From 125f059267830d8f81694edb3ca83c6257f608f3 Mon Sep 17 00:00:00 2001
From: yum <yum.food.vr@gmail.com>
Date: Wed, 14 Jan 2026 14:31:13 -0800
Subject: begin work on hemi octahedral article

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+# hemi-octahedral impostors
+14 Jan 2026
+
+Ryan Brucks published [an
+article](https://shaderbits.com/blog/octahedral-impostors) describing
+"octahedral impostors" in 2018. The basic idea is to to take photos of some
+subject at octahedral lattice points, record them to an atlas, then reconstruct
+those photos in a particle.
+
+![Octahedral lattice points around some object.](./images/2026_01_14/Screenshot from 2026-01-14
+13-35-40.png)
+
+Why octahedrons? The octahedral mapping is simply one way to convert between a
+flat coordinate system and a spherical coordinate system. It is notable because
+it does not use any trig functions, making it suitable for use in realtime
+graphics.
+
+This is what an octahedron looks like:
+
+![Unit octahedron.](./images/2026_01_14/Screenshot from 2026-01-14
+13-54-05.png)
+
+It is a polyhedron with 8 triangular faces and 6 vertices. The equator is a
+square.
+
+Let's work out how we'd convert this octahedron to a plane.
+First, we project the upper hemisphere onto the xz plane:
+
+![Octahedron with upper hemisphere projected onto xz
+plane.](./images/2026_01_14/Screenshot from 2026-01-14 13-50-27.png)
+
+Next, we effectively need to "rotate" the triangles in the lower half around
+those diagonal edges. We can cheat by first *reflecting* the bottom vertex of each
+triangle about its diagonal edge:
+
+![Octahedron with reflected lower hemisphere.](./images/2026_01_14/Screenshot from 2026-01-14 13-58-18.png)
+
+Finally, we can just project those points in the lower hemisphere onto the xz
+plane:
+
+![Fully unwrapped octahedron.](./images/2026_01_14/Screenshot from 2026-01-14 13-59-24.png)
+
+Viewed head on, we can see a very beautifully symmetric unwrapping:
+
+![Unwrapped octahedron, head on.](./images/2026_01_14/Screenshot from 2026-01-14 14-00-16.png)
+
+Note that we never actually did any rotations, so there's no trig! Here's the
+same procedure in code:
+
+```c
+// Convert unit octahedron to a [-1,1] x [-1,1] patch on xz plane.
+float3 octahedron_to_plane(float3 p) {
+  if (p.y >= 0) {
+    // Project upper hemispher onto xz plane.
+    p.y = 0;
+    return p;
+  }
+  // First, reflect the lower hemisphere's points about their diagonal.
+  p.x = sign(p.x) * (1 - abs(p.x));
+  p.z = sign(p.z) * (1 - abs(p.z));
+  // Then project onto the xz plane.
+  p.y = 0;
+  return p;
+}
+```
+
+We can generalize this procedure to unwrap *any* spherical object by just
+switching norms:
+
+```c
+// Convert unit sphere to a [-1,1] x [-1,1] patch on xz plane.
+float3 octahedron_to_plane(float3 p) {
+  // Switch from L2 to L1 norm. This basically bends a sphere to an octahedron.
+  float l1_norm = abs(p.x) + abs(p.y) + abs(p.z);
+  p /= l1_norm;
+  // Then unwrap.
+  if (p.y >= 0) {
+    // Project upper hemispher onto xz plane.
+    p.y = 0;
+    return p;
+  }
+  // First, reflect the lower hemisphere's points about their diagonal.
+  p.x = sign(p.x) * (1 - abs(p.x));
+  p.z = sign(p.z) * (1 - abs(p.z));
+  // Then project onto the xz plane.
+  p.y = 0;
+  return p;
+}
+```
+
+Here's a quick demo showing what that norm conversion does to a unit sphere:
+
+![Converting a sphere to an octahedron via norm conversion.](./images/2026_01_14/hemi_octahedral_04.mp4)
+
+If you'd like more discussion on this topic, I recommend the spherical geometry section in [the PBR book](https://www.pbr-book.org/4ed/Geometry_and_Transformations/Spherical_Geometry#x3-OctahedralEncoding).)
+
 # 6 wave dispersion relations with derivatives
 21 Sep 2025
 
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