Definition

An oblique projection is a parallel projection where the projectors are not perpendicular to the view plane. Faces parallel to the view plane keep their true size and shape; depth lines recede at an angle.

Setup

Suppose the view plane is the z = 0 plane and the projection direction makes angle alpha with the plane (and angle phi in the plane). A point (x, y, z) maps to: x' = x + z L cos(phi) y' = y + z L sin(phi) where L = cot(alpha) is the depth scale.

Cavalier

Depth scale L = 1. Lines along z preserve length.

• Pros: simple to draw; orthogonal vertices align cleanly.
• Cons: appears elongated and unnatural for cubes.
• Standard receding angle phi = 30 or 45 degrees.

Cabinet

Depth scale L = 0.5. Depth is halved, simulating roughly the way furniture catalogs depict cabinets.

• Pros: looks more realistic than cavalier; minimal distortion.
• Cons: not a true projection of any single physical viewpoint - depth measurements are not literal.

Comparison Table

Worked Example

Cube with side 1 at origin. Oblique with phi = 45, cabinet (L = 0.5).

• (0,0,0) -> (0, 0).
• (1,0,0) -> (1, 0).
• (1,1,0) -> (1, 1).
• (0,1,0) -> (0, 1).
• (0,0,1) -> (0.5cos45, 0.5sin45) = (0.354, 0.354).
• (1,0,1) -> (1 + 0.354, 0.354) = (1.354, 0.354).

The cube becomes a hexagonal silhouette in the plane.

Matrix Form

M_oblique = [ 1 0 Lcos(phi) 0 ] [ 0 1 Lsin(phi) 0 ] [ 0 0 0 0 ] [ 0 0 0 1 ]

When to Use

Oblique projections are not common in modern 3D engines (which use perspective). They survive in:

• Educational diagrams to depict 3D shape with simple geometry.
• Vintage video games (Q*bert, SimCity 1) and pixel-art style.
• Architectural sketches.

Caveats

Oblique projection is a parallel projection - parallel lines stay parallel, no foreshortening except along the depth axis. Combining oblique projection with perspective is unusual and produces visual oddities.