Add "Converting from 3D to 2D" blog post

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 <?xml version='1.0' encoding='UTF-8'?>
-<rss xmlns:atom="http://www.w3.org/2005/Atom" version="2.0"><channel><title>Fryzek Concepts</title><atom:link href="https://fryzekconcepts.com/feed.xml" rel="self" type="application/rss+xml"/><link>https://fryzekconcepts.com</link><description>Lucas is a developer working on cool things</description><lastBuildDate>Mon, 22 May 2023 23:23:21 -0000</lastBuildDate><item><title>2022 Graphics Team Contributions at Igalia</title><link>https://fryzekconcepts.com/notes/2022_igalia_graphics_team.html</link><description>&lt;p&gt;This year I started a new job working with &lt;a
+<rss xmlns:atom="http://www.w3.org/2005/Atom" version="2.0"><channel><title>Fryzek Concepts</title><atom:link href="https://fryzekconcepts.com/feed.xml" rel="self" type="application/rss+xml"/><link>https://fryzekconcepts.com</link><description>Lucas is a developer working on cool things</description><lastBuildDate>Mon, 25 Sep 2023 19:42:55 -0000</lastBuildDate><item><title>2022 Graphics Team Contributions at Igalia</title><link>https://fryzekconcepts.com/notes/2022_igalia_graphics_team.html</link><description>&lt;p&gt;This year I started a new job working with &lt;a
 href="https://www.igalia.com/technology/graphics"&gt;Igalia’s Graphics
 Team&lt;/a&gt;. For those of you who don’t know &lt;a
 href="https://www.igalia.com/"&gt;Igalia&lt;/a&gt; they are a &lt;a
@@ -552,4 +552,116 @@ href="https://gitlab.freedesktop.org/mesa/mesa/-/merge_requests/21558"&gt;https:
 &lt;li&gt;&lt;a
 href="https://gitlab.freedesktop.org/mesa/mesa/-/merge_requests/20180"&gt;https://gitlab.freedesktop.org/mesa/mesa/-/merge_requests/20180&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
-</description><pubDate>Thu, 11 May 2023 04:00:00 -0000</pubDate><guid>https://fryzekconcepts.com/notes/mesa_23_1_contributions_behind_the_scenes.html</guid></item></channel></rss>
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+</description><pubDate>Thu, 11 May 2023 04:00:00 -0000</pubDate><guid>https://fryzekconcepts.com/notes/mesa_23_1_contributions_behind_the_scenes.html</guid></item><item><title>Converting from 3D to 2D</title><link>https://fryzekconcepts.com/notes/converting_from_3d_to_2d.html</link><description>&lt;p&gt;Recently I’ve been working on a project where I needed to convert an
+application written in OpenGL to a software renderer. The matrix
+transformation code in OpenGL made use of the GLM library for matrix
+math, and I needed to convert the 4x4 matrices to be 3x3 matrices to
+work with the software renderer. There was some existing code to do this
+that was broken, and looked something like this:&lt;/p&gt;
+&lt;pre&gt;&lt;code&gt;glm::mat3 mat3x3 = glm::mat3(mat4x4);&lt;/code&gt;&lt;/pre&gt;
+&lt;p&gt;Don’t worry if you don’t see the problem already, I’m going to
+illustrate in more detail with the example of a translation matrix. In
+3D a standard translation matrix to translate by a vector
+&lt;code&gt;(x, y, z)&lt;/code&gt; looks something like this:&lt;/p&gt;
+&lt;pre&gt;&lt;code&gt;[1 0 0 x]
+[0 1 0 y]
+[0 0 1 z]
+[0 0 0 1]&lt;/code&gt;&lt;/pre&gt;
+&lt;p&gt;Then when we multiply this matrix by a vector like
+&lt;code&gt;(a, b, c, 1)&lt;/code&gt; the result is
+&lt;code&gt;(a + x, b + y, c + z, 1)&lt;/code&gt;. If you don’t understand why the
+matrix is 4x4 or why we have that extra 1 at the end don’t worry, I’ll
+explain that in more detail later.&lt;/p&gt;
+&lt;p&gt;Now using the existing conversion code to get a 3x3 matrix will
+simply take the first 3 columns and first 3 rows of the matrix and
+produce a 3x3 matrix from those. Converting the translation matrix above
+using this code produces the following matrix:&lt;/p&gt;
+&lt;pre&gt;&lt;code&gt;[1 0 0]
+[0 1 0]
+[0 0 1]&lt;/code&gt;&lt;/pre&gt;
+&lt;p&gt;See the problem now? The &lt;code&gt;(x, y, z)&lt;/code&gt; values disappeared!
