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Talk:Ray tracing (graphics)

Too specific for subject?
This page is perhaps too specific.
 * Ray tracing itself is not concerned with rendering, only ray-object intersections. Shading models (phong, etc) are not a part of ray tracing.
 * Most renderers now use a hybrid solution e.g a fast scan-line or REYES algorithm to "draw" the visible parts, and ray tracing to determine shadows, reflections, and refrations. They often even allow selection between ray tracing and faster but more primitive methods (e.g shadow/environment maps) on a per-light/material basis.
 * Ray tracing is even used in various radiosity implementations to determine the contribution of one patch on another. In this capacity it's simply used to determine visibility, much like light rays in a traditional ray tracer.

Thus I suggest you make a distinction between the core of "ray tracing" as an algorithm and the various ways in which it is used to render images. -- —Preceding unsigned comment added by Imroy (talk • contribs) 10:37, 26 September 2004

 * "Ray tracing itself is not concerned with rendering, only ray-object intersections. Shading models (phong, etc) are not a part of ray tracing."

 * Not sure I agree. Ray tracing is the process of rendering an image by tracing rays through a scene and seeing how the energy propegates. I think you are thinking about ray casting, not ray tracing.

 * Maybe the article itself would benefit from having sections going into depth with the various issues with ray tracing, eg. the light transportation equation, trajectory splitting (eg. shadow rays), the monte carlo solution to light transport, the various basic sampling methods and their variance and uniformity, and possible even a section describing quasi-monte carlo sampling. --Kolibri 13:17, 19 October 2005 (UTC)

 * I agree with Kolibri that having sections regarding issues of raytracing should have their own section within the article, but I think that particularly large portions (Monte Carlo) should have their own article. Right now, this article needs some formating; it is not very well organized. --Osmaker 19:21, 20 October 2005 (UTC)

Possible linkspam
Ad potential link spam in the article:

the ffconsultancy.com link in the "external links" section actually refers to a study that primarily deals *not* with raytracing, but rather with a strongly biased programming language comparison. Presumably should be removed as it is of no real relevance to raytracing. - Thomas Fischbacher -- —Preceding unsigned comment added by 86.135.98.42 (talk • contribs) 23:27, 27 October 2005
 * The link is to a page entirely about ray tracing, comparing equivalent ray tracers written in different languages. The page includes and links to further pages that contain detailed information about the construction of these programs and explanations of how they act as ray tracers. The page also acts as a language comparison specifically in the context of ray tracing. - Jon Harrop -- —Preceding unsigned comment added by 80.229.56.224 (talk • contribs) 22:36, 31 October 2005

More and more links seem to point to commercial applications rather than web pages or documents that people can look at to learn about ray-tracing. For example lately Arauna & Photon Studio. Would everybody agree to remove them ? Or separate links in two categories. One for papers/tutorials and another one for any link that points to a product or a company ? --83.112.48.158 (talk) 06:34, 23 June 2008 (UTC)

Agreed; unless it is particularly significant, I think that we should remove these commercial links; I've never heard of Arauna or Photon Studio before, despite being professionally involved in this field, I cant imagine that there could be any sort of consensus for viewing these as significant. I will remove. Cdecoro (talk) 08:15, 23 June 2008 (UTC)

I don't want to remove any links but it seems to me that a couple of them are not greatly contributing to the article namely, "A series of tutorials on implementing a ray tracer using C++", "Tutorial on implementing a ray tracer in PHP". The "chapters" for the first article are not working, and an example of very basic ray tracing in PHP is really not adding any value to the article. I would like to suggest they get removed. There's tons of webpages of people putting the code of their ray tracer online. Only professional links should be kept in this article. jeancolasp (talk) 19:21, 16 November 2014 (UTC)

Rewrite and formatting
I was researching Ray Tracing and immediately came to Wikipedia for a start to finesse my definition and realized this definition needed finessing. I reformatted and re-arranged the article today (11/18). It seems a little more linear now. I also agree that the links section needs bolstering - and is a more appropriate place for deeper explorations of the Ray Tracing topic (Monte Carlo, SIGGRAPH papers, etc.). Whoever put in all the stuff I edited seems to have included the most appropriate historical and technical balance for the definition. Thanks!kcdot -- —Preceding unsigned comment added by 12.7.137.83 (talk • contribs) 19:38, 18 November 2005

Ray coherence
Scanline algorithms and other algorithms use data coherence to share computations between pixels, while ray tracing normally starts the process anew, treating each eye ray separately.

Coherence is exploited in high performance parallel ray tracing, even if mostly to get better cache behaviour. Such a system typically shots a set of (hopefully) highly coherent rays together. As they have coherent memory access patterns, this greatly reduces memory bandwidth compared to shoting rays at random. This is done for example in the Ray Processing Unit. --Taw 12:53, 18 December 2005 (UTC)

Ray casting VS Ray Tracing
I think that the differences between ray casting and ray tracing are not made clear in this article. Ray casting can be thought of as a subclass of ray tracing. Both techniques cast rays from the eye into the scene through the pixel to intersect with objects/surfaces in the scene.

The general idea is that each surface is illuminated by lights and other surfaces. So the algorithm traces rays from the surface to other surfaces recursively. This is ray tracing.

Ray casting only considers the initial ray-object intersection. The color / shading of the pixel is dependant on the characteristics of the initial surface that the ray intersected with.

Please note that I am using the terms objects and surfaces interchangebly above. -- —Preceding unsigned comment added by 81.97.15.130 (talk • contribs) 00:30, 14 January 2006

Example
Is the example that was just added too simple to be of any use? There is a lot more going on in raytracing than just finding the point where a line intersects a sphere. BTW, please check the math. I think it was dead wrong as originally posted, since the author confused vector and scalar math. I think I've fixed it but a second opinion would be useful.--Srleffler 23:10, 4 February 2006 (UTC)

Sorry if I confused things - that's my first mathematical contribution here, and I'm not used to all the standards! The maths, however, was correct as it stood if one correctly interpreted dot product vs. multiplication - I've implemented it in a raytracer I wrote myself. Also, I know it was specific, and hence I was doubtful myself as to whether to add it - as I attmepted to make clear, it is only a example/taster of one of a particular algorithm used. "be brave" is the motto here - I just thought I'd give it a go. If you still dislike it, remove it. I was only trying to provide something of some interest, so if that was not the case, it would be better were it removed. -- Wrayal

 * No problem. I only asked because I wasn't sure. Nobody else seems to feel strongly about it, so it might as well stay. Yes, by all means be bold in editing! In retrospect, I agree the math was correct except for the omission of the sign for dot product. It confused the heck out of me, though, because there was nothing at the beginning to indicate that the quantities indicated were vectors.--Srleffler 04:14, 16 February 2006 (UTC)

I think the example is useful in the case of an optics application using a spherical lens. Anyway, I cleaned up the equations a bit. The boldface on the 'd' had been dropped halfway through, so I fixed that. I added a note saying it was vector notation. Also, I put the equation in quadratic form so it's easier to read and understand. Mikiemike 16:32, 8 November 2006 (UTC)

Does this equation really have more than one solution? Now this quadratic equation has solution(s):

 * $$t=\frac{-(2\mathbf{V}\cdot\mathbf{d})\pm\sqrt{(2\mathbf{V}\cdot\mathbf{d})^2-4(\mathbf{d}^2)(\mathbf{V}^2-r^2)}}{2(\mathbf{d}^2)}\ $$

Unless t (time) can be negative, there will be either one or no solutions at all. --Niks 11:34, 2 April 2007 (UTC)

It might be worth noting that 1) t isn't time, it's just a scalar distance (effectively); 2) you can have two positive solutions (V.d may be negative). Out of interest, should one root be positive, one negative, you would discard the latter, though in this instance you'd be inside the sphere. 131.111.245.246 20:56, 21 April 2007 (UTC) This example is just what I needed for a college assignment. Thanks guys/gals! Yes, the equation can have up to two solutions. If you intersect a line with a sphere you might get an "entry" and "exit" point. Yang (talk) 20:33, 4 March 2008 (UTC)

Added Simple Example of Algorithm
I have added an image and a short description describing the algorithm in process. It should make the pseudocode easier to understand. --Kolibri 10:57, 27 October 2006 (UTC)

Shouldn't the article be explicit that the example algorithm is pseudocode? Some people may actually think computer languages look like this. - 15:57, 18 January 2008 (UTC) —Preceding unsigned comment added by 137.99.115.235 (talk)

External trimmage
The section below is mercilessly removed. The article already has the "See also / Software" subsection of wikipedia articles about ray-soft. The idea of the deletion is simple: if a software is notable, make an article (survive AfD :-) and wikilink it ito the abovementioned section. Otherwise the corresponding ext link is spam. Good luck, `'mikka 02:24, 19 May 2007 (UTC)

Raytracing software

 * Blender
 * BRL-CAD
 * Bryce (software)
 * Indigo - an unbiased render engine
 * jawray - A portable raytracer written in C++
 * Kray - global illumination renderer
 * OpenRT - realtime raytracing library
 * Optis - Straylight and illumination software with full CAD integration (Solidworks and Catia V5)
 * OSLO - Lens design and optimization software; OSLO-EDU is a free download
 * PBRT - a Physically Based Raytracer
 * Pixie
 * POV-Ray
 * Radiance
 * Raster3D
 * Rayshade
 * RayTrace - open source C++ software
 * RealStorm Engine - a realtime raytracing engine
 * RPS Ray Tace - AccuRender Ray Trace for SketchUp
 * SSRT - C++ source code for a Monte-carlo pathtracer - written with ease of understanding in mind.
 * Sunflow - Written in Java (platform independent)
 * Tachyon
 * TracePro - Straylight and illumination software with a CAD-like interface
 * TropLux - Daylighting simulation
 * Yafray
 * Zemax - Well Known Commercial Software for Optics design
 * More ray tracing source code links

Should be renamed Optical Ray Tracing
This article should be renamed 'optical ray tracing', because ray tracing has, for decades, also been done at RF and much of the argument here would need to be modified for that. `---Aussienick

I agree. Ray tracing is used in many areas apart from the CG applications mentioned in the article; for example, I came here looking for ray tracing as related to acoustics. --Devnevyn 11:32, 1 November 2007 (UTC)

I also agree. Ray tracing is a technique used to determine the path a wave will take, given its frequency and wavenumber; as the above authors mention (RF & acoustics) it is applicable to science beyond just computer graphics; for example, I came in search of earth/planetary wave ray tracing. 07:22, 10 February 2008 (UTC) —Preceding unsigned comment added by 131.128.73.5 (talk)

 * Well, I'm pretty sure that this is the "most common" use. So the best thing to do would probably be to create Ray tracing (disambiguation), link it from the header, and point to the other uses there. Adam McCormick (talk) 13:17, 18 April 2008 (UTC)

Mention Raytracing is an Embarassingly Parallel Algo?
Not entirely sure where to put this, but I believe a mention should be made about the fact that the ray tracing algorithm is embarrassingly parallel - an example from that page, "In ray tracing, each pixel may be rendered independently. In computer animation, each frame may be rendered independently."

