Homogeneous Coordinates

Key idea

Represent translations with matrix multiplication.

In depth

Add an extra coordinate (x, y, 1) so translations become matrix multiplications: [1 0 tx; 0 1 ty; 0 0 1]. Then rotation, scaling, and translation compose in one framework.

Worked example

Translating (2, 3) by (4, −1): [1 0 4; 0 1 −1; 0 0 1](2, 3, 1) = (6, 2, 1).

Real-world uses

• Applying translations and perspective in 3D rendering with a single matrix multiplication

Common mistake

Forgetting to divide by the last coordinate in projective cases.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Linear Algebra

• Triple Product – Scalar triple product gives parallelepiped volume.
• Homogeneous Systems – Ax = 0 always has the trivial solution.
• Cofactor Expansion – Compute determinants recursively.
• Properties of Determinants – Multiplicative; row swaps change sign.
• Geometric Meaning of Determinants – Area or volume scaling factor.
• Cramer’s Rule for Systems – Solve systems using determinants.