+In the conversion process we lost these critical values from the
+translation matrix, and now if we multiply by this matrix nothing will
+happen since we are just left with the identity matrix. So if we can’t
+use this simple “cast” function in GLM, what can we use?&lt;/p&gt;
+&lt;p&gt;Well one thing we can do is preserve the last column and last row of
+the matrix. So assume we have a 4x4 matrix like this:&lt;/p&gt;
+&lt;pre&gt;&lt;code&gt;[a b c d]
+[e f g h]
+[i j k l]
+[m n o p]&lt;/code&gt;&lt;/pre&gt;
+&lt;p&gt;Then preserving the last row and column we should get a matrix like
+this:&lt;/p&gt;
+&lt;pre&gt;&lt;code&gt;[a b d]
+[e f h]
+[m n p]&lt;/code&gt;&lt;/pre&gt;
+&lt;p&gt;And if we use this conversion process for the same translation matrix
+we will get:&lt;/p&gt;
+&lt;pre&gt;&lt;code&gt;[1 0 x]
+[0 1 y]
+[0 0 1]&lt;/code&gt;&lt;/pre&gt;
+&lt;p&gt;Now we see that the &lt;code&gt;(x, y)&lt;/code&gt; part of the translation is
+preserved, and if we try to multiply this matrix by the vector
+&lt;code&gt;(a, b, 1)&lt;/code&gt; the result will be
+&lt;code&gt;(a + x, b + y, 1)&lt;/code&gt;. The translation is preserved in the
+conversion!&lt;/p&gt;
+&lt;h2 id="why-do-we-have-to-use-this-conversion"&gt;Why do we have to use
+this conversion?&lt;/h2&gt;
+&lt;p&gt;The reason the conversion is more complicated is hidden in how we
+defined the translation matrix and vector we wanted to translate. The
+vector was actually a 4D vector with the final component set to 1. The
+reason we do this is that we actually want to represent an affine space
+instead of just a vector space. An affine space being a type of space
+where you can have both points and vectors. A point is exactly what you
+would expect it to be just a point in space from some origin, and vector
+is a direction with magnitude but no origin. This is important because
+strictly speaking translation isn’t actually defined for vectors in a
+normal vector space. Additionally if you try to construct a matrix to
+represent translation for a vector space you’ll find that its impossible
+to derive a matrix to do this and that operation is not a linear
+function. On the other hand operations like translation are well defined
+in an affine space and do what you would expect.&lt;/p&gt;
+&lt;p&gt;To get around the problem of vector spaces, mathematicians more
+clever than I figured out you can implement an affine space in a normal
+vector space by increasing the dimension of the vector space by one, and
+by adding an extra row and column to the transformation matrices used.
+They called this a &lt;strong&gt;homogeneous coordinate system&lt;/strong&gt;. This
+lets you say that a vector is actually just a point if the 4th component
+is 1, but if its 0 its just a vector. Using this abstraction one can
+implement all the well defined operations for an affine space (like
+translation!).&lt;/p&gt;
+&lt;p&gt;So using the “homogeneous coordinate system” abstraction, translation
+is an operation that defined by taking a point and moving it by a
+vector. Lets look at how that works with the translation matrix I used
+as an example above. If you multiply that matrix by a 4D vector where
+the 4th component is 0, it will just return the same vector. Now if we
+multiply by a 4D vector where the 4th component is 1, it will return the
+point translated by the vector we used to construct that translation
+matrix. This implements the translation operation as its defined in an
+affine space!&lt;/p&gt;
+&lt;p&gt;If you’re interested in understanding more about homogeneous
+coordinate spaces, (like how the translation matrix is derived in the
+first place) I would encourage you to look at resources like &lt;a
+href="https://books.google.ca/books/about/Mathematics_for_Computer_Graphics_Applic.html?id=YmQy799flPkC&amp;amp;redir_esc=y"&gt;“Mathematics
+for Computer Graphics Applications”&lt;/a&gt;. They provide a much more
+detailed explanation than I am providing here. (The homogeneous
+coordinate system also has some benefits for representing projections
+which I won’t get into here, but are explained in that text book.)&lt;/p&gt;
+&lt;p&gt;Now to finally answer the question about why we needed to preserve
+those final columns and vectors. Based on what we now know, we weren’t
+actually just converting from a “3D space” to a “2D space” we were
+converting from a “3D homogeneous space” to a “2D homogeneous space”.
+The process of converting from a higher dimension matrix to a lower
+dimensional matrix is lossy and some transformation details are going to
+be lost in process (like for example the translation along the z-axis).
+There is no way to tell what kind of space a given matrix is supposed to
+transform just by looking at the matrix itself. The matrix does not
+carry any information about about what space its operating in and any
+conversion function would need to know that information to properly
+convert that matrix. Therefore we need develop our own conversion
+function that preserves the transformations that are important to our
+application when moving from a “3D homogeneous space” to a “2D
+homogeneous space”.&lt;/p&gt;
+&lt;p&gt;Hopefully this explanation helps if you are every working on
+converting 3D transformation code to 2D.&lt;/p&gt;
+</description><pubDate>Mon, 25 Sep 2023 04:00:00 -0000</pubDate><guid>https://fryzekconcepts.com/notes/converting_from_3d_to_2d.html</guid></item></channel></rss>
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