Cartoons in London, Francisco Goya's tapestry cartoons, and examples by Leonardo da Vinci, are highly prized in their own right. Tapestry cartoons, usually colored, could be placed behind the loom, where the weaver would replicate the design. As tapestries are worked from behind, a mirror could be placed behind the loom to allow the weaver to see their work; in such cases the cartoon was placed behind the weaver.

Mass media

In print media, a cartoon is a drawing or series of drawings, usually humorous in intent. This usage dates from 1843, when Punch magazine applied the term to satirical drawings in its pages, particularly sketches by John Leech. The first of these parodied the preparatory cartoons for grand historical frescoes in the then-new Palace of Westminster in London.

Sir John Tenniel—illustrator of Alice's Adventures in Wonderland—joined Punch in 1850, and over 50 years contributed over two thousand cartoons.

Cartoons can be divided into gag cartoons, which include editorial cartoons, and comic strips.

Modern single-panel gag cartoons, found in magazines, generally consist of a single drawing with a typeset caption positioned beneath, or, less often, a speech balloon. Newspaper syndicates have also distributed single-panel gag cartoons by Mel Calman, Bill Holman, Gary Larson, George Lichty, Fred Neher and others. Many consider New Yorker cartoonist Peter Arno the father of the modern gag cartoon (as did Arno himself). The roster of magazine gag cartoonists includes Charles Addams, Charles Barsotti, and Chon Day.

Bill Hoest, Jerry Marcus, and Virgil Partch began as magazine gag cartoonists and moved to syndicated comic strips. Richard Thompson illustrated numerous feature articles in The Washington Post before creating his Cul de Sac comic strip. The sports section of newspapers usually featured cartoons, sometimes including syndicated features such as Chester "Chet" Brown's All in Sport.

Editorial cartoons are found almost exclusively in news publications and news websites. Although they also employ humor, they are more serious in tone, commonly using irony or satire. The art usually acts as a visual metaphor to illustrate a point of view on current social or political topics. Editorial cartoons often include speech balloons and sometimes use multiple panels. Editorial cartoonists of note include Herblock, David Low, Jeff MacNelly, Mike Peters, and Gerald Scarfe.

Comic strips, also known as cartoon strips in the United Kingdom, are found daily in newspapers worldwide, and are usually a short series of cartoon illustrations in sequence. In the United States, they are not commonly called "cartoons" themselves, but rather "comics" or "funnies". Nonetheless, the creators of comic strips—as well as comic books and graphic novels—are usually referred to as "cartoonists". Although humor is the most prevalent subject matter, adventure and drama are also represented in this medium. Some noteworthy cartoonists of humorous comic strips are Scott Adams, Charles Schulz, E. C. Segar, Mort Walker and Bill Watterson.

Political
Political cartoons are like illustrated editorials that serve visual commentaries on political events. They offer subtle criticism which are cleverly quoted with humour and satire to the extent that the criticized does not get embittered.

The pictorial satire of William Hogarth is regarded as a precursor to the development of political cartoons in 18th century England. George Townshend produced some of the first overtly political cartoons and caricatures in the 1750s. The medium began to develop in the latter part of the 18th century under the direction of its great exponents, James Gillray and Thomas Rowlandson, both from London. Gillray explored the use of the medium for lampooning and caricature, and has been referred to as the father of the political cartoon. By calling the king, prime ministers and generals to account for their behaviour, many of Gillray's satires were directed against George III, depicting him as a pretentious buffoon, while the bulk of his work was dedicated to ridiculing the ambitions of revolutionary France and Napoleon. George Cruikshank became the leading cartoonist in the period following Gillray, from 1815 until the 1840s. His career was renowned for his social caricatures of English life for popular publications.

By the mid 19th century, major political newspapers in many other countries featured cartoons commenting on the politics of the day. Thomas Nast, in New York City, showed how realistic German drawing techniques could redefine American cartooning. His 160 cartoons relentlessly pursued the criminal characteristic of the Tweed machine in New York City, and helped bring it down. Indeed, Tweed was arrested in Spain when police identified him from Nast's cartoons. In Britain, Sir John Tenniel was the toast of London. In France under the July Monarchy, Honoré Daumier took up the new genre of political and social caricature, most famously lampooning the rotund King Louis Philippe.

