1. rcrnstn/glslisometry
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Add implementation

Robert Cranston authored on 21/12/2021 12:41:46
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README.md
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 A [GLSL][]/[OpenGL][] [\>=2.1][] [isometry][] matrix library.
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+If a `mat4` matrix represents an [isometry][], which in this case means that it
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+encodes only rotations and translations (no scaling or shearing, and a
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+projective `w` entry equal to `1`), it is easy and performant to extract those
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+rotations and translations, and to compute the [inverse][]. This library
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+provides functions to do so.
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+
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 [`gliso`]: https://git.rcrnstn.net/rcrnstn/gliso
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 [GLSL]: https://en.wikipedia.org/wiki/OpenGL_Shading_Language
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 [OpenGL]: https://en.wikipedia.org/wiki/OpenGL
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 [\>=2.1]: https://en.wikipedia.org/wiki/OpenGL#Version_history
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 [isometry]: https://en.wikipedia.org/wiki/Isometry
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+[inverse]: https://en.wikipedia.org/wiki/Invertible_matrix
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+
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+## Requirements
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+
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+Support for `#include` directives is required. This can be provided by e.g. the
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+standardized [`ARB_shading_language_include`][] extension or some third party
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+library.
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+
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+[`ARB_shading_language_include`]: https://www.khronos.org/registry/OpenGL/extensions/ARB/ARB_shading_language_include.txt
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+
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+## Usage
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+
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+Link shader programs with the provided `iso.glsl`.
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+
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+Include the provided `iso.h` from shaders. It declares the following functions:
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+
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+-   `mat{3,4} isorot{inv}{3,4}(mat4 iso)`
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+
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+    Extract (inverse) rotation.
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+
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+-   `vec{3,4} isotrans{inv}{3,4}(mat4 iso)`
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+
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+    Extract (inverse) translation.
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+
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+    The `z` components of the `4` versions are set to `0`.
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+
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+-   `mat4 isoinv(mat4 iso)`
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+
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+    Compute inverse.
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+
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+## Theory
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+
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+For simplicity, we use the same name for both the [projective][] and
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+[Euclidean][] versions of vectors and transformations below.
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+
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+Assume $M$ is an isometry,
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+
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+$$
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+M v
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+=
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+\begin{pmatrix}
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+    r_{xx} & r_{yx} & r_{zx} & t_x \\
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+    r_{xx} & r_{yx} & r_{zx} & t_y \\
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+    r_{xx} & r_{yx} & r_{zx} & t_z \\
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+         0 &      0 &      0 &   1 \\
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+\end{pmatrix}
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+\begin{pmatrix}
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+    v_x \\
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+    v_y \\
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+    v_z \\
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+      1 \\
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+\end{pmatrix}
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+=
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+R v + t
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+$$
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+
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+It is trivial to extract the rotation $R$ and translation $t$ directly from the
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+components of $M$.
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+
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+Since the rotation part $R$ is [orthonormal][], its inverse is its own
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+transpose. The inverse of the translation is its own negation.
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+
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+$$
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+\begin{aligned}
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+    R^{-1} &= R^T \\
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+    t^{-1} &= -t \\
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+\end{aligned}
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+$$
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+
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+Assuming the $3 \mathsf{x} 3$ sub-matrix of $M^{-1}$ is $R^{-1}$, finding the
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+full inverse is simply a matter of finding $M^{-1}$'s fourth column $m^{-1}$:
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+
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+$$
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+\begin{aligned}
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+    v
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+    &= M^{-1} M v \\
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+    &= R^{-1} (R v + t) + m^{-1} \\
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+    &= v + R^{-1} t + m^{-1} \\
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+    &\iff \\
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+    m^{-1} &= -R^{-1} t = R^{-1} t^{-1} \\
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+\end{aligned}
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+$$
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+
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+Computing this is much faster than the built-in, general, [`inverse`][].
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+
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+[projective]: https://en.wikipedia.org/wiki/Homogeneous_coordinates
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+[Euclidean]: https://en.wikipedia.org/wiki/Cartesian_coordinates
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+[orthonormal]: https://en.wikipedia.org/wiki/Orthogonal_matrix
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+[`inverse`]: https://www.khronos.org/registry/OpenGL-Refpages/gl4/html/inverse.xhtml
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 ## Build system
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assets/shaders/iso.glsl
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+#version 120
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+
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+
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+mat3 isorot3     (mat4 iso) { return mat3(iso); }
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+vec3 isotrans3   (mat4 iso) { return iso[3].xyz; }
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+
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+mat3 isorotinv3  (mat4 iso) { return transpose(isorot3(iso)); }
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+vec3 isotransinv3(mat4 iso) { return -isotrans3(iso); }
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+
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+mat4 isorot4     (mat4 iso) { return mat4(isorot3     (iso)); }
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+mat4 isorotinv4  (mat4 iso) { return mat4(isorotinv3  (iso)); }
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+vec4 isotrans4   (mat4 iso) { return vec4(isotrans3   (iso), 0); }
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+vec4 isotransinv4(mat4 iso) { return vec4(isotransinv3(iso), 0); }
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+
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+mat4 isoinv(mat4 iso)
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+{
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+    // mat4 inv = isorotinv4(iso);
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+    // inv[3] += inv * isotransinv4(iso);
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+    mat3 rotinv = isorotinv3(iso);
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+    mat4 inv = mat4(rotinv);
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+    inv[3].xyz = rotinv * isotransinv3(iso);
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+    return inv;
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+}
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+
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+
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+#ifdef GLSLRUN
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+void main()
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+{
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+    float phi = 1;
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+    mat4 iso = transpose(mat4(
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+        +cos(phi), -sin(phi), 0, 1,
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+        +sin(phi), +cos(phi), 0, 2,
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+        0,         0,         1, 3,
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+        0,         0,         0, 1
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+    ));
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+    vec4 vec = vec4(4, 5, 6, 1);
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+
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+    cout
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+        << iso * vec << endl
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+        << isorot4(iso) *      vec  + isotrans4(iso) << endl
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+        << isorot3(iso) * vec3(vec) + isotrans3(iso) << endl;
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+
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+    cout
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+        << inverse(iso) << endl
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+        << isoinv (iso) << endl;
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+}
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+#endif // GLSLRUN
assets/shaders/iso.h
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+mat3 isorot3(mat4 iso);
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+mat4 isorot4(mat4 iso);
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+
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+mat3 isorotinv3(mat4 iso);
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+mat4 isorotinv4(mat4 iso);
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+
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+vec3 isotrans3(mat4 iso);
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+vec4 isotrans4(mat4 iso);
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+
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+vec3 isotransinv3(mat4 iso);
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+vec4 isotransinv4(mat4 iso);
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+
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+mat4 isoinv(mat4 iso);