Anyone concur?

Smerity 09:30, 22 October 2007 (UTC)

raytrace is best thing KOSOVO —Preceding unsigned comment added by 80.80.161.147 (talk) 20:24, 27 February 2008 (UTC)

Possible misleading figure PathOfRays.svg
The figure PathOfRays.svg, and the associated text (below the code sample, beginning with "First, a ray is created at an eyepoint"), seems to imply that when a ray intersects a diffuse surface, that ray is reflected onto another diffuse surface, and so on, until it reaches a light source.

While this is correct for reflective, mirror-like surfaces, or transparent glassy surfaces, it is not true for *diffuse* surfaces. When a ray intersects a diffuse, non-reflective surface, the recursion stops, and shadow rays are projected only to light sources in order to compute the direct illumination term. The recursion does not continue. This is why ray-tracing cannot do indirect illumination, which is precisely what inter-reflections among diffuse surfaces would entail.

Note that while the descriptive text and the figure are incorrect, the pseudo-code excerpt is in fact correct. Note that it specifies "generate ... ray: recurse" only if the surface is either "reflective" or "transparent". If neither of these is true, the surface is diffuse and the recursion stops.

The statement that "For example if the light source emitted white light and the two diffuse surfaces were blue, then the resulting color of the pixel is blue" implies color blending from one surface to another, which is not true for ray-tracing (but is true for radiosity or other general illumination methods)

-- marciot

 * I've removed the offending text and replied on the user's talk. Adam McCormick (talk) 13:24, 18 April 2008 (UTC)

 * That is completely wrong. Global illumination with a ray tracer is possible, and works by recursing upon contact with a diffuse surface. This models the effect of light scattering off of diffuse surfaces, and it works great. Timrb (talk) 08:40, 22 May 2008 (UTC)

 * Yes, as the writer of that part, reflection from diffuse surfaces does indeed happen. See Diffuse reflection and Diffuse Shading. --Kolibri (talk) 12:05, 9 January 2010 (UTC)

Suggest separating into two articles
Ray tracing in computer graphics and ray tracing in other applications are similar in concept, but quite different topics. I suggest creating "Ray Tracing" and "Ray Tracing (physics)". Ray tracing in graphics is quite a deep topic, and I don't think it makes sense to group it with discussion about tracing radio waves through the ionosphere, etc.

The former of the two pages would focus on graphics applications, with a "if you are looking for..." thingie at the top pointing to the physics page. A quick google search for "ray trace" shows almost entirely graphics results, so it should probably be the main article. This would give the article more space to talk about some of the different variations and non-physical tricks that ray tracing algorithms use.

The "Ray Tracing (physics)" page would have a similar redirect, and would discuss purely scientific applications of ray tracing, like the part about radio waves & lens design. Ray tracing is also used to trace acoustic signals through the varying density of the ocean, and the physics page would be a good place to put this as well.

It seems like other people on this talk page agree that the two topics are somehow fighting with each other. Thoughts?

Timrb (talk) 09:11, 18 April 2008 (UTC)

 * I could agree with that, if physics is the major other use and you have some expertise in the matter feel free to be WP:BOLD and pull some content for physics. I'd be happy to help reorganize and clean up. Adam McCormick (talk) 13:20, 18 April 2008 (UTC)

 * The split was a great idea -- thanks for that! —Preceding unsigned comment added by 99.251.254.165 (talk) 18:50, 23 April 2008 (UTC)

Wording of main image caption
There seems to be some dispute over whether the main image uses diffuse interreflection and area light sources. I can tell you as the creator of the image and programmer of the renderer that it does, and this is documented in the image's description page. If you still don't believe me, I can render it again with global illumination turned off and you can see the difference for yourself. Timrb (talk) 08:40, 22 May 2008 (UTC)
 * I for one would like to see examples. One with neither diffuse interreflection nor area light sources, one with just the interreflection and one with just the area light sources. Would you be willing to create those so we can see the difference? It would probably improve the article to show what those things are (both here and in those articles) rather than just stating that they are used. Adam McCormick (talk) 22:52, 29 May 2008 (UTC)

 * Very well:

 * Alan has suggested adding this stuff to the article; if anyone wants to write it up, feel free. I'm planning on doing a major re-write for this whole article anyway, so I'll pass. Timrb (talk) 09:42, 16 June 2008 (UTC)

 * I second that a major rewrite is definitely needed; that would be quite admirable of you if you have the time! I would strongly suggest that you look at the German Wikipedia page on Raytracing (linked at the top).  The article is very well put together, and I think it correctly focuses more on the ray-tracing aspect as opposed to the more general physically-based rendering (which of course relies nearly exclusivly on ray tracing, but is certainly a broader term).  Cdecoro (talk) 04:00, 17 June 2008 (UTC)

 * Oh BTW, its quite clear to me that the main image is displaying diffuse color bleeding and area light sources; perhaps I'm just more sensitive since I look at this stuff all day, but it certainly doesn't require the pictures to justify it. If you are planning a major rewrite, perhaps you can start it on a user page and let us know about it here?  I'd be interested in giving feedback as you put it together. Cdecoro (talk) 04:03, 17 June 2008 (UTC)
 * What is really clear is that both this image and the cups and bottles one from Gilles Tran do not quite pertain to an article about pure raytracing as they involve global illumination algorithms as well. They are stunningly good looking no doubt, but pure raytracing should be a lot more flat. 189.27.20.223 (talk) 03:09, 11 October 2008 (UTC)

 * See User:Timrb/Ray tracing for a (heavily annotated) work in progress. Timrb (talk) 03:11, 21 June 2008 (UTC)

Page move broke everything
When this page was moved to "Ray tracing (graphics)" from just plain "Ray tracing", all the links to this page were suddenly broken, and now every article that was supposed to link to an article about ray tracing in graphics now points to a disambig page.

This Should Not Be and I think drives the point home that the (graphics) part of the title is not necessary and just messes things up, as anyone who wrote ray tracing would reasonably expect it to point to the graphics article. A quick google search for "ray trace" verifies that the vast majority of pages are about ray tracing in graphics. The Disambiguation page allows for a non-disambiguated title if there is a clear primary topic. If there are no objections, I would like to move this page back to its original title. Timrb (talk) 09:09, 22 May 2008 (UTC)

 * Ray tracing is a common technique in optics, and this is the primary and original use of this technique. Any physicist or optical engineer who writes "ray tracing" will expect it to link to the other article. The problem here is that the person who moved the page and created the disambiguation page should have gone through all of the links to the dab page and corrected them to point to the appropriate article. The correct solution here is not to move the article back, but to fix the links. On the other hand, dab pages are not supposed to have only two entries. Disambiguation with only two articles is handled via redirects and hattext. I'll fix it.--Srleffler (talk) 17:24, 29 May 2008 (UTC)
 * OK, done. Variations of Ray tracing link here. Ray tracer and Ray trace link to the physics article. Someone still should go through all of the links and ensure that they point to the correct article.--Srleffler (talk) 17:35, 29 May 2008 (UTC)

 * I don't agree with that, ray tracer is the name given to the vast majority of raytracing graphics renderers, and I think that this page is still the most likely thing being referenced when someone wikilinks ray tracer. Adam McCormick (talk) 19:36, 29 May 2008 (UTC)
 * I've gone through all the links to Ray tracing and only about 15 of the 110 or were used in a physics context so I've changed them to point to the graphics article directly. I think ray tracer is probably similar in that it is used in more graphics articles than physics articles. In fact, of the six links, none are physics-based, so I'm going to point it here not there. Adam McCormick (talk) 20:24, 29 May 2008 (UTC)
 * OK. Thanks for checking and updating links.--Srleffler (talk) 22:25, 29 May 2008 (UTC)

 * The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the proposal was Do not move. Looks like someone is messing with the target redirect now. Everyone play nice please. —Wknight94 (talk) 17:19, 11 June 2008 (UTC)

Requested move
Ray tracing redirects here. As this is the primary topic, there is no need to add (graphics) to the article name. 199.125.109.107 (talk) 06:01, 30 May 2008 (UTC)

 * Oppose – The classical and most broadly used meaning is the general meaning that is covered in ray tracing (physics). The use of ray tracing in graphics is just one currently hot application of that general principal, and should not be the "main" interpretation.  The fact that ray tracing was recently changed to be a redirect to here has now been undone by me; it is again a redirect to the disambig page.  Dicklyon (talk) 06:16, 30 May 2008 (UTC)
 * On the other hand, I hadn't read all the above discussion and the history before commenting. If it's true that this article had the name Ray Tracing before, and it was hijacked without discussion, then I'd be OK with fixing it back to the way it was. I don't, however, buy srleffler's logic that a dab page needs more than two entries; I don't find support for that at WP:MOSDAB. Dicklyon (talk) 06:21, 30 May 2008 (UTC)
 * You're right. The MOS recommends using hatnotes instead of a two-item dab page, but does not forbid the latter. My mistake. The article wasn't hijacked without discussion. See . What happened was the physics and graphics content in the original raytracing article were split out into two separate articles. The problem is that the editor who did that did not go through and fix all the links to the page. Most of the physics links were subsequently fixed to point to the physics article, but many graphics-related links still point to pages that now redirect to the dab page. Someone should go through and fix all those links.--Srleffler (talk) 08:13, 11 June 2008 (UTC)
 * That's not quite right. I was the one who separated the two articles, into "ray tracing" and "ray tracing (physics)". I did go through all the links and fix everything up properly. Then penubag came along and made a disambig page, and redirected "ray tracing" toward it, and did so without any discussion. That was the move that broke everything. If you ask me, the hatnotes were working just fine, the disambig page is redundant step, and the (graphics) suffix isn't needed. Timrb (talk) 12:15, 11 June 2008 (UTC)