Political cartoons can be humorous or satirical, sometimes with piercing effect. The target of the humor may complain, but can seldom fight back. Lawsuits have been very rare; the first successful lawsuit against a cartoonist in over a century in Britain came in 1921, when J. H. Thomas, the leader of the National Union of Railwaymen (NUR), initiated libel proceedings against the magazine of the British Communist Party. Thomas claimed defamation in the form of cartoons and words depicting the events of "Black Friday", when he allegedly betrayed the locked-out Miners' Federation. To Thomas, the framing of his image by the far left threatened to grievously degrade his character in the popular imagination. Soviet-inspired communism was a new element in European politics, and cartoonists unrestrained by tradition tested the boundaries of libel law. Thomas won the lawsuit and restored his reputation.

Scientific
Cartoons such as xkcd have also found their place in the world of science, mathematics, and technology. For example, the cartoon Wonderlab looked at daily life in the chemistry lab. In the U.S., one well-known cartoonist for these fields is Sidney Harris. Many of Gary Larson's cartoons have a scientific flavor.

Comic books
The first comic-strip cartoons were of a humorous tone. Notable early humor comics include the Swiss comic-strip book Mr. Vieux Bois (1837), the British strip Ally Sloper (first appearing in 1867) and the American strip Yellow Kid (first appearing in 1895).

In the United States in the 1930s, books with cartoons were magazine-format "American comic books" with original material, or occasionally reprints of newspaper comic strips.

In Britain in the 1930s, adventure comic magazines became quite popular, especially those published by DC Thomson; the publisher sent observers around the country to talk to boys and learn what they wanted to read about. The story line in magazines, comic books and cinema that most appealed to boys was the glamorous heroism of British soldiers fighting wars that were exciting and just. DC Thomson issued the first The Dandy Comic in December 1937. It had a revolutionary design that broke away from the usual children's comics that were published broadsheet in size and not very colourful. Thomson capitalized on its success with a similar product The Beano in 1938.

On some occasions, new gag cartoons have been created for book publication.

Animation

Because of the stylistic similarities between comic strips and early animated films, cartoon came to refer to animation, and the word cartoon is currently used in reference to both animated cartoons and gag cartoons. While animation designates any style of illustrated images seen in rapid succession to give the impression of movement, the word "cartoon" is most often used as a descriptor for television programs and short films aimed at children, possibly featuring anthropomorphized animals, superheroes, the adventures of child protagonists or related themes.

In the 1980s, cartoon was shortened to toon, referring to characters in animated productions. This term was popularized in 1988 by the combined live-action/animated film Who Framed Roger Rabbit, followed in 1990 by the animated TV series Tiny Toon Adventures.

title: Cartoon


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Chief Minister of the Northern Territory

The chief minister of the Northern Territory is the head of government of the Northern Territory. The office is the equivalent of a state premier. When the Northern Territory Legislative Assembly was created in 1974, the head of government was officially known as majority leader. This title was used in the first parliament (1974–1977) and the first eighteen months of the second. When self-government was granted the Northern Territory in 1978, the title of the head of government became chief minister.

The chief minister is formally appointed by the administrator, who in normal circumstances will appoint the head of whichever party holds the majority of seats in the unicameral Legislative Assembly. In times of constitutional crisis, the administrator can appoint someone else as chief minister, though this has never occurred.

Since 21 December 2023, following the resignation of Natasha Fyles, the chief minister is Eva Lawler of the Labor Party. She is the third female chief minister of the Northern Territory.

History
The Country Liberal Party won the first Northern Territory election on 19 October 1974 and elected Goff Letts majority leader. He headed an Executive that carried out most of the functions of a ministry at the state level. At the 1977 election Letts lost his seat and party leadership. He was succeeded on 13 August 1977 by Paul Everingham (CLP) as Majority Leader. When the Territory attained self-government on 1 July 1978, Everingham became chief minister with greatly expanded powers.

