Homogeneous Coordinates

Implement a conversion between Cartesian coordinates and homogeneous coordinates, a fundamental concept in projective geometry. This conversion is crucial in computer vision for representing and applying projective transformations.

In Cartesian coordinates, a 2D point is represented as $(x, y)$ . In contrast, homogeneous coordinates introduce an additional dimension, representing the same point as $(x, y, 1)$ . This extension enables the use of linear algebra techniques, such as matrix multiplications, to perform projective transformations.

Here are the steps to convert between these coordinate systems:

To convert from Cartesian to homogeneous, append a third coordinate with value $1$ .
To convert from homogeneous to Cartesian, divide the first two coordinates by the third.

\begin{aligned} (x, y) &\rightarrow (x, y, 1) \\ (X, Y, W) &\rightarrow \left(\frac{X}{W}, \frac{Y}{W}\right) \end{aligned}

This technique is widely used in image processing and computer vision applications.

Example:

Input:

to_cartesian([6, 8, 2])

Output:

[3.0, 4.0]

Reasoning:

6/2 = 3, 8/2 = 4

Constraints:

For to_homogeneous: input is [x, y], output is [x, y, 1]
For to_cartesian: input is [X, Y, W], output is [X/W, Y/W] rounded to 4 decimals
W is guaranteed to be non-zero

Example:

Constraints:

Test Results