2D Linear Transforms

Translate as Vector Add

Move every vertex of a triangle by adding the same translation vector. Translation is vector addition here, not a 2x2 matrix.

Example

Move a triangle by adding the same translation vector to each vertex.

highlighted = computed this step

Step 1 — Set up

Use vector addition; translation is not a two-by-two matrix.

t=[2−1]t=\begin{bmatrix}2\\-1\end{bmatrix}
2D transform triangleA dim original triangle and transformed vertices computed from the algebra.

Step 2 — Move vertex A

Add the translation vector to vertex A.

A′=[11]+[2−1]=[30]A'=\begin{bmatrix}1\\1\end{bmatrix}+\begin{bmatrix}2\\-1\end{bmatrix}=\hlmath{\begin{bmatrix}3\\0\end{bmatrix}}
2D transform triangleA dim original triangle and transformed vertices computed from the algebra.A'

Step 3 — Move vertex B

Add the translation vector to vertex B.

B′=[31]+[2−1]=[50]B'=\begin{bmatrix}3\\1\end{bmatrix}+\begin{bmatrix}2\\-1\end{bmatrix}=\hlmath{\begin{bmatrix}5\\0\end{bmatrix}}
2D transform triangleA dim original triangle and transformed vertices computed from the algebra.A'B'

Step 4 — Move vertex C

Add the translation vector to vertex C.

C′=[12]+[2−1]=[31]C'=\begin{bmatrix}1\\2\end{bmatrix}+\begin{bmatrix}2\\-1\end{bmatrix}=\hlmath{\begin{bmatrix}3\\1\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'
translate-as-vector-add Translation adds a vector to each point. This is why homogeneous coordinates are needed later for matrix-based translation.