In 2001, Clare Martin became the first Labor and female chief minister of the Northern Territory. Until 2004 the conduct of elections and drawing of electoral boundaries was performed by the Northern Territory Electoral Office, a unit of the Department of the chief minister. In March 2004 the independent Northern Territory Electoral Commission was established.

In 2013, Mills was replaced as chief minister and CLP leader by Adam Giles at the 2013 CLP leadership ballot on 13 March to become the first indigenous Australian to lead a state or territory government in Australia.

Following the 2016 election landslide outcome, Labor's Michael Gunner became chief minister; he was the first Chief Minister who was born in the Northern Territory. On 10 May 2022, Gunner announced his intention to resign. On 13 May 2022, Natasha Fyles was elected to the position by the Labor caucus. On 19 December 2023, Fyles resigned following controversy over undeclared shares in mining company South32. On 21 December 2023, Eva Lawler replaced Fyles by a unanimous decision of the Labor caucus.

List of chief ministers of the Northern Territory
From the foundation of the Northern Territory Legislative Assembly in 1974 until the granting of self-government in 1978, the head of government was known as the majority leader:

From 1978, the position was known as the chief minister:


text:
Talk:Color temperature

Applications
The applications list in the lead is currently

 * lighting, photography, videography, publishing, manufacturing, astrophysics and other fields

I dropped horticulture, because it's really an abuse of the concept--color temperature is specifically with respect to the human eye, and photosynethesis has a different spectral response.

I have my doubts about the manufacturing and astrophysics items in the list. Certainly, a manufacturing plant needs lighting, but so do lots of other spaces and they all fall under lighting. And if you are manufacturing something that is supposed to be a particular color, you need to make sure you look at it under a specified light source(s), but color temperature isn't an adequate spec for that.

[Incorrect comments about astrophysics deleted; see new comment below]

For the rest, of the applications, it would be cool to link to where color temperature is discussed in articles about each. But that could be a project for another day. Ccrrccrr (talk) 23:22, 14 April 2023 (UTC)

 * My initial comment on astrophysics was ill informed. What I should have said is that the parameter defined as color temperature in astrophysics is a little different from the definition generally discussed in this article.  I will edit it to make that clearer in the lead when I get a chance. Ccrrccrr (talk) 10:24, 15 April 2023 (UTC)

Also for temperature measurement. North8000 (talk) 00:02, 15 April 2023 (UTC)

I'm afraid I don't agree that it is "really an abuse of the concept" to discuss the horticultural applications. I actually looked to this article to learn something pertinent to my job, only to find that even a hint of such uses has been deliberately removed, based on what seems to me to be classic WP:OR. I was, at the time, at work holding a very expensive grow light bulb in my hand, that listed and also graphed the color temrature of the unit on the packaging, and I wanted to learn more as I am taking over supervising lighting systems at the cultivation facility I work at, and we're trying to figure out why previous persons in charge of these things made the decisions they did. The article was indeed helpful to me and I plan to use some of the images in it to make a presentation to the owners of the company, but I have to believe there must be some sources that discuss the uses of color temperature in grow lights. While plants obviously don't have eyes, the quality of the light you provide to them in different stages of their life is critically important to the final product in terms of vigor and yield. Beeblebrox (talk) 04:38, 7 August 2023 (UTC)

 * @Beeblebrox Absolutely, the quality of the light that you provide them is important for lots of reasons. The light spectrum is important in ways that are not captured by the single number color temperature.  when you discuss something being graphed, it sounds like you are talking about the spectrum, not the color temperature. Ccrrccrr (talk) 05:01, 8 August 2023 (UTC)
 * I think the most productive approach to this would be to remember that wikipedia standards state that the lede should summarise the main article. For any application to be mentioned in the lede, it should contain cited text in main article. This is the best and clearest way to decide what applications should appear in this list. So, we need to set about finding sources for all these applications. WP:OR and WP:RS are fundamental principles of wikipedia. Ashmoo (talk) 13:34, 22 November 2023 (UTC)

