Order Matters

Matrix multiplication is associative but not commutative in general. Translating then rotating is different from rotating then translating.

Composition Rule (column-vector convention)

If a point is transformed first by A, then by B, then by C, the combined matrix is M = C B A. Read right to left: closest to P is applied first.

Common Composite: Rotate about Arbitrary Pivot (px, py)

M = T(px,py) R(theta) T(-px,-py). Walk through:

1. Translate so pivot moves to origin.
2. Rotate.
3. Translate back so pivot returns to (px, py).

Common Composite: Scale about Arbitrary Pivot

M = T(px,py) S(sx,sy) T(-px,-py).

Common Composite: Reflect Across Arbitrary Line ax+by+c=0

1. Translate so line passes through origin.
2. Rotate so line aligns with X-axis.
3. Reflect across X-axis.
4. Inverse rotate.
5. Inverse translate.

Numerical Example

Rotate point (5, 0) by 90 deg about pivot (2, 0): M = T(2,0) R(90) T(-2,0).

• T(-2,0): (5,0) -> (3,0).
• R(90): (3,0) -> (0,3).
• T(2,0): (0,3) -> (2,3). Final answer: (2, 3).

If we naively rotated first then translated: R(90)(5,0)=(0,5); T(2,0)(0,5)=(2,5). Different result -> shows non-commutativity.

Why Compose at All

Composing into one matrix means each vertex is transformed by one 3x3 (or 4x4) multiplication regardless of how many logical operations you wanted. For modeling hierarchies (a hand attached to an arm attached to a torso), each child concatenates its local matrix with its parent's, producing a single world-space matrix per object.

Inverse and Identity

• Identity I leaves points unchanged; M * I = M.
• Inverse undoes a transform; useful for converting between coordinate systems.
• Inverses: T^-1(tx,ty) = T(-tx,-ty); R^-1(theta) = R(-theta); S^-1(sx,sy) = S(1/sx, 1/sy) (sx, sy != 0).