This chapter visualizes 3D linear transformations through the unit cube. Scaling changes coordinate lengths independently, while shear keeps some coordinates fixed and adds controlled multiples of other coordinates. The matrices are shown together with before-and-after cube views so the algebra and geometry can be read side by side.
In 3D, a matrix can be read as a rule for changing the three coordinates of every point.
The goal is to make the entries of a 3×3 transformation matrix visible as concrete changes in a 3D shape.
A 3D shear keeps most coordinates unchanged and replaces one coordinate by that coordinate plus a multiple of another. For example, shearing x₁ by x₂ sends x₁′ = x₁ + sx₂ while x₂ and x₃ stay fixed. Geometrically, the cube tilts, but the origin remains fixed and parallel lines remain parallel.
The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.
| Topic | Pages |
|---|---|
| Scaling | 1–2 |
| Elementary shear of x₁ by x₂ | 2–3 |
| Elementary shear of x₁ by x₃ | 3–4 |
| Elementary shear of x₂ by x₁ | 4–5 |
| Elementary shear of x₂ by x₃ | 6–7 |
| Elementary shear of x₃ by x₁ | 7–8 |
| Elementary shear of x₃ by x₂ | 8–9 |
| Shear x₃ by x₁ and x₂ | 10–11 |
| Shear x₂ by x₁ and x₃ | 11–12 |
| Shear x₁ by x₂ and x₃ | 12–13 |
A shear is still linear, so it sends lines to lines and keeps the origin fixed. But it mixes coordinates, so points farther in one direction are shifted more in another direction. That coordinate mixing turns the cube into a tilted parallelepiped.