Long term lack of sources
This article contains some useful information, but sadly contains a large amount of text that has had "citation needed" tags for 11 years. We really need find sources for this text or remove the text from the article (no matter how reasonable or accurate it seems). Ashmoo (talk) 13:36, 22 November 2023 (UTC)

White point = 6500K?
White point is assumed as 6500K in both scale picture and Template:Color temperature, but 6500K is a really obviously bluish light in fact. I think 5500K will be a more accurate white point. For example, my all screens are obviously less bluish than 6500K lamps but also less warm than 4000K lamps, i don't have a 5000K/5500K lamp to compare but i think they are about 5500K. Can anyone update this image and template to make the white point 5500K (or more preferably make the template allow custom white points, and even gammas)? (The lowest and highest values given must also be changed, and even the current lowest value is wrong (it does not give pure red). This discussion is more about the image and template but i think discussing it there can be make it more centralized.) RuzDD (talk) 13:18, 6 February 2024 (UTC)

 * Addition: Colors of File:CFBulbs.jpg looks very realistic to me with my screens, and i think most of LCD (and probably ?LED) screens have ~5500K white points like mine displays. RuzDD (talk) 13:26, 6 February 2024 (UTC)
 * The use of 6500K is very much intentional here. Due to color constancy, there is no "true white", and even standard illuminant E (equal energy white), which has a uniform spectrum, is not special when it comes to white points. Like the D-series standard illuminants on the daylight locus, like Illuminant D65, E is also not on the planckian locus, and is not directly defined by temperature, but only by correlated color temperature, where it is approximated by 5455K. However, most color science uses D65 (6504K) as the default illuminant, since it is generally under this illuminant that human color vision evolved (average midday light in Western Europe... so mind the eurocentricity), and therefore is expected to represent a true neutral illuminant, if such a thing can even be said to exist. You could almost say that if D65 looks blue, then you may have normalized to the color temp of (typically warmer) indoor lighting! Curran919 (talk) 20:23, 6 February 2024 (UTC)
 * Isn't D65 deliberately bluish because it includes illumination from the blue sky rather than being a definition of white? Where the sun alone in that condition is around 5700K? Sincerely, North8000 (talk) 20:47, 6 February 2024 (UTC)
 * @North8000 essentially, and when something is illuminated under daylight, that includes direct and scattered sunlight. You have to add direct sunlight and the blue sky light integrated over the sky's visible solid angle (ideally, a hemisphere) to get the daylight standard illuminant, which is estimated by D65. If you block out the scattered, indirect skylight at noon, you will get close to 5700K with direct sunlight alone (though I don't know if that changes after passing through the atmospheric window). At sunset, the direct sunlight passes through so much atmosphere, that it drops to 2000K, which is why so many people characterize the sun as being yellow instead of white, as it is (almost) by definition. That's why it's called your golden hour. Curran919 (talk) 22:43, 6 February 2024 (UTC)
 * I know the physics better than the related terminology and standards. The question is does daylight standard (illuminant) define "white"?  But I guess the more specific question need to start by exactly defining the proposed change.  Sincerely, North8000 (talk) 23:12, 6 February 2024 (UTC)
 * The temperature of sun is 5772K. I know this is for photosphere, and we need the temperature of it because we are seeing it, not the core of the sun. Due to rayleigh scattering, temperature of sunlight will be change for most parameters but the average temperature of the sunlight will be about 5772K. So, if we get sun as the reference, we must assume the whitepoint is about 5772K otherwise our value will not be average. RuzDD (talk) 20:56, 6 February 2024 (UTC)
 * I can see that (sunlight is warmer than 6500K indoor lighting even in noon) when i turn on a 6500K lamp at noon. RuzDD (talk) 20:59, 6 February 2024 (UTC)
 * @RuzDD CCT of ambient light changes significantly depending on latitude, season, time of day, portion of indirect to direct illumination (how much blue sky is seen) and atmospheric content. D65 is just an "average". Like most standards, it is to an extent arbitrary, but it is the standard now and that's why it's used when defining the graphics. Curran919 (talk) 22:33, 6 February 2024 (UTC)
 * If the average is D65, where is the warmer part of the light? If temperature of the sun is 5772K, the average must be 5772K otherwise this means sun gives warmer lights to some directions and colder lights to some directions.