Oppose Ray tracing has a long history in fields other than CG; Gauss was tracing rays in his studies of optics long before computers. The title should stay as it is. Cdecoro (talk) 17:26, 10 June 2008 (UTC)

Does anyone know how this move happened, even though it had only opposition here? Dicklyon (talk) 03:25, 11 June 2008 (UTC)

 * Never mind, I see what happened now. It's good. Dicklyon (talk) 03:32, 11 June 2008 (UTC)

Oppose. Independent of frequency of use, the physics sense of the term is primary. The graphics technique is an application of physics-based ray tracing.--Srleffler (talk) 08:13, 11 June 2008 (UTC)
 * The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Unfocused Article
This article really doesn't know what its focus is on; I agree with one of the first posters: is it about visibility determination along a ray, specifically expressed as ray-object intersection? Or is it about rendering algorithms that use ray tracing as a component, which is basically equivalent to saying ALL physically-based rendering algorithms (even radiosity algorithms generally use ray tracing to compute form factors, and frequently use ray tracing to perform the final image synthesis to pixels). I suggest that this article be primarily focused on the former (visibility determination by intersecting rays) and specific articles be made to cover the rendering algorithms that use it. Cdecoro (talk) 17:26, 10 June 2008 (UTC)

Cdecoro's section rewrite
User:Cdecoro says "Replaced incorrect and misleading information on radiosity, photon mapping, and MLT with a (hopefully!) clearer and more informed version." But how can we tell if it's less misleading, clearer, or more informed, if you don't tell us what it's informed by? To encourage the citation of a source or two, I think I'll just revert it for now. Dicklyon (talk) 17:32, 10 June 2008 (UTC)

I've now added additional references to the original papers on bidir path tracing and photon mapping; in conjunction with the references by Timrb, this is sufficient. Cdecoro (talk) 21:41, 11 June 2008 (UTC)

 * I converted the first two to proper refs using Template:cite journal and Template:cite web. Take a look at those for what other parameters are available, and do a few more; let me know here if you need help. Dicklyon (talk) 04:50, 12 June 2008 (UTC)

Complete re-write coming
I've got a work in progress here: User:Timrb/Ray tracing. Some comments on structure and content could be helpful. (In particular, I'm now wondering about all that lighting model stuff-- perhaps it belongs in an article of its own? On the other hand, it will make formally talking about things like path tracing a lot easier.)Timrb (talk) 12:18, 23 June 2008 (UTC)

Hey, I think its looking really good. I think the light modeling section is very good to have; if we're going to talk about ray tracing as the rendering algorithm, as opposed to merely a visibility determination method, than I think theres no better place than here. I would strongly suggest (for the Rendering equation article also) using $$\omega_i$$ and $$\omega_o$$ instead of w and w' for the exitant and incident directions, respectivly. It's now mostly the standard usage, and I think its a lot clearer, both in indicating that we are talking about a direction (with omega) and disambiguating the directions. The illustrations look really good btw, I just figured I'd mention that sooner or later someone is going to attach one of those "should be SVG" tags, though. Cdecoro (talk) 20:31, 23 June 2008 (UTC)

This new version is interesting and agree, illustrations are good. However I feel it is not justified to switch from the existing version to this one. I believe it would have been much better to improve the existing document which is already rather good. The main difference between the two articles is your section on illumination models. However illumination models are not specific to ray-tracing. It would be therefore, as suggested, a much better idea to move that content to a specific article on CG illumination models. As for the article itself. The two articles are still not focused enough on the concept of rays and how they can be used in a rendering program. We should included example such as shooting rays to compute indirect diffuse, shooting rays to simulate area lights, shooting rays to compute BSSRDF, etc. and have clearer references to topics in which rays can be used (importance sampling, advanced light transport models, etc). There's no note on 'acne' which is a typical rendering artifact of ray-tracers. A last note, citation list could be cleaned with at last proper citation of Appel & Whited's papers, I would suggest to clean the Software list, keep renderers' names which have been known to contribute to make this algorithm popular (MRay, POV, lately PRman) and remove the others (this is not an article on 3D rendering programs). I am unsure as why the External Links is different in the suggested article than the existing one (links can't be left to purely personal choices). Also the pseudo code version is good but reference to Java could be removed and it would make it really pseudo code then. For example the code is readable without the public declaration... plus floats could be used instead of doubles (which sticks to what most render programs use). I don't want to discourage any attempt in improving this article. On the contrary it's great it has some attention. We are just a bit off track here while a lot can still be done to add more useful information on the subject. ---Mast4as (talk) 07:01, 27 July 2008 (UTC)

Calculations are wrong
It's true, I just checked them myself, 3 times and I know what I'm talking about. I have a PhD in Advanced Mathematics from MIT. 89.123.87.161 (talk)M.Johnson —Preceding undated comment was added at 17:59, 19 January 2009 (UTC).

Calculation for beginners
I might only be in algebra, but that doesn't mean I can't try to make a ray-tracer! The only problem is, I can't find equations to re-create reflection of a line off any surface! Could someone maybe edit the page to describe an equation to allow you to MAKE reflective objects? Becuase, until then, my ray-tracer can only be diffused. —Preceding unsigned comment added by GMEncrypt (talk • contribs) 01:32, 10 July 2009 (UTC)

 * If p is the direction of the incident ray and q is the surface normal (both being unit vectors), then the reflected ray's direction is p - 2*(p·q)*q, if I remember right, where p·q is a dot product. —Tamfang (talk) 03:47, 13 July 2009 (UTC)

Foveated_imaging
More of a question than a comment - is there any research into combining foveated imaging with ray tracing to cut down on the calculation overheads? I am imagining some sort of view-direction detection for animated ray-traced graphics, so that the image rendered is high-resolution only where the viewer is actually looking at that moment. Simon Oliver (nli) 212.125.69.106 (talk) 13:59, 26 January 2010 (UTC)

Ray tracing will never be correct on computer until size of virtual camera (or eye) matrix will not be 1000 times smaller
I mean glass, water light refraction and even for computer graphics, if object is big like human at 1 meter distance, then you will not see him like real 3D person but with some unrealistic 3D effects. Only small objects like bugs or mouse from closely (from close look) can be viewed correctly, but even mouse is too big. So for ray tracing (glass, water light refraction) this "BIG EYE"-"BIG VIRTUAL CAMERA" effect will be deadly and distortion and wrong ray tracing will appear. But since not many of us, think what can be correct ray tracing and what can be wrong ray trancing, then many can do not notice difference, between correct ray tracing and wrong ray tracing for big objects (bigger than size of eye; bigger than 1 cm size).

What is solution? Solution is, that need to change all 3D drawing algorithms, because they are all wrong even not for ray tracing, but for all big and not distance objects, but this will require addition computing power of about 2-10 times more.

In current games iris size is about 1 square meter.

The exact solution would be to shrink virtual camera (in game or rendering) size or to maximize size of all objects around and then in front of virtual camera [matrix] need to put glass lens, which have correctly chosen light refraction of about 1 cm of size in front of virtual camera (or virtual eye). This is possible for 3D rendering like in 3Dmax, but just need to put on camera correct lens of glass and need to magnifie objects, which need to render. But in current 3D games, where you can't have light refraction in glass - is impossible to see correct 3D shapes/objects.

— Preceding unsigned comment added by Versatranitsonlywaytofly (talk • contribs)

 * This could make an interesting addition to the article is it has been noted by a reliable source. GDallimore (Talk) 13:23, 30 January 2011 (UTC)

 * What the heck are you guys talking about? Dicklyon (talk) 22:08, 30 January 2011 (UTC)

 * Is the point here that the real pupil is not an ideal pinhole? Or perhaps that the real retina is not a flat raster?  How should correct ray-tracings look different from the present incorrect ray-tracings? —Tamfang (talk) 00:07, 31 January 2011 (UTC)

 * I can to provide pictures-screenshots, to show the problem. If object is in center of the screen, then object shape is good, but if you move mouse to right, then object will be on left side and will be longer horizontaly (wider). And if you move computer mouse down, then object will go up and will be longer (taller) than when object was in center of screen. And if you move mouse to corner, then object will be in wrong shape and will be bigger than on center. And those shape distortions do not depending on how far or how close object is. Need somehow to remove some part of vertical and horizontal lines from rendered image and need remove proportionally depending on how far this visible lines are from center. Or need to change all 3D algorithms. So if objects is not correctly rendered if his position is far from screen center, then is possibility in ray trancing when need to refract rays in glass there also can be serious distortions or all rays can go wrong, because algorithm pretty stupid and don't have real lens and iris of real certain size, but some perspective only in one axis (say this axis x into deep, and over axis do not have perspective - distance felling, like sloping line to corner is the same distance for algorithm, like line along center into deep/far distance and thats why objects become with distorted shape).
 * Car in center of screen: http://img404.imageshack.us/i/screenshot0001n.png/
 * Car on top of screen looks longer verticaly: http://img831.imageshack.us/i/screenshot0002.png/
 * Car in the bottom of the screen looks taller: http://img225.imageshack.us/i/screenshot0003o.png/
 * Car on the left of screen is wider, than in screen center: http://img21.imageshack.us/i/screenshot0004jpg.jpg/
 * Car on the right of screen [[file:Screenshot0005jpg.jpg|thumb]]

 * Looks to me like you've got the wrong end of the stick. Firstly, I don't think these are raytraced images, just 3D graphics in general. Secondly, these distortion effects could quite easily be deliberate and caused by a fish-eye-lens style view to increase the field of view and aid in gameplay. Don't see anything fundamentally wrong with either ray-tracing or these graphics.
 * Ultimately, there is no point continuing this conversation unless you can find a source which says the same thing as you. Saying "look at these images" is not providing a source. This talk page is for improving the article and we can't improve the article without sources. GDallimore (Talk) 22:07, 2 February 2011 (UTC)
 * Yes it's indeed about sources. But sources can be any site with images (how you feel with sources about Progression of the bench press world record?). And it (distortions) can not be corrected by changing angle of view in console (~ like in Counter Strike game, I don't know about new games). In reality objects (in filmed or photographed image) on left side or on right side looks smaller, than in center. But in 3D games or in rendering 3D is vice versa - objects on left and on right side looking bigger and wider. If they only looks bigger, then it would be vice versa, but they looks wider and this is not exactly vice versa, but even probably worse. And objects existing in say left bottom corner are, I think, not only bigger, but also with wrong shape. — Preceding unsigned comment added by Versatranitsonlywaytofly (talk • contribs) 10:08, 5 February 2011 (UTC)

