Orthographic Projection

Parallel Projectors

All projectors (projection lines) are parallel to each other:

• No vanishing point
• Parallel lines remain parallel
• No perspective distortion

Mathematical Form

Simply discard the $z$ coordinate:

$(x, y, z) \rightarrow (x, y)$

Matrix Form

Orthographic projection maps a box to the NDC cube $[-1, 1]^3$:

$\mathbf{P}_{\text{ortho}} = \begin{bmatrix} \frac{2}{r-l} & 0 & 0 & -\frac{r+l}{r-l} \\ 0 & \frac{2}{t-b} & 0 & -\frac{t+b}{t-b} \\ 0 & 0 & \frac{2}{f-n} & -\frac{f+n}{f-n} \\ 0 & 0 & 0 & 1 \end{bmatrix}$

where $l, r, b, t, n, f$ are left, right, bottom, top, near, far planes.

Properties

• Preserves measurements (no perspective distortion)
• Parallel lines stay parallel
• Used in CAD, architecture, technical drawings
• Looks “flat” and unrealistic for natural scenes

When to Use

• Technical drawings where accuracy matters
• 2D games and UI
• Isometric view games
• Architectural plans

Summary

• Orthographic discards $z$: parallel projection
• No perspective: distant objects same size
• Preserves parallel lines and measurements
• Used for technical applications