 * When i copy the hexcode of 3000K and paste it to make my screen glow in that color under a 3000K light, the screen is looked more yellow than my white objects. When i copy the hexcode of 4000K and do the same thing, my screen and white objects looked more smilar (i also have a 4000K light source and it's obviously less yellow). So, the whitepoint of my screens are more near to 5500K than 6500K and the white of my screens are really white to me.. RuzDD (talk) 23:21, 6 February 2024 (UTC)
 * Let me try out an analogy that came to me as a shower thought. Trying to define "white" as a specific spectrum is like trying to define room temperature as a specific temperature. There is not one real answer, because what feels like the ideal neutral ambient temperature is going to change depending on circumstance. It can be useful to define it at 25*C to have a standard, but that doesn't mean it is a ground truth. Someone can easily come and say, "actually, I feel like room temp should be 24 or 26*C", which may feel more neutral to them given their local humidity, windchill, etc. as well as them subjectively due to their base body temperature... but if they try to write an exam by saying "I'm going to assume ambient temperature as 26*C instead of the standard ambient 25*C", its going to confused the prof and probably get marked wrong. Thats why we have standards. Not only is your perception of the white point highly dependent on all of your environmental conditions (as I explained), its also quite subjective, given the variance in color vision, such as the ratio of blue to green/red cones, which determines the blue-yellow color axis that is approximately parallel to the daylight locus near the white point. Of course, that cone ratio shouldn't matter much due to color constancy and chromatic adaptation (you haven't acknowledged these terms, which are critical to the understanding of this concept, so I don't know if I'm screaming into the void), but that's exactly the point, that white is... fluid... and we just need to agree on a standard, which the world has largely treated as defaut D65 for most color appearance models. Curran919 (talk) 15:55, 7 February 2024 (UTC)
 * I know those, but i i think there's a better way to define the ideal "room temperature". We find the average air temperature of the earth for all places and all times, and this will be a more good standard. Same way, i think getting the average sunlight temperature (average, in all places and all times, for example not only noon's) of the earth will give a more accurate average (and, physics says this must be same with the photosphere temperature of the sun otherwise sun must have very different photosphere temperatures in different positions). That's 5772K, and i think it will be better to round it to 5500K so the difference will be very rounded. RuzDD (talk) 16:12, 7 February 2024 (UTC)
 * @RuzDD well then you need to take it up with CIE, not Wikipedia! Curran919 (talk) 16:19, 7 February 2024 (UTC)
 * According to standard illuminant CIE already created many different whitepoints including D55. RuzDD (talk) 16:33, 7 February 2024 (UTC)
 * @RuzDD That article is a bit outdated. See the discussion on its talk page: Talk%3AStandard_illuminant Curran919 (talk) 22:06, 7 February 2024 (UTC)
 * D55 is not standard but it exists (if i understood it correctly), and i think that'll not require move this discussion to CIE (i think we don't have to watch the standards those are given by CIE in all circumstances). RuzDD (talk) 23:43, 7 February 2024 (UTC)
 * By the way, looks like [[File:Black_body_visible_spectrum.gif]] kinda accepts me because 5000K-5500K temperatures have a very flat curve. RuzDD (talk) 01:45, 8 February 2024 (UTC)
 * @RuzDD Maybe to reiterate how arbitrary white is and how equal energy white (based on the spectrum, related to "white" noise) is a completely separate concept from perceptual white, imagine the star Vega, which has an average photosphere temp of 9600K. The star looks blue to us now, but if we evolved on a planet around Vega, the most useful perceptual white point, that which would allow us to see the most colors on Vega would be 9600K and not equal energy white (~5405K). We'd likely evolve to have that as a "default" white point, and equal energy white would look very yellow to us. We probably wouldn't even call it equal energy "white" because by the time we invent light spectra measurement, the equal energy point would look nothing like the "white" we had been using as our term for a perceptually neutral color. Curran919 (talk) 10:48, 8 February 2024 (UTC)
 * To add a note, this only applies to an illuminant or self-luminous object. The definition of white for an object (surface color, in terms of reflectance spectrum) is still uniform, as you'd expect, regardless of where your eye evolved. Curran919 (talk) 11:04, 8 February 2024 (UTC)