 * I'm convinced that game graphics are not as true as one might wish, but that's no surprise. Why bring it up here? What realtime game uses ray tracing? —Tamfang (talk) 10:25, 6 February 2011 (UTC)

 * I'm quite sure none of the images presented is ray traced. Apart from that, I never came across this distortion problem in any reliable literature and from the principle of ray tracing, I don't think the virtual camera size matters. Niky cz 17:08, 8 February 2011 (UTC)

 * They're certainly not raytraced, but the underlying mathematical principles are probably the same - since it's all going to be based on well-known techniques for representing 3D scenes on a 2D plane (the screen). I still think it's a deliberate effect to increase field of view for improved gameplay, which naturally results in distortion around the edges and "unreal" effects because the field of view is "unreal".
 * I've gone back to my basic books on perspective to check. This is paraphrased from Gills "Perspective: From Basic to Creative" (to keep it brief and to avoid copyright infringement):
 * The field of vision (of a person) is over 180 degress, but it is not possible to see clearly over this whole range, and a cone of 90 degrees is the maximum range that can be seen clearly. Perspective drawing is usually limited to 60 degrees or less. Any object which would not normally be seen clearly because it is lying outside of the cone of vision will be distorted if it is drawn.
 * I think it could be an interesting point in game design about finding the balance between an adequate cone of vision for gameplay purposes versus realism. This is why I've asked the editor to find some reliable sources to back up what he's saying, since he appears to have some knowledge of the topic. But, as people do keep on pointing out, it's a discussion that belongs at the much more general levels of 3D graphics or perspective drawing in general and it can be taken there if any sources are found. GDallimore (Talk) 18:33, 8 February 2011 (UTC)

 * Here some sources:
 * http://www.rjdown.co.uk/projects/bfbc2/fovcalculator.php
 * http://gaming.stackexchange.com/questions/17917/how-to-change-fov-in-crysis-2-demo

And by the way, glass refraction is (I don't know how correctly) is possible even on Radeon 9700 card (about 2004 year) in ATI Debevec 9700 demo. But multiple times of reflections in current games are still imposible.
 * http://www.youtube.com/watch?v=n5Lf5FMhvS4&feature=related
 * http://www.youtube.com/watch?v=fW_GPCR9_GU
 * http://www.youtube.com/watch?v=dTAc2CTQcLw

(glass balls are purple, green for instance and big in center) — Preceding unsigned comment added by Versatranitsonlywaytofly (talk • contribs) 11:52, 14 March 2011 (UTC)
 * This is negative possibly effect (do not must be holes in object, when refracted through glass sphere):
 * http://img703.imageshack.us/i/atisushi00.jpg/
 * http://img820.imageshack.us/i/atisushi01.jpg/
 * http://img809.imageshack.us/i/atisushi02.jpg/
 * http://img130.imageshack.us/i/atisushi03.jpg/
 * http://img717.imageshack.us/i/atisushi04.jpg/
 * This is positive thing:
 * http://img859.imageshack.us/i/atisushi05.jpg/
 * http://img692.imageshack.us/i/atisushi06.jpg/
 * http://img28.imageshack.us/g/atisushi07.jpg/
 * positive thing is that at bigger distance refracted object is diferent size, than from close looking. Like in real for example through glasses, if put glasses close to eyes and far, then object at 1-3 metters distance will be bigger or smaller.

 * I try everything to change angle with "fov" or "fov_default", but it only writes on screen "fov 0" or "default_fov 90" (for counter-strike and half-life1) and puting over number do not changing anything. It's probably Valve marketing trick and lie. It is not possible to change angle of view nor aspect ratio with console command "~" in any game. — Preceding unsigned comment added by Versatranitsonlywaytofly (talk • contribs) 17:43, 14 March 2011 (UTC)

 * You're wasting your time linking to all these images. They are not reliable sources which can be used to add anything to the article. Find a computer graphics magazine article or something similar if you want to provide useful contributions to wikipedia. GDallimore (Talk) 19:21, 14 March 2011 (UTC)
 * I supose, that if you sitting near at big TV, when playing game or at 23" display, then distance [from eye] to corners is bigger and it compensates each over and in this case, even positive effect makes longer and wider objects on display sides/corners. — Preceding unsigned comment added by Versatranitsonlywaytofly (talk • contribs) 22:04, 14 March 2011 (UTC)
 * http://img51.imageshack.us/i/screenshotallcars4.jpg/
 * http://imageupload.org/?d=4D8DE6D31
 * All screenshots was taken do not quiting from game and do not touching keyboard move keys, but only rotating mouse. In the middle car is car, when it was in center. In the right up corner car is from screenshot, when car was in the right up corner and so on. So forget about any possible shape fixing tricks. There need new algorithm. So ray-tracing never will be even close to real not distorted shapes. Because ray-tracing using the same algorithms, which was used for rendering this car.
 * There is one possible solution to fix all things completly in rendering. This solution is based on fact, that filmed or photographed image from webcam (or any photocamera, which have small photo-matrix and very small glass lenses; webcam actualy have two lenses and bigger one is inside, close to photomatrix) is not blured and enough big resolution (like 640*480), from whatever distance you taking picture with webcam, the picture is always not blured and correct. So if maximize all objects in scene, which will be rendered, and to put on top of camera the same 3d-glass lenses of same shape like in web cam, then possible rendering will be as real, as picture and very realistic, without any distorted shapes.

 * Are you listening? Screenshots are useless! Stop wasting your and everyone else's time. GDallimore (Talk) 15:19, 26 March 2011 (UTC)
 * You got to see this from real life:
 * http://img710.imageshack.us/i/questionlensokoradjuste.jpg
 * http://www.imageupload.org/?d=4DC2765B1
 * It is the same as here http://img51.imageshack.us/i/screenshotallcars4.jpg/ and here http://img807.imageshack.us/i/screenshot0008jpg.jpg/ in game. But I don't think, that in real life on edges objects becoming bigger like in 3D game. But what if become bigger even in real life? Well I don't believe it. Also can be, that lenses was adjusted in this real life picture for more objects to fit in into frame. Or maybe even cheap lenses is used in photoaprat with which was taked picture. Or all lenses are wrong shaped, or wrong to put many lenses together (unlike on eye is only one lense). But it's unlikely that all linses are wrong shaped, because in overs photographs I everything see correct - on edges objects are little bit smaller than in center of picture and no shape distortions if I see horizont line in center. If in 3D games objects on edges is just bigger than in center - then very big monitor in front of you and very close sitting to it can fix all problems almost exactly (just hard to adjust distance to monitor and size of monitor). Display of 30 inches and 30 cm eyes from display would be nice, pretty correct solution (if you sit close to big monitor and look only with one eye, perpendicular to monitor screen center, then theoretically there is possible to adjust exactly correct distance between eye and monitor and this correct distance would mean, that 3D rendered image or in 3D-game action is exactly the same like in real world looking with one eye, just need hold one eye perpendicular to monitor center and only you can look in monitor screen edges by using eye muscles, but not moving body; problem is only to choose exact distance between your one eye and monitor; and if you using two eyes there will be only almost correct like in real world). I can explain, why 3D computer graphics is not exactly like in real life. Say there is stone 10 meters from you and you looking straight to stone. So stone is in front of your vision. Now imagine that one line going through from one you ear (A) to another ear (B). Line from your eye (point C) to stone (point D) let's call CD. So CD line is perpendicular to AB line. Now there is going third line EF through stone, which is parallel to line from your one ear to another ear. So line EF is going through stone and is parallel to line AB and perpendicular to line CD. Now if you, by moving eye muscles, will look at line EF right (or left) edge, which going through stone, then distance between EF line edge/corner and eye will be bigger than distance between eye and stone. So in real life distance from eye to EF line edge is bigger than from eye to stone. So if you will put stone on EF line edge then it will look smaller than in point, where intersects CD and EF lines. But not in 3D graphics. In 3D graphics stone on EF line edge is the same size like stone in EF and CD lines intersection point and this is wrong. Because of it, 3D game graphics is not exactly real life 3D.

 * In previous images one up car and one down car is not from the same distance in screenshots snapshoted (and thats why looks the same size like in center car). So here all cars are from same distance snapshoted (game capable to write into folder "documtents/my games/far cry 2/screenshots" all screenshots automaticly, without exiting game). So here all cars are snapshoted without quiting game and do not changing player position, but only moving mouse: http://imageshack.us/g/29/screenshot0014oh.jpg/ . Here all cars cuted from each screenshots to compare they size, when player was looking around, but standing in the same point: http://imageshack.us/photo/my-images/96/allcarscomparition3.jpg/  . As we see cars on edges are bigger than in center. Should be the same size, when looking from one point, if distance from player to car is always the same.

To what rendering algorithm class does ray tracing belong?
I wonder what the class of rendering algorithms is called that traces paths of particles, either it is from the eye or from a light source, and lets the particles bounce around in the scene (ray tracing and path tracing (what's the difference?), Metropolis light transport, photon mapping, instant radiosity etc.). "Path tracing" seems to be a suitable name for such a family since all of those algorithms basically trace paths, and the article says it is a generalization of conventional ray tracing, but path tracing refers to a specific algorithm in which paths are traced from the eye and samples the light input to the eye in Monte Carlo fashion (or so I've heard). Besides, the article talks about the path tracing algorithm. Ray tracing on the other hand, can refer to Whitted-style ray tracing, which simulates direct illumination from delta (point and directional) light sources and specular reflection and refraction. So what should you say if you want to talk about all these algorithms at once?

