christianermann.dev-hugo

commit 4a539c327ac93e063ee59f4ccb02f36bf50dc0a1
parent 3f35cd53900c2d982af5c21fb866f2895779363b
Author: Christian Ermann <christianermann@gmail.com>
Date:   Tue, 30 Nov 2021 19:22:44 -0800

Wrote Deriving the View Matrix article

Diffstat:
A
content/posts/view_matrix.md
 | 
160
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

1 file changed, 160 insertions(+), 0 deletions(-)
diff --git a/content/posts/view_matrix.md b/content/posts/view_matrix.md
@@ -0,0 +1,160 @@
+---
+title: "Deriving the View Matrix"
+date: 2021-11-21T13:34:08-08:00
+draft: false
+tags: ["Graphics", "Matrices", "OpenGL"]
+math: true
+---
+
+In computer graphics, the view matrix is used to transform points from
+world space into camera space. Below is an explanation of how to derive this
+matrix.
+
+Let $V$ be the view matrix.
+If $\vec{r}$, $\vec{u}$, and $\vec{f}$ denote the right, up, and forward
+vectors of the camera and $\vec{p}$ is the camera's position, $V$ should
+satisfy the following properties:
+$$
+    \begin{matrix}
+        V_{1*} \times
+        \begin{bmatrix}
+            r_{x} \\\
+            r_{y} \\\
+            r_{z} \\\
+            0
+        \end{bmatrix}
+        =
+        \begin{bmatrix}
+            1 \\\
+            0 \\\
+            0 \\\
+            0
+        \end{bmatrix},&
+        V_{2*} \times
+        \begin{bmatrix}
+            u_{x} \\\
+            u_{y} \\\
+            u_{z} \\\
+            0
+        \end{bmatrix}
+        =
+        \begin{bmatrix}
+            0 \\\
+            1 \\\
+            0 \\\
+            0
+        \end{bmatrix},&
+        V_{3*} \times
+        \begin{bmatrix}
+            f_{x} \\\
+            f_{y} \\\
+            f_{z} \\\
+            0
+        \end{bmatrix}
+        =
+        \begin{bmatrix}
+            0 \\\
+            0 \\\
+            1 \\\
+            0
+        \end{bmatrix},&
+        V_{4*} \times
+        \begin{bmatrix}
+            p_{x} \\\
+            p_{y} \\\
+            p_{z} \\\
+            1
+        \end{bmatrix}
+        =
+        \begin{bmatrix}
+            0 \\\
+            0 \\\
+            0 \\\
+            1
+        \end{bmatrix}
+    \end{matrix}
+$$
+In words, the view matrix should translate the camera to the origin and rotate
+it to align with the $\vec{x}$, $\vec{y}$, and $\vec{z}$ axes.
+
+This can be expressed more concisely as the product of two matrices:
+$$
+    V \times
+    \begin{bmatrix}
+        r_{x} & u_{x} & f_{x} & p_{x} \\\
+        r_{y} & u_{y} & f_{y} & p_{y} \\\
+        r_{z} & u_{z} & f_{z} & p_{z} \\\
+        0     & 0     & 0     & 1
+    \end{bmatrix}
+    =
+    \begin{bmatrix}
+        1 & 0 & 0 & 0 \\\
+        0 & 1 & 0 & 0 \\\
+        0 & 0 & 1 & 0 \\\
+        0 & 0 & 0 & 1
+    \end{bmatrix}
+$$
+
+To make this equation easier to solve, we can split the matrix containing our
+camera parameters into the product of a translation matrix $T$ and a rotation
+matrix $R$:
+$$
+    V \times (T \times R) = I
+$$
+where
+$$
+    \begin{matrix}
+        T =
+        \begin{bmatrix}
+            1 & 0 & 0 & p_{x} \\\
+            0 & 1 & 0 & p_{y} \\\
+            0 & 0 & 1 & p_{z} \\\
+            0 & 0 & 0 & 1
+        \end{bmatrix},&
+        R =
+        \begin{bmatrix}
+            r_{x} & u_{x} & f_{x} & 0 \\\
+            r_{y} & u_{y} & f_{y} & 0 \\\
+            r_{z} & u_{z} & f_{z} & 0 \\\
+            0     & 0     & 0     & 1
+        \end{bmatrix}
+    \end{matrix}
+$$
+
+Solving for $V$ reveals that $V=R^{-1} \times T^{-1}$:
+$$
+    V =
+    \begin{bmatrix}
+        r_{x} & r_{y} & r_{z} & 0 \\\
+        u_{x} & u_{y} & u_{z} & 0 \\\
+        f_{x} & f_{y} & f_{z} & 0 \\\
+        0     & 0     & 0     & 1
+    \end{bmatrix}
+    \times
+    \begin{bmatrix}
+        1 & 0 & 0 & -p_{x} \\\
+        0 & 1 & 0 & -p_{y} \\\
+        0 & 0 & 1 & -p_{z} \\\
+        0 & 0 & 0 & 1
+    \end{bmatrix}
+$$
+
+And with some minor simplification,
+$$
+    V
+    =
+    \begin{bmatrix}
+        r_{x} & r_{y} & r_{z} & -(\vec{r} \cdot \vec{p}) \\\
+        u_{x} & u_{y} & u_{z} & -(\vec{u} \cdot \vec{p}) \\\
+        f_{x} & f_{y} & f_{z} & -(\vec{f} \cdot \vec{p}) \\\
+        0     & 0     & 0     & 1
+    \end{bmatrix}
+$$
+
+When we're ready to render our scene, we can apply $V$ to each vertex $\vec{q}$
+by calculating
+$$
+    V \times \vec{q}
+$$
+in our vertex shader.
+