Compose Order Matters - Computer Graphics Step by Step

Compare two transform orders on the same triangle and see that the results differ.

compose-order-matters With column vectors, the rightmost matrix acts first. Scaling after rotation and rotating after scaling can produce different shapes.

Example

Compare S R and R S to see why transform order changes the result.

highlighted = computed this step

Step 1 — Set up

Set up rotation and scale; rightmost matrix acts first.

R=\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}\qquad S=\begin{bmatrix}2&0&0\\0&1&0\\0&0&1\end{bmatrix}

Step 2 — Compose both orders

Multiply in both orders; the products differ.

\begin{aligned}S R&=\begin{bmatrix}0&-2&0\\1&0&0\\0&0&1\end{bmatrix}\\R S&=\begin{bmatrix}0&-1&0\\2&0&0\\0&0&1\end{bmatrix}\end{aligned}

Step 3 — Compare vertex A

Apply both orders to vertex A.

\begin{aligned}(S R)A&=\begin{bmatrix}1\\1\\1\end{bmatrix}\to\hlmath{\begin{bmatrix}-2\\1\\1\end{bmatrix}}\\(R S)A&=\begin{bmatrix}1\\1\\1\end{bmatrix}\to\hlmath{\begin{bmatrix}-1\\2\\1\end{bmatrix}}\end{aligned}

Step 4 — Compare vertex B

Apply both orders to vertex B.

\begin{aligned}(S R)B&=\begin{bmatrix}3\\1\\1\end{bmatrix}\to\hlmath{\begin{bmatrix}-2\\3\\1\end{bmatrix}}\\(R S)B&=\begin{bmatrix}3\\1\\1\end{bmatrix}\to\hlmath{\begin{bmatrix}-1\\6\\1\end{bmatrix}}\end{aligned}

Step 5 — Compare vertex C

Apply both orders to vertex C.

\begin{aligned}(S R)C&=\begin{bmatrix}1\\2\\1\end{bmatrix}\to\hlmath{\begin{bmatrix}-4\\1\\1\end{bmatrix}}\\(R S)C&=\begin{bmatrix}1\\2\\1\end{bmatrix}\to\hlmath{\begin{bmatrix}-2\\2\\1\end{bmatrix}}\end{aligned}

Step 6 — Result

The two composite orders make different triangles.

S R\ne R S