If you leave an answer, please notify me on my talk page. —Kri (talk) 00:58, 14 August 2011 (UTC)

What is relationship between this method and Monte Carlo method for photon transport? 142.196.169.88 (talk) 05:19, 29 July 2013 (UTC)

example image does not meet WP standards
Nobody can verify if the example image was in fact produced using the techniques described in the article. We have to trust the author. That's not the way an open encyclopedia works. --84.177.32.89 (talk) 08:24, 19 August 2011 (UTC)

Disagree with move.
I disagree with the move from "Ray tracing (graphics)" to "Ray tracing". As discussed previously, the graphics usage is not unambiguously primary.--Srleffler (talk) 05:18, 9 December 2011 (UTC)

 * I disagree too, and have discussed this unilateral move with him at User_talk:Thumperward. Thumperward, I request that you put it back to where it was and make a two-way disambig page.  Dicklyon (talk) 05:23, 9 December 2011 (UTC)
 * Actually, I was able to move it back myself. But "Ray tracing (disambiguation)" is not as easy to move back to "Ray tracing" now.  I notice their are several articles linking to Ray tracing from when it was a disambig, which is wrong, but they include both meanings; some disambiguating of links is in order.   Dicklyon (talk) 05:25, 9 December 2011 (UTC)

 * Having a disambiguation page for two links is actively harmful. I can just about see the rationale behind leaving the root title as a redirect to one of the two topics, though as I've said to Dicklyon I consider this to be a distinctly minority reading of the weight of the "primary importance" section of WP:PRIMARYTOPIC, but a disambiguation page simply means that rather than inconveniencing 50% (say) of readers with an additional mouse click to get to their desired page we inconvenience 100% of readers. Article titles are a means to an end and are not canonical representations of importance. Similarly, dab pages are technical matters of convenience and not arbitrators of equality. Chris Cunningham (user:thumperward) (talk) 07:30, 9 December 2011 (UTC)

 * You appear to be in conflict with the policy you cite, so I suggest you take it up there if you want a change in policy. I have no preference either way, but you've shot yourself in the foot here. GDallimore (Talk) 12:43, 9 December 2011 (UTC)

 * I'm not even sure that comment was supposed to be addressed to me, such is the level of vagueness. Who and what is in conflict with what part of what policy? Chris Cunningham (user:thumperward) (talk) 12:45, 9 December 2011 (UTC)

 * You're the only person citing policies and you're the only person to be misrepresenting them in your comments: "if an ambiguous term is considered to have no primary topic, then that term should lead to a disambiguation page". GDallimore (Talk) 12:54, 9 December 2011 (UTC)

 * I'm not arguing that there's no primary topic. I am arguing that of the two methods given at WP:PRIMARYTOPIC, it is far more useful to optimise for usage and thus readers (who are over six times more likely to visit the graphics page than the physics page) than for alleged importance, and thus nobody. Of the two methods, the former strongly indicates that the graphics article is the primary topic, while the latter has no consensus. We should therefore consider the graphics page to be primary. My comments should be taken in that light. As to the current situation, where the root title redirects to the graphics page, I feel that it may have some diplomatic value in hopefully placating physicists upset with the idea of usage being favoured over importance, though I'd be just as happy with the graphics page being at the root title personally. Chris Cunningham (user:thumperward) (talk) 13:12, 9 December 2011 (UTC)

I've added Category:Computer graphics algorithms
If this is wrong to do, please explain to me why. AFAIK this is one of the most classic graphics algorithms there is... ToolmakerSteve (talk) 06:56, 27 October 2012 (UTC)

 * Well, it's wrongish in that it's already in Category:Global illumination algorithms, which is itself a member of Category:Computer graphics algorithms. —Tamfang (talk) 07:03, 27 October 2012 (UTC)

 * That doesn't make it "wrongish", it makes it plain wrong. Removed. GDallimore (Talk) 16:57, 27 October 2012 (UTC)

Disadvantages clarity
I am not an expert in ray tracing, but even I can tell that there is a problem with the disadvantages section as it is written. I think the problem stems from ambiguity between "ray tracing in the narrow, historical sense of Whitted's ray tracing algorithm" and "ray tracing as a technique, that is used in a wide variety of algorithms". Correct me if I'm wrong, but without the concept of tracing rays, most (performance practical) global illumination algorithms would not exist. Photon mapping & path tracing - couldn't these (oversimplified) be described as casting additional rays (in a particular manner)? Or am I confusing the basic act of "ray casting" with a specific use of ray casting, that is considered "ray tracing"? Is there consensus on the meaning of "ray tracing"? ToolmakerSteve (talk) 07:36, 27 October 2012 (UTC)
 * According to the wiki article on ray casting, I am not confusing the two. Paraphrasing, "ray casting" refers to a technique WITH NO RECURSION (no secondary rays). As soon as there is recursion / secondary rays, a technique is categorized as a "ray tracing" technique. Therefore, these advanced Global Illumination (GI) algorithms that are mentioned, fall into the category of "ray tracing techniques". That is, they are algorithms, one of whose central characteristics is, they involve secondary rays. Therefore, I consider the sentence "Other methods, including photon mapping, are based upon ray tracing for certain parts of the algorithm, yet give far better results." to be inappropriately placed in the disadvantages section. What I mean is, this is *NOT* a "disadvantage of ray-tracing". Indeed, there is no such thing as photon mapping WITHOUT ray-tracing. So I am going to be bold, and instead reword this as a need to EXTEND ray-tracing with additional algorithms. ToolmakerSteve (talk) 07:44, 27 October 2012 (UTC)
 * I am now going to go even further, and reference a recent GI technique, that is not generally (AFAIK) classified as a ray tracing technique, though it involves ray casting. To me, it falls in a grey area (classification-wise). But again, I'm not an expert -- perhaps I am over-relying on the radically different implementation approach (parallel projection; rasterization; depth peeling). Perhaps underneath, the mathematics SHOULD categorize this as a ray-tracing technique. The bottom line is that this single paper fundamentally altered my personal understanding of how to approach high-performance GI. It challenged my underlying assumptions (as a technical evaluator examining competing vendors and future directions) of the discussion of "rasterization techniques versus ray-tracing techniques", as we attempt to have more realistic lighting in computer graphics. The existence of such a technique to me is fundamentally important in discussing pros and cons of rasterization and ray-tracing, hence my attempt to insert it into this discussion. http://http.developer.nvidia.com/GPUGems2/gpugems2_chapter38.html ToolmakerSteve (talk) 08:07, 27 October 2012 (UTC)
 * And for anyone who is skeptical that the above paper does describe a true GI approach (when I first read it, it seemed "too good to be true"), here are the two images that together I found compelling http://www.bee-www.com/parthenon/stairs.png and  http://www.bee-www.com/parthenon/ibl_sample2.png  and the page containing downloadable code and other images http://www.bee-www.com/parthenon/ ToolmakerSteve (talk) 08:34, 27 October 2012 (UTC)

Illustration needed - Bounding volumes
The section https://en.wikipedia.org/wiki/Ray_tracing_(graphics)#Bounding_volumes would be far more understandable to non-mathamaticians if there was an illustration of the pieces that make up the bounding volume.

Proposed merge with Ray-tracing hardware
Hardware raytracing is just a subtopic of raytracing Ethanpet113 (talk) 04:20, 20 April 2019 (UTC)

 * My view is that WP:BOLD merge not be appropriate as the references suggest that there is evidence for independent notability. Therefore, proposing first, giving a reason for the proposal and starting a discussion, is the way forward as per WP:MERGEPROP. My view would then be against a merge unless the argument was strong. Klbrain (talk) 09:25, 8 April 2020 (UTC)

Ray Tracing can be applied to any phenomenon that behave linearly
The last paragraph of the introduction has this sentence: "In fact, any physical wave or particle phenomenon with approximately linear motion can be simulated with ray tracing." I am wondering what would be an example of a physical phenomenon that behaves nonlinearly? I am wondering if there is a phenomenon that behaves in a spiral shape or something else? ScientistBuilder (talk) 02:14, 27 January 2022 (UTC)ScientistBuilderScientistBuilder (talk) 02:14, 27 January 2022 (UTC)

Limitations of Ray Tracing Source Addition without Copyright Infringement from Close Paraphrasing
I would like to add this source https://reference.wolfram.com/language/tutorial/PhysicallyBasedRendering.html. It would be good to add a few details about the limitations of ray tracing. ScientistBuilder (talk) 21:08, 27 January 2022 (UTC)ScientistBuilderScientistBuilder (talk) 21:08, 27 January 2022 (UTC)

Disadvantages
Is there a way to measure how much more computing power is needed for every increase in resolution? I am curious about this part of the Disadvantages section: "A serious disadvantage of ray tracing is performance (though it can in theory be faster than traditional scanline rendering depending on scene complexity vs. number of pixels on-screen). Until the late 2010s, ray tracing in real time was usually considered impossible on consumer hardware for nontrivial tasks." given a finite amount of computing power, which is more realistic: a lower resolution image with more ray tracing calculations done or a higher resolution image (4k for example) with reduced ray tracing calculations? Is there a way to approximate using Big O or asymptotic notation the number of additional calculations needed? ScientistBuilder (talk) 21:12, 27 January 2022 (UTC)ScientistBuilderScientistBuilder (talk) 21:12, 27 January 2022 (UTC)

"Ray tracing (graphics" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Ray tracing (graphics and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 October 27 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Steel1943 (talk) 19:24, 27 October 2022 (UTC)

Emissions theory of sight
It turns out that ray tracing in the most common form is reminiscent of the emission theory as set forth by Greek philosophers, notably Empedocles. This ancient, now scientifically obsolete theory postulated that our eyes emitted rays lighting up things for us to see. Remember, Empedocles was the same man who theorized that everything was composed of air, earth, fire, and water, with love pulling the elements together and hate tearing them apart. In his theory of light, the fire in our heads emitted beams that would bounce off objects before entering the eye again. Essentially, our eyes were like flashlights. Never mind that a startling number of people continue to adhere to the obsolete theory. It has been pointed out that it has practical use in computer graphics as recursive ray tracing since it is computationally orders of magnitude more efficient than to have the light sources emitting rays in all directions, most of which never reach the camera. Does this mention have a place in the article, perhaps under history?  Free Media  Kid$  15:54, 16 November 2022 (UTC)

Interactive Ray Tracing: What's BVH?
Section uses acronym "BVH" without that being defined. Marcusmueller ettus (talk) 14:18, 16 January 2023 (UTC)

title: Talk:Ray tracing (graphics)


text:
Max Pruss

Max Pruss (13 September 1891 – 28 November 1960) was the commanding captain of the zeppelin LZ 129 Hindenburg on its last voyage and a surviving crew member of the disaster.

