How Perspective Works

Perspective projection mirrors how lenses (and eyes) work: parallel lines that recede from the viewer converge at one or more vanishing points on the horizon. Objects further from the eye appear smaller (foreshortening proportional to 1/z).

Number of Vanishing Points

• 1-Point: View plane is parallel to two principal axes. Lines along the third axis meet at one vanishing point. Example: looking straight down a railroad track or a hallway.
• 2-Point: View plane is parallel to one principal axis (typically Y, vertical). The other two axes each yield a vanishing point on the horizon. Example: standing on a sidewalk looking at a corner of a building.
• 3-Point: View plane is not parallel to any principal axis. Three vanishing points - one on the horizon for each receding direction, plus a third above (looking up) or below (looking down). Example: photographing a skyscraper from street level (vertical lines converge upward).

Math: Perspective Matrix (COP at origin, plane at z = -n)

M_per = [ n 0 0 0 ] [ 0 n 0 0 ] [ 0 0 -(f+n)/(f-n) -2fn/(f-n) ] [ 0 0 -1 0 ] where n and f are near and far clipping plane distances. After multiplying [x,y,z,1]^T we divide x and y by w (= -z) -> the perspective divide that creates foreshortening.

In simpler 1-point form on plane z = d with COP at origin: x' = d x / z y' = d y / z

Field of View

The horizontal FOV theta is set by the aspect ratio and focal length: tan(theta/2) = (image_width/2) / d. Wide FOV (90+) feels distorted at edges; narrow FOV (~30) looks zoomed-in. Cinematic FOV is around 50-60.

Worked Example

Camera at origin, plane at d = 5. Cube with corners (1,1,5) and (1,1,15).

• Corner at z=5: x' = 5*1/5 = 1, y' = 1.
• Corner at z=15: x' = 5*1/15 = 0.333, y' = 0.333.

The corner three times further appears at one third the apparent size - exactly 1/z scaling.

Comparison

Pros and Cons of Perspective

• Pros: Realistic depth perception; matches photographic and human vision.
• Cons: Distorts measurements (parallel lines bend), making it unsuitable for engineering documentation; objects directly between viewer and frustum walls can be partially clipped strangely.

Far/Near Plane Tradeoffs

• Near plane too small -> z-fighting (depth precision loss) on far geometry.
• Far plane too large -> z-buffer wastes precision on emptiness.
• Common rule: keep far/near ratio under ~1000 for 24-bit depth buffer.