View Matrix and Look-At

Purpose

The view matrix transforms world coordinates to camera coordinates:

• Camera at origin
• Looking down $-z$ axis
• $+y$ up, $+x$ right

Standard Camera Assumptions

For uniform projection mathematics:

• Camera at origin $(0, 0, 0)$
• Screen centre at $(0, 0, -d)$
• Screen parallel to $xy$-plane

Look-At Construction

Given:

• Camera position $\mathbf{c}$
• Look-at point $\mathbf{l}$ (where camera points)
• Up direction $\mathbf{u}$ (which way is “up”)

Build Orthonormal Basis

1. Forward: $\hat{\mathbf{v}} = \frac{\mathbf{c} - \mathbf{l}}{|\mathbf{c} - \mathbf{l}|}$

2. Right: $\hat{\mathbf{r}} = \frac{\mathbf{u} \times \hat{\mathbf{v}}}{|\mathbf{u} \times \hat{\mathbf{v}}|}$

3. True up: $\hat{\mathbf{u}} = \hat{\mathbf{v}} \times \hat{\mathbf{r}}$

View Matrix

$\mathbf{V} = \begin{bmatrix} \hat{r}_x & \hat{r}_y & \hat{r}_z & -\mathbf{c} \cdot \hat{\mathbf{r}} \\ \hat{u}_x & \hat{u}_y & \hat{u}_z & -\mathbf{c} \cdot \hat{\mathbf{u}} \\ \hat{v}_x & \hat{v}_y & \hat{v}_z & -\mathbf{c} \cdot \hat{\mathbf{v}} \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Coordinate System Conventions

Space	Handedness	z-axis
Camera (OpenGL)	Right-handed	Look down $-z$
NDC	Left-handed	$+z$ into screen

The projection matrix flips the handedness.

Summary

• View matrix places camera at origin, looking down $-z$
• Constructed from position, look-at point, and up vector
• Build orthonormal basis using cross products
• Translation to origin + rotation to align axes

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Past Paper Questions

2024 Paper 3 Question 4(b): Derive the look-at view matrix for a camera at position $\mathbf{c}$, looking at point $\mathbf{l}$, with up vector $\mathbf{u}$.