Biography
Max Pruss was born in 1891 in Sgonn, East Prussia (now Zgon, Warmian-Masurian Voivodeship, Poland). He joined the German Navy in 1906 and completed airship training during World War I, serving as an elevatorman on the German Zeppelins. Pruss became part of the Hindenburg crew in 1936 on the third flight to Rio de Janeiro. During his career, he flew 171 times over the Atlantic. The final flight of the Hindenburg was May 3–6, 1937, and it was Pruss' first flight as commanding Captain of the Hindenburg. According to Airships.net he was a member of the NSDAP. He died at age 69.

Hindenburg disaster
Pruss was commander of the airship during the Hindenburg disaster of 6 May 1937. This was his first time commanding a trip to Lakehurst. Pruss and several crew members rode the Hindenburg down to the ground as it burned, then ordered everybody out. He carried radio operator Willy Speck out of the wreckage, then looked for survivors until rescuers were forced to restrain him. Pruss suffered extensive burns and had to be taken out by ambulance to Paul Kimball Hospital in Lakewood. The burns were so severe that he was given last rites, but although his face was disfigured for the rest of his life, his condition improved over the next few months. Pruss was unable to testify at investigative committees, but officially he was not held responsible.

Pruss, along with other airship crewmen, maintained that the disaster was caused by sabotage, and dismissed the possibility that it was sparked by lightning or static electricity. Although master zeppelin captain Hugo Eckener did not rule out other causes, he criticized Pruss' decision to carry out the landing in poor weather conditions, expressing his belief that sharp turns ordered by Pruss during the landing approach may have caused gas to leak, which could have been ignited by static electricity. Pruss insisted that such turns were normal procedure, and that the stern heaviness experienced during the approach was normal due to rainwater being displaced at the tail. It has been suggested that Pruss maintained his belief of sabotage because of guilt or to maintain the credibility of himself and the airship business.

After the Hindenburg
Pruss returned to Germany around October 1937, where he served as commandant of Frankfurt Airport as World War II broke out. By this time he was already urging the modernization of Germany's remaining Zeppelin fleet, and during a 1940 visit by Hermann Göring to Frankfurt Airport this was the subject of an alleged quarrel between Pruss and Göring. In the 1950s Pruss tried to raise money for new Zeppelin construction, citing the comfort and luxury of this mode of transportation. He died in 1960 of pneumonia after a stomach operation. Pruss did not see his dream realized, as his death was over 30 years before the construction of a new airship at the Friedrichshafen complex by Zeppelin Neue Technologie (NT).

Portrayals
In the 1975 film, The Hindenburg, Pruss was portrayed by Charles Durning. This portrayal is inaccurate because Pruss is portrayed as ignoring the advice of the 2nd Captain Ernst Lehmann, who only traveled as an observer, saying "I'll do the worrying on this trip". In the docudrama Hindenburg: The Untold Story he was portrayed by Albert Welling. In the 2011 RTL television movie he is portrayed by Jürgen Schornagel.

The Hindenburg is featured in the series Beyond Belief: Fact or Fiction as the setting of a story in which a married couple ride on the airship while recounting to friends of theirs how years ago they escaped certain death when they missed a trip on the doomed Passenger Liner RMS Lusitania.

In the pilot episode of the TV series Timeless Pruss was featured but not mentioned by name.

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

title: Max Pruss


text:
Talk:Tidal harness

Someone might want to merge tidal power with this article or visa versa as they appear to be talking about the same thing.--GD 10:28, 26 Sep 2004 (UTC)

title: Talk:Tidal harness


text:
Talk:National Catfish Day

Is it annual?
hmm, is it june 25 of every year, or only 1987? -- John Joseph Bachir
 * haha, i dunno John, seems like a strange holiday, but it's on my birthday so ill celebrate it! -Julian S

Really a holiday?
Does this fit the definition of a "holiday" ? It seems that the holiday category should only include things that a large number of people regularly celebrate. There are hundreds of "National X Days" and I don't think most people would call those "holidays". (I have nothing against catfish, however, they are quite good!) Jpp42 23:06, 23 April 2007 (UTC)

Is this whole page a joke?
...or only the part about tying up members of Congress and forcing them to watch Bedtime for Bonzo (which would violate the US Constitutional prohibition against cruel and unusual punishment).

Maybe this page should be moved to the satirical version of Wikipedia (forgot the name).RandallC 12:24, 23 June 2007 (UTC)

Is the date for this holiday correct?
I was looking into the reason WHY our government wasted time and money making this a real thing, and in doing so, I looked up Joint House Resolution 178 and it says April 4th is National Catfish Day. Did this date later get changed and I'm not seeing this?

https://www.congress.gov/bill/100th-congress/house-joint-resolution/178 — Preceding unsigned comment added by 192.96.192.5 (talk) 10:49, 25 June 2016 (UTC)

title: Talk:National Catfish Day


text:
Arthur Boyd

Arthur Merric Bloomfield Boyd (24 July 1920 – 24 April 1999) was a leading Australian painter of the middle to late 20th century. Boyd's work ranges from impressionist renderings of Australian landscape to starkly expressionist figuration, and many canvases feature both. Several famous works set Biblical stories against the Australian landscape, such as The Expulsion (1947–48), now at the Art Gallery of New South Wales. Having a strong social conscience, Boyd's work deals with humanitarian issues and universal themes of love, loss and shame.

Boyd was a member of the Antipodeans, a group of Melbourne painters that also included Clifton Pugh, David Boyd, John Brack, Robert Dickerson, John Perceval and Charles Blackman.

The Boyd family line of successive and connective artists includes painters, sculptors, architects and other arts professionals, commencing with Boyd's grandmother Emma Minnie Boyd and her husband Arthur Merric Boyd, Boyd's father Merric and mother Doris; 'She was the backbone of the family' recalls Boyd "without her, the entire family would have fallen apart", uncle Penleigh Boyd (and his son, Arthur's cousin, Robin), uncle Martin Boyd, and siblings Guy, David and Lucy. His other sister Mary Boyd, married first John Perceval, and then later Sidney Nolan, both artists. Boyd's wife, Yvonne Boyd (née Lennie ) is also a painter; as are their children Jamie, Polly, and Lucy.

In 1993, Arthur and Yvonne Boyd gave family properties comprising 1100 ha at Bundanon on the Shoalhaven River to the people of Australia. Held in trust, Boyd later donated further property, artwork, and the copyright to all of his work.

Early years and background
Boyd was born at Murrumbeena, Victoria, the son of Doris Boyd and her husband Merric, both potters and painters. Boyd's sisters Lucy and Mary were both artists as well as both of Boyd's younger brothers; David was a painter, and Guy a sculptor. After leaving school aged 14 years, Boyd briefly attended night classes at the National Gallery School in Melbourne where Jewish immigrant artist Yosl Bergner introduced Boyd to writers such as Dostoyevsky and Kafka and influenced his humanitarian values and social conscience. Boyd later spent some time living on the Mornington Peninsula at Rosebud with his grandfather, the landscape painter Arthur Merric Boyd, a primary guide to the formation of his talent. Early paintings were portraits and of seascapes of Port Phillip created while he was an adolescent, living in the suburbs of Melbourne. He moved to the inner city where he was influenced by his contact with European refugees. Reflecting this move in the late 1930s, his work moved into a distinct period of depictions of fanciful characters in urban settings.

Military service
Boyd (Service number V101720) was 20 when conscripted to serve in the militia from 12 May 1941 until 25 March 1944. Initially serving with the 2nd Cavalry Division, which later became the 2nd Armoured Division (Australia), Boyd was transferred to the 4th Division (Australia) for full time service, and then Army Headquarters Cartography Company. Boyd predominantly served in the Bendigo area as a cartographer. Boyd's expressionistic wartime paintings included images of disabled people and those considered unfit to serve. These works have been described as depicting "the dispossessed and the outcast".

Artistic career
Following the war, Boyd, together with John Perceval founded a workshop at Murrumbeena and turned his hand to what he since childhood had seen his father's hands occupied with, pottery. Later came ceramic painting and sculpture. Although Boyd was the closest of friends with Albert Tucker, Joy Hester and Sidney Nolan and the art patrons John and Sunday Reed, the modernist Heide Circle and its hierarchical structure did not beckon him overtly as his position in the Boyd family gave him the fullest identity in itself.

In the late 1940s and early 1950s, Boyd traveled to Victoria's Wimmera country and to Central Australia including Alice Springs and his work turned towards landscape paintings. During this period, perhaps his best-known work comes from his Love, Marriage and Death of a Half-Caste Bride series of 31 paintings, also known as The Bride, that imagined an Aboriginal person of mixed descent as a neglected outsider. First exhibited in Melbourne in April 1958, the series met a mixed reaction, as it did later that year in Adelaide and Sydney. Following the 1999 acquisition of Reflected Bride 1 by the National Gallery of Australia, the gallery's director Brian Kennedy commented in 2002: "The Bride paintings are among the greatest expressions of conscience by an Australian artist. Brilliantly executed and of sustained quality, Reflected Bride 1 speaks to contemporary Australia, beseeching reconciliation, understanding and a tolerant, compassionate meeting of old and new cultures.

Boyd’s paintings are not pretty, however, and carry a pervasive magical and somewhat menacing atmosphere. It is as if the figures and the landscape are one. The bride rises from the stream, an Ophelia caught by a groom whose foot hooks a tree. The bride is staring at an absurd mask-like white bride’s head which appears to glow out of the forest. This is a surreal wilderness, a strange place of nightmarish dreams."

In 1956, Boyd's ceramic sculpture 'Olympic Pylon' was installed in the forecourt of the Melbourne Olympic Swimming Pool.

Relocates to Europe
Boyd represented Australia with Arthur Streeton at the Venice Biennale in 1958, where his Bride series was well received.

