commit aa3e8755304f3f07979d434e3e24dc5f6f49bd35
parent b35ede9814395ac0d8e4533d6defe17faf438417
Author: Christian Ermann <christianermann@gmail.com>
Date: Mon, 10 Oct 2022 18:26:52 -0400
Added camera look-at post
Diffstat:
1 file changed, 44 insertions(+), 0 deletions(-)
diff --git a/content/posts/look-at.md b/content/posts/look-at.md
@@ -0,0 +1,44 @@
+---
+title: "Understanding Camera Look-At."
+date: 2022-10-10T00:31:00-04:00
+draft: false
+tags: ["Graphics", "Matrices", "OpenGL"]
+math: true
+---
+
+Often times, we want to position the camera in our scene at a particular point,
+such as the head of the character in a first-person shooter or the cockpit of a
+plane in a flight simulator. We also want the camera to look in the direction
+that the player is trying to move.
+
+To do this, we need to find the forward, right, and up vectors that correspond
+to the desired view.
+
+If we let $\vec{p}$ be the camera's position and $\vec{t}$ be the position of
+what we want to look at, the camera's look direction is
+$$\vec{l}=\vec{t}-\vec{p}\\,.$$
+Conventionally, the camera looks down the negative forward axis so
+$$\vec{f}=-\vec{l}\\,.$$
+
+In order to determine the right vector, we need a reference up direction,
+$\vec{up}$ (which isn't necessarily orthogonal to $\vec{f}$). This enables us
+to calculate
+$$\vec{r} = \vec{f} \times \vec{up}$$
+and
+$$\vec{u} = \vec{r} \times \vec{f}\\,.$$
+
+Now we can use the camera position $\vec{p}$ and normalized forms of $\vec{f}$,
+$\vec{r}$, and $\vec{u}$ to fill in our
+[view matrix]({{< relref "view-matrix" >}}):
+
+$$
+ V
+ =
+ \begin{bmatrix}
+ r_{x} & r_{y} & r_{z} & -(\vec{r} \cdot \vec{p}) \\\
+ u_{x} & u_{y} & u_{z} & -(\vec{u} \cdot \vec{p}) \\\
+ f_{x} & f_{y} & f_{z} & -(\vec{f} \cdot \vec{p}) \\\
+ 0 & 0 & 0 & 1
+ \end{bmatrix}
+$$
+