Of course "white" is just a word with varying definitions. Maybe clear day noon sun is a good one. But I do agree that 5500 is closer to the overall norm than 6500. North8000 (talk) 15:28, 6 February 2024 (UTC)
 * Strike a part of my post. Maybe I'm wrong. North8000 (talk) 20:51, 6 February 2024 (UTC)

I think we need to focus this discussion. To do that, could you specify exactly what your proposed change is? Sincerely, North8000 (talk) 03:39, 7 February 2024 (UTC)

 * Mathematical equations those used in this template and for making that gradient photo. RuzDD (talk) 12:48, 7 February 2024 (UTC)
 * [[File:Color temperature black body 800-12200K.svg]]
 * There is a good description for that graphic you can read on its | wikicommons page that explains the derivation. I see nothing wrong with it. If you still disagree, you should post about it on the color wikiproject where there are smarter people than myself when it comes to color models. Curran919 (talk) 15:44, 7 February 2024 (UTC)
 * One problem with this picture is that the device-dependent lightness artifacts are misleading: 6500k is shown as significantly lighter than the colors further away because that's what an sRGB display can show the brightest, when the intention of the diagram is ostensibly to show the differences in color. It might work better to put a narrower strip on a black background and then at least roughly equalize the lightness across the gradient. Even better might be to limit it to, say, 3000–9000K CCT, and just show a circular spot of the same lightness for each example, against a black background, instead of a gradient. If these are all representing "white" then it's to some extent inherently misleading to show a smooth gradient at all. –jacobolus (t) 17:31, 7 February 2024 (UTC)
 * @Jacobolus I think gradient is better bacause it allows users to select much more colors, and i think narrower things will be less understandable. And, i think lightness operation is not needed right now but if it should be made i think it'll be better to doing it in a new image and looking at the blackbody curve to make brightness increasing instead of staying the same (for example, 8000K is lighter than 6500K, just like the incendescent bulbs does). I think this kelvin scale (1000-12000) is good and even not enough, i don't think narrower will be better. But i think we must make different images for different gammas (1.0, 1.2, 1.4, 1.6, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, etc). RuzDD (talk) 18:35, 7 February 2024 (UTC)
 * Addition: but i think putting colors on black will be better as you say. RuzDD (talk) 18:36, 7 February 2024 (UTC)
 * Any particular CCT does not involve any inherent intensity or perceived lightness. It only describes chromaticity. I have no idea what you are getting at with "different images for different gammas". The output device is always presumed to be an sRGB display with prescribed characteristics. –jacobolus (t) 18:41, 7 February 2024 (UTC)
 * Yes, but blackbodies involves and i think creating some grays is not a very great idea unless they are representating something (blackbody in my case). RuzDD (talk) 18:46, 7 February 2024 (UTC)
 * @Jacobolus the spectrum here appears to be more isoluminant, if that's what you mean. I see the benefit. I wonder if turning the white to gray would lead to confusion? I do like the divergent nature of the "emphasized" white, but maybe after this whole conversation of white not being fixed, deemphasizing it would convey that idea. Curran919 (talk) 00:43, 8 February 2024 (UTC)
 * This photo is a lot harder to understand for me (you didn't pinged me but i want to answer anyway). But, creating a photo with increasing brightness (higher kelvin = brighter color) may be good for some circumstances (like an incendescent bulb, both brightness and color temperature increases as you force it to work harder and harder). RuzDD (talk) 00:51, 8 February 2024 (UTC)
 * Note: this photo is even not very color balanced for me, for example yellow is significiantly brighter than white/gray for me. RuzDD (talk) 00:52, 8 February 2024 (UTC)
 * Yeah, something like that, but it ideally needs to have as much black or dark gray background as possible for the gradient to really read as something like "white". –jacobolus (t) 01:08, 8 February 2024 (UTC)

title: Talk:Color temperature