He was affiliated with the Antipodeans, a group of painters founded in 1959 and supported by Australian art historian Bernard Smith, who tried to promote figurative art when abstract painting and sculpture was dominant. The group exhibited at the Whitechapel gallery in London. In 1959 Boyd and his family moved to London, where he remained until 1971. In London, he started receiving commissions for ballet and opera set designs, and, after taking up etching and returning to ceramic painting, in 1966 he began the Nebuchadnezzar series in response to the Vietnam War as a statement of the human condition. While in London, Boyd entered another distinct period with his works themed around the idea of metamorphosis.

He produced several series of works, including a collection of fifteen biblical paintings based on the teaching of his mother, Doris. Later he produced a tempera series about large areas of sky and land, called the Wimmera series.

Return to Australia
The recipient of a Creative Arts Fellowship from the Australian National University, in 1971 Boyd and his family returned to Australia as one of Australia's most highly regarded artists. In 1975, Boyd donated several thousand works including pastels, sculptures, ceramics, etchings, tapestries, paintings and drawings to the National Gallery of Australia.

In 1978, Arthur and Yvonne Boyd purchased properties and settled permanently at Bundanon on the Shoalhaven River. The following year, ABC TV and BBC TV co-produced the television documentary film, A Man of Two Worlds, based on Boyd's life and work. During the latter part of Boyd's painting career, his landscape works were based on the Shoalhaven River. At first encounter, Boyd was a little overwhelmed to paint the area; he found the scenery rugged and wild, vastly different from the landscapes he knew. But over the years, painting scenes of the Shoalhaven River and the surrounding bushland, he befriended the formidable landscape. This resulted in a significant series of paintings that are not simply landscapes but rather, a fusion of Boyd's European and Australian backgrounds.

"His Australian scapegoat paintings of the 1980s explored constructions of Australian identity in the lead up to the bicentenary of the arrival of the First Fleet in 1988. With their violent imagery and aggressive colouring they draw on archetypes of Australian military history to suggest the futility of war. In addition to painting, Boyd worked prolifically in ceramics, designed sets for the theatre, and provided illustrations for the poems of Australian poet Peter Porter." Boyd donated a villa in Tuscany to the Australia Council for an artist-in-residence program in 1982. In 1984, he was commissioned to design a tapestry based on the painting Untitled (Shoalhaven Landscape) for the Great Hall at the new Parliament House in Canberra. The work is one of the world's largest tapestries. Yet Boyd could not find the strength to fight for the retention of the lower rectangle ablated by the building consultants. This tapestry was produced in the Victorian Tapestry Workshop, Melbourne. He also produced sixteen canvasses for the foyer of the Victorian Arts Centre in the same year.

Boyd again represented Australia at the 1988 Venice Biennale with eight major works; and at the 2000 Venice Biennale. Boyd was commissioned to paint Earth and Fire for the cover of the 28 November 1988 Time magazine special issue dealing with environmental conservation in Australia.

Final years

A major retrospective of Boyd's work was exhibited at the Art Gallery of New South Wales in 1993.

In 1997 for the first time Boyd exhibited together with the six members of his artistic dynasty under one roof; with brothers David and Guy, son Jamie, and nieces Lenore and Tessa Perceval. The exhibition entitled the Best of Boyd comprised 80 paintings and 40 bronze sculptures. The exhibition was held in Galeria Aniela Fine Art Gallery and Sculpture Park, NSW. Documentary reviews were shown on the ABC TV Australian National News, 18 May 1997 and the ABC TV Sunday Afternoon, June 1997. Boyd died in 1999 at 78 years of age, and was buried at Brighton General Cemetery. He was survived by his wife Yvonne, their son Jamie, and daughters Polly and Lucy.

Bundanon Trust
In 1993 the Australian Government accepted the gift of Bundanon, valued at the time at A$20 million. The 1100 ha property owned by Arthur and Yvonne Boyd was given to the people of Australia. Located approximately 20 km west of Nowra, the gift was entrusted to the Bundanon Trust, along with further gifts by Boyd, including copyright of all of his artwork, and several thousand works of art from five generations of Boyds, and other Australian artists. These properties provide an environment that promotes visual arts, writing, music and other performing arts, and the promotion of education and research in the arts.

Directors
Current and past directors and board members include:
 * Alice Spigelman
 * David Gonski
 * James Spigelman
 * Gene Sherman 1995–2002
 * Jim Service

Bundanon Trust artists in residence

 * Steve Lopes
 * Josh Foley 2016
 * Jennifer Joseph 2003

Bundanon Trust musicians in residence

 * Marshall McGuire 2003

Bundanon Trust poets in residence

 * Eileen Chong 2016

Technique
Boyd was a master at manipulating elements to express himself. He developed new techniques when he was still a teenager and later changed technique depending on his preferred style, media, location and what he was depicting.

He would often use loose strokes of thickly coated brushes. He applied paint with his fingers and palm because it is quicker, while the body contact directly connected him with the painting. He believed this allowed for a greater sense of freedom and pleasure from the act of painting.

Honours and awards
Boyd was appointed an Officer of the Order of the British Empire on 1 January 1970 for services to art. On 26 January 1979, Boyd was appointed an Officer of the Order of Australia for service to the visual arts. In recognition of his service to the visual arts and to the development of Australian artists and crafts people, Boyd was appointed a Companion of the Order of Australia on 8 June 1992.

In 1995, the Prime Minister announced Boyd as Australian of the Year for his contribution to Australian art and his generosity to the Australian people.

Australia Post honoured Boyd in 1998 with a series of postage stamps produced with his photo and examples of his work.

On 24 July 2010, Google celebrated Arthur Boyd's 90th Birthday with a doodle.

Notable sales of works

 * 2000: Dreaming Bridegroom I (1957) sold for A$957,000.
 * 2000: Mourning Bride I (1958) sold for A$833,000.
 * 2011: Frightened Bridegroom (1958) sold for A$1.2 million.
 * 2012: Dry Creek Bed, Alice Springs (1953–54) sold for A$1.2 million.
 * 2012: Bride Running Away (1957) sold for A$1.68 million, setting a new record for a Boyd work.

Selected works
Boyd's subjects were often mythical, realistic, malformed people and monsters, depicting a tragic drama.

 * Creek near Rosebud, 1937
 * The Seasons, 1944
 * The Lovers, 1944
 * David and Saul, 1946
 * Melbourne Burning, 1946–1947
 * Wimmera Landscape, 1950
 * Merric Boyd, 1952
 * Half Cast Child, 1957
 * Moses leading the People, 1957
 * Lovers with a Bluebird, 1962
 * Figure supporting Back Legs, 1973
 * Riversdale Bushland, 1976
 * Flood receding one Winter Evening, 1981
 * Bathers Pulpit Rock, 1985

title: Arthur Boyd


text:
Talk:Autonomous Republic of Crimea

Discussion about the listing as level-5-vital article
Please note that I just suggested to remove this article from the list of level-5-vital articles. You may add your opinion at Wikipedia_talk:Vital_articles/Level/5. Rsk6400 (talk) 07:04, 18 September 2023 (UTC)

title: Talk:Autonomous Republic of Crimea


text:
General Leo

Leo Cristophe (レオ・クリストフ) more commonly referred to as General Leo, is a character introduced in the 1994 role-playing video game Final Fantasy VI by Square Enix. An honorable soldier in the game's Gestahlian Empire that refused the magical energy infusion given to his compatriots, he appears in the first half of the game helping to ensure his Emperor's expansionist goals and is betrayed in the end. He has since appeared in other games related to the Final Fantasy franchise, including Dissidia Final Fantasy Opera Omnia, where he is voiced by Masashi Sugawara.

Despite being a temporarily playable character, Leo was well received for his characterization and charisma, with players trying to find ways to revive him or make him a fully playable character. He was additionally seen as a contrasting element to the game's final villain, Kefka, representing hope for a better world in light of the latter's representation of despair. His role as a catalyst for the story's progression has also been analyzed, and how it shaped both how Kefka was perceived but also the resolve of the protagonists to oppose the Empire. His appearance meanwhile has been read by the former staff of 1UP.com as racially coded as a black person, and they considered him one of the first examples of a black character in the Final Fantasy franchise.

Appearances
General Leo Christophe is introduced in the 1994 role-playing video game Final Fantasy VI. He is one of the three top generals of the game's antagonist Gestahlian Empire, the others being fellow characters Celes Chere and Kefka Palazzo. He has the unique ability Shock, which creates a large area of effect explosion. Unlike his compatriots, he refused to be infused with magical energy, and instead relies on his own talent. He is first seen at the Empire's siege of the kingdom of Doma. Not wanting to see men on either side needlessly killed, he refuses an all out attack against Doma. Kefka on the other hand sees this as weakness and argues with him, and when Leo is called back to the Empire on urgent business, Kefka poisons the kingdom's water supply, killing almost everyone inside and ending the conflict.

Later in the game he appears when the Empire offers peace to help find the home of the Espers, a race of magical beings. Attempting to atone for the Empire's past crimes against the protagonists, he joins with them as they travel to the town of Thamasa to open the gate to the Esper's domain. However, they discover it was a ruse by the Emperor, as Kefka and his soldiers arrive to capture the Espers for their magical energy and attack the town. Enraged, Leo attacks and defeats Kefka, only to discover it was an illusion as he is stabbed in the back by the real Kefka and killed. After the attack, the survivors bury his body, and the protagonists resolve to stop the Kefka and the Empire.

In other games, he appears as a playable character in both Final Fantasy Brave Exvius and Final Fantasy Record Keeper. He is additionally a playable character in Dissidia Final Fantasy Opera Omnia, where he is voiced by Masashi Sugawara. Set after the events of VI, he reflects on his death and past with the Empire, now that he has no ruler to follow. Outside of video games, cards representing Leo have been produced for the Final Fantasy Trading Card Game.

Conception and design
Final Fantasy VI was developed with the mindset that none of the playable cast was the protagonist, and that each of them were equally the "main character". The cast of characters were selected from submissions from across the development team. Once the cast was selected, each individual would write their character's story, with Yoshinori Kitase balancing the plot as things developed. The character was originally intended to be a full party member, but the idea was cut during development.

