2D Linear Transforms

Scale About Origin

Scale a triangle by multiplying each vertex by a diagonal 2x2 matrix using column vectors.

Example

Scale each vertex with a diagonal matrix using column vectors.

highlighted = computed this step

Step 1 — Set up

Use column vectors with matrix times vertex.

M=[2003]M=\begin{bmatrix}2&0\\0&3\end{bmatrix}
2D transform triangleA dim original triangle and transformed vertices computed from the algebra.

Step 2 — Move vertex A

Multiply the matrix by vertex A.

A′=[2003][11]=[2⋅1+0⋅10⋅1+3⋅1]=[23]A'=\begin{bmatrix}2&0\\0&3\end{bmatrix}\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}2\cdot1+0\cdot1\\0\cdot1+3\cdot1\end{bmatrix}=\hlmath{\begin{bmatrix}2\\3\end{bmatrix}}
2D transform triangleA dim original triangle and transformed vertices computed from the algebra.A'

Step 3 — Move vertex B

Multiply the matrix by vertex B.

B′=[2003][31]=[2⋅3+0⋅10⋅3+3⋅1]=[63]B'=\begin{bmatrix}2&0\\0&3\end{bmatrix}\begin{bmatrix}3\\1\end{bmatrix}=\begin{bmatrix}2\cdot3+0\cdot1\\0\cdot3+3\cdot1\end{bmatrix}=\hlmath{\begin{bmatrix}6\\3\end{bmatrix}}
2D transform triangleA dim original triangle and transformed vertices computed from the algebra.A'B'

Step 4 — Move vertex C

Multiply the matrix by vertex C.

C′=[2003][12]=[2⋅1+0⋅20⋅1+3⋅2]=[26]C'=\begin{bmatrix}2&0\\0&3\end{bmatrix}\begin{bmatrix}1\\2\end{bmatrix}=\begin{bmatrix}2\cdot1+0\cdot2\\0\cdot1+3\cdot2\end{bmatrix}=\hlmath{\begin{bmatrix}2\\6\end{bmatrix}}
2D transform triangleA dim original triangle and transformed vertices computed from the algebra.A'B'C'

Step 5 — Result

The transformed triangle is complete.

transformed triangle complete\text{transformed triangle complete}
2D transform triangleA dim original triangle and transformed vertices computed from the algebra.A'B'C'
scale-about-origin With column vectors, a 2x2 matrix multiplies a vertex on the left: M v. Scaling about the origin uses a diagonal matrix.