Question

In 3D transformations, what is the purpose of homogeneous coordinates?

• A. To represent points using only two coordinates
• B. To simplify calculations for translation only
• C. To perform transformations using matrix multiplication
• D. To remove the need for scaling transformations

Hint:

Adding an extra coordinate lets even translation become a matrix operation.

Answer 1

The Correct Option is C

The core problem is that plain translation cannot be expressed as a $3 \times 3$ matrix product, while rotation and scaling can. This blocks combining them cleanly.

Homogeneous coordinates fix this by adding an extra coordinate, so a point becomes $(x, y, z, 1)$ and every transformation becomes a $4 \times 4$ matrix. Now translation, rotation, and scaling are all matrix multiplications.

Because they are all products, a chain of operations composes into a single matrix, applied by one multiplication.

\[\boxed{\text{To perform transformations using matrix multiplication}}\]