Standing 188 cm (6 ft 2 in) tall, artist Yoshitaka Amano was commissioned to design the characters from the brief outlines provided. Given full creative freedom, Amano wanted to make "real" and "alive" characters, though with consideration for their representation in-game as small computer sprites. Said graphics were designed by Nomura, who drew the chibi artwork of the characters, and Kazuko Shibuya, who designed the in-game sprites. His appearance is as a tall, dark skinned man with a yellow mohawk, prominent yellow lips, and a green military uniform with yellow and orange highlights.

Critical reception
Since his debut, the character was positively received. Dengeki Online's Kawachi expressed how much he enjoyed Leo as a character, finding him very charismatic and expecting he would be a full party member in the game. However, Leo's sacrifice took him by surprise, and as much as he enjoyed using the character briefly, he felt it served as a powerful motivator for the game's story due to how it encouraged Terra and the other protagonists' resolve to overthrow the Empire. Meanwhile the staff of Futabanet Manga Plus, in a retrospective, cited him as one of the franchise's poignant examples of self-sacrifice. They further praised how much humanity his character demonstrated in the story, and how much it left a lasting impact on players. Leo's portrayal and role as a temporary party member lead to players trying to figure out how to revive him or make him a permanent companion, resulting in players trying to glitch the game to keep him alive and usable beyond the sequence.

In a podcast discussion examining the characters of Final Fantasy VI for RPGFan, journalist Michael Sollosi compared Leo's character to Kefka's, describing him as an "extremely likable soldier" trying to make the world a better place and atone for the Empire's actions, while Kefka revels in them. He saw Leo as representing a recurring theme of "hope" in Final Fantasy VI, in his context the hope for a better unified world. Through this scope, Kefka's killing of him in a brutal manner represented despair killing hope for Sollosi, elevating both his character but also the heroes as a last light of hope against Kefka. Fellow writer Alana Hagues liked his conflicted character of being honorable while serving the game's villain, and expressed confusion as to why the Empire would back Kefka over him in their ambitions. In later podcasts discussing the game, the staff of RPGFan further analyzed his character, noting that he was an important element of Terra accepting her nature, but also served as a more humanizing aspect of the Empire, giving some faith to the player that they may be able to reform.

Sebastian Deken in his book examining Final Fantasy VI and its themes cited Leo's death as a turning point in how players perceived Kefka as a character. While previously he was portrayed more comedic and bumbling personality, killing Leo made him "irredeemable", and any "comic relief or lighthearted mischief—or humanity—is excised" from Kefka's character afterwards. Most notability too Leo's death is the last use of Kefka's more melancholic theme, an aspect that Deken felt reflected the end of the comedic aspect and his character's shift to full villainy in the eyes of the audience.

Leo's appearance has also been of some discussion. Kawachi upon controlling the character and seeing his portrait admitted they were shocked, moreso than by his death. He had expected a more handsome character due to how Leo acted, and joked "you could say that it taught me that a man is not a face. A man's charm isn't his appearance!" Meanwhile, the former staff of 1UP.com in their Retronauts podcast read him to be racially coded as black due to his skin tone and prominent lips, which would have made him in their eyes one of the first instances of a black character in the Final Fantasy franchise.

title: General Leo


text:
Category:Marshals of the Soviet Union

See: Marshal of the Soviet Union

title: Category:Marshals of the Soviet Union


text:
Dirac measure

In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element x or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.

Definition
A Dirac measure is a measure ${x,y,z}$ on a set $δ_{x}$ (with any $δ_{x}$-algebra of subsets of $X$) defined for a given $σ$ and any (measurable) set $X$ by
 * $$\delta_x (A) = 1_A(x)= \begin{cases} 0, & x \not \in A; \\ 1, & x \in A. \end{cases}$$

where $x ∈ X$ is the indicator function of $A ⊆ X$.

The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome $1_{A}$ in the sample space $A$. We can also say that the measure is a single atom at $x$; however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a delta sequence. The Dirac measures are the extreme points of the convex set of probability measures on $X$.

The name is a back-formation from the Dirac delta function; considered as a Schwartz distribution, for example on the real line, measures can be taken to be a special kind of distribution. The identity
 * $$\int_{X} f(y) \, \mathrm{d} \delta_x (y) = f(x),$$

which, in the form
 * $$\int_X f(y) \delta_x (y) \, \mathrm{d} y = f(x),$$

is often taken to be part of the definition of the "delta function", holds as a theorem of Lebesgue integration.

Properties of the Dirac measure
Let $x$ denote the Dirac measure centred on some fixed point $X$ in some measurable space $δ_{x}$.
 * $x$ is a probability measure, and hence a finite measure.

Suppose that $(X, Σ)$ is a topological space and that $δ_{x}$ is at least as fine as the Borel $(X, T)$-algebra $Σ$ on $σ$.
 * $σ(T)$ is a strictly positive measure if and only if the topology $X$ is such that $δ_{x}$ lies within every non-empty open set, e.g. in the case of the trivial topology $T$.
 * Since $x$ is probability measure, it is also a locally finite measure.
 * If ${∅, X}$ is a Hausdorff topological space with its Borel $δ_{x}$-algebra, then $X$ satisfies the condition to be an inner regular measure, since singleton sets such as $σ$ are always compact. Hence, $δ_{x}$ is also a Radon measure.
 * Assuming that the topology ${x}$ is fine enough that $δ_{x}$ is closed, which is the case in most applications, the support of $T$ is ${x}$. (Otherwise, $δ_{x}$ is the closure of ${x}$ in $supp(δ_{x})$.) Furthermore, ${x}$ is the only probability measure whose support is $(X, T)$.
 * If $δ_{x}$ is ${x}$-dimensional Euclidean space $X$ with its usual $n$-algebra and $R^{n}$-dimensional Lebesgue measure $σ$, then $n$ is a singular measure with respect to $λ^{n}$: simply decompose $δ_{x}$ as $λ^{n}$ and $R^{n}$ and observe that $A = R^{n} \ {x}$.
 * The Dirac measure is a sigma-finite measure.

Generalizations
A discrete measure is similar to the Dirac measure, except that it is concentrated at countably many points instead of a single point. More formally, a measure  on the real line is called a discrete measure (in respect to the Lebesgue measure) if its support is at most a countable set.

title: Dirac measure


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Marpa Lotsawa

Marpa Lotsāwa (མར་པ་ལོ་ཙཱ་བ་ཆོས་ཀྱི་བློ་གྲོས་, 1012–1097), sometimes known fully as Marpa Chökyi Lodrö (Wylie: mar pa chos kyi blo gros) or commonly as Marpa the Translator (Marpa Lotsāwa), was a Tibetan Buddhist teacher credited with the transmission of many Vajrayana teachings from India, including the teachings and lineages of Mahamudra. Due to this, the Kagyu lineage, which he founded, is often called Marpa Kagyu in his honour.

Although some accounts relate that the Mahasiddha Naropa was the personal teacher of Marpa, other accounts suggest that Marpa held Naropa's lineage through intermediary disciples only. Either way, Marpa was a personal student of the Mahasiddha Maitripa and of the dakini Niguma.

Biography
Born as Marpa Chökyi Lodrö, in Lhodrak Chukhyer in the southern part of Tibet, to an affluent family, he began studying at a young age but was wild and untamed compared to other children. Marpa first received instruction for three years at Mangkhar with Drokmi Shakya Yeshe and mastered Sanskrit. He decided to travel to India to study with renowned Indian Buddhist masters. His first trip to Nepal and India was in the company of Nyo Lōtsawa, whom he attended as a servant. Marpa returned home to Lhodrak and converted his entire inheritance into gold to fund his travel expenses and to make offerings to teachers.

Marpa journeyed first to Nepal where he studied with Paindapa and Chitherpa, two famous students of Naropa. Paindapa later accompanied Marpa to Pullahari, near Nalanda University, where Naropa taught. Marpa spent twelve years studying with Naropa and other great Indian gurus, most prominently Maitripada. After twelve years he set forth on his journey back to Tibet to teach and continue his dharma activities. Marpa was to travel to India twice more and Nepal three more times and studied with Naropa and other great teachers including Maitripa. He is said to have stayed in the cave at Phugtal Monastery. On his third visit to India, Naropa, who was engaged in tantric practices, proved difficult to find. However eventually Marpa found him and received the final teachings and instructions from Naropa. It was then that Naropa prophesied that a family lineage would not continue for Marpa, but that his lineage would be carried on by his disciples. Marpa now had received the full transmission, so Naropa formally declared Marpa to be his successor although he had other major disciples including Paindapa, Chitherpa, Shri Shantibhadra or Kukuripa, and Maitripa.

Upon his return to Tibet, Marpa spent many years translating Buddhist scriptures and made a major contribution to the transmission of the complete buddhadharma to Tibet. Marpa continued to practice and give teachings and transmissions to many students in Tibet. After his second visit to India Milarepa became his disciple. After the death of Marpa's son, Darma Dode, Milarepa inherited his lineage in full. Marpa lived with his wife Dakmema and their sons in Lhodrak in the southern part of Tibet. Marpa is said to have founded Stongdey Monastery in Zanskar in 1052 CE.

Marpa's lineage emphasized the paired systems of the "Six Teachings of Naropa" and the "Great Seal" (Mahamudra). The "Six Teachings of Naropa" included refining techniques of lucid dreaming.

title: Marpa Lotsawa


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Spaarndam

Spaarndam is a village in the province of North Holland, the Netherlands, on the Spaarne river and the IJ lake. The oldest part of the village, on the western side of the Spaarne, belongs to the municipality of Haarlem; the newer part on the eastern side is a part of the municipality of Haarlemmermeer. The village is built around a dam in the river, which is also the division line of the two municipalities.

Spaarndam was created around a dam where the river Spaarne flows into the IJ. This dam was built here by count Floris V of Holland in 1285. The village collected tolls at this dam, and people made their living from fishing. From 1812 to 1927, the western part of Spaarndam was an independent municipality.

The village has some tourism, and many people commute to Amsterdam and Haarlem.

Spaarndam has always been strongly connected to water. Besides the river and the fishing, it is also famous because of a story within the 1865 American novel, Hans Brinker or the Silver Skates, about a Dutch boy, "The Hero of Haarlem," who stuck his finger in a dike to prevent the town from flooding. The story became a popular legend in America, and in 1950 the local tourist bureau put a statue of this character in Spaarndam.

title: Spaarndam


text: Category:Proverbs

title: Category:Proverbs


text: Category:Histology

title: Category:Histology