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Apply 4×4 Homogeneous Transform

3D Geometry

Medium

A homogeneous transformation combines a three-dimensional linear transformation and a translation in one $4 \times 4$ matrix:

T = \begin{bmatrix} R & t \\ 0 & 1 \end{bmatrix}

Here, $R$ is a $3 \times 3$ matrix and $t$ is a three-dimensional translation vector. For a point $p = (x, y, z)$ , form its homogeneous coordinate:

p_h = \begin{bmatrix} x & y & z & 1 \end{bmatrix}^{\mathsf T}

Apply the transformation:

p'_h = T p_h

Return the first three coordinates of every transformed point. A single input point must produce an array of shape $(3,)$ , while a batch must produce an array of shape $(N, 3)$ .

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Examples

Input: T = [[1, 0, 0, 1], [0, 1, 0, 2], [0, 0, 1, 3], [0, 0, 0, 1]], points = [0, 0, 0]

Output: [1.0, 2.0, 3.0]

Explanation: Appending 1 allows the last column of T to translate the point by (1, 2, 3).

Input: T = [[0, -1, 0, 1], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], points = [[1, 0, 0], [0, 1, 0]]

Output: [[1.0, 1.0, 0.0], [0.0, 0.0, 0.0]]

Hint 1

np.ones((points.shape[0], 1)) creates the homogeneous coordinate for a batch.

Hint 2

(T @ points_h.T).T applies one transform to every point at once.

Requirements

Accept T as a list representing a $4 \times 4$ homogeneous transformation
Accept either one three-dimensional point or a batch of points
Return a NumPy array with shape $(3,)$ for one point or $(N, 3)$ for a batch
Process a batch without a Python loop over its points

Constraints

T has shape $(4, 4)$
Points has shape $(3,)$ or $(N, 3)$
$1 \leq N \leq 100{,}000$
Use NumPy only

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