Homogeneous and Compose

Homogeneous Coordinates

Lift a 2D point into homogeneous form by adding a final 1.

Example

Lift 2D points to (x, y, 1) so matrix transforms can include translation.

highlighted = computed this step

Step 1 — Set up

Start with the plane triangle before lifting points.

points (x,y)\text{points }(x,y)
Homogeneous coordinate setupThe same 2D triangle while each vertex is lifted to a homogeneous coordinate.

Step 2 — Lift vertex A

Add a one to vertex A.

A:[11]→[111]A:\begin{bmatrix}1\\1\end{bmatrix}\to\hlmath{\begin{bmatrix}1\\1\\1\end{bmatrix}}
Homogeneous coordinate setupThe same 2D triangle while each vertex is lifted to a homogeneous coordinate.A

Step 3 — Lift vertex B

Add a one to vertex B.

B:[31]→[311]B:\begin{bmatrix}3\\1\end{bmatrix}\to\hlmath{\begin{bmatrix}3\\1\\1\end{bmatrix}}
Homogeneous coordinate setupThe same 2D triangle while each vertex is lifted to a homogeneous coordinate.AB

Step 4 — Lift vertex C

Add a one to vertex C.

C:[12]→[121]C:\begin{bmatrix}1\\2\end{bmatrix}\to\hlmath{\begin{bmatrix}1\\2\\1\end{bmatrix}}
Homogeneous coordinate setupThe same 2D triangle while each vertex is lifted to a homogeneous coordinate.ABC

Step 5 — Result

Each plane point now has a homogeneous one.

homogeneous vertices ready\text{homogeneous vertices ready}
Homogeneous coordinate setupThe same 2D triangle while each vertex is lifted to a homogeneous coordinate.ABC
homogeneous-coordinates Homogeneous coordinates write a 2D point as (x, y, 1). That extra 1 lets translation become matrix multiplication in the next lesson.