Orthographic

Projectors are perpendicular to the view plane. No foreshortening along axes parallel to the plane.

Multiview

Views generated by projecting along the principal axes:

• Front view (project along +Z): (x, y, z) -> (x, y).
• Top view (project along -Y): (x, y, z) -> (x, z).
• Side view (project along -X): (x, y, z) -> (z, y).

These three together (plus optional rear, bottom, other side) form first-angle or third-angle engineering drawings, the lingua franca of mechanical design.

Axonometric

The view plane is not perpendicular to a principal axis, so all three axes appear in the projection. This gives a sense of 3D while keeping parallel lines parallel.

Three Subtypes

Isometric Detail

The standard isometric projection rotates the world so each axis makes equal angles (~120 degrees) with each other axis on the projection plane. The actual foreshortening factor is cos(35.26 deg) ~= 0.8165 along each axis. In retro game art ("isometric" pixel games), the simplified 2:1 dimetric is often used because aligning to integer pixels is easier.

Matrix Sketch (Isometric)

Apply rotations Rx and Ry to align object axes equally to the view plane, then orthographic project: M_iso = OrthoZ Rx(35.26 deg) Ry(45 deg).

Worked Mini-Example

Cube with vertices (0,0,0) and (1,1,1).

• Front view: (0,0) and (1,1).
• Top view: (0,0) and (1,1).
• Isometric (after rotation): edges appear at 30 deg above horizontal; cube has 6 visible edges forming a hexagon outline.

Pros and Cons

• Pros: Measurements preserved (great for blueprints and pixel art games).
• Cons: No depth cue from foreshortening; objects of the same size look the same regardless of distance, which can mislead the eye.

Where You See It

• Architectural plans (multiview) and CAD model documentation.
• Strategy games (XCOM, Civilization isometric tiles) and pixel art.
• Technical illustrations in engineering manuals.