Transform Point World to Camera

easyPROPose EstimationPoseCV: Structure from Motion and SLAM

Transform a 3D point from world coordinates to camera coordinates.

The transformation from world to camera frame uses a rotation matrix R and translation vector t:

$X_{cam} = R \cdot X_{world} + t$

where:

$X_{world}$ is the point in world coordinates
$R$ is the 3×3 rotation matrix (camera orientation)
$t$ is the translation vector (camera position)
$X_{cam}$ is the point in camera coordinates

This is a rigid transformation - it preserves distances and angles. The camera "looks down" its Z-axis, with X pointing right and Y pointing down (typical convention).

Example

Input

transform_point([[1,0,0],[0,1,0],[0,0,1]], [1, 2, 3], [0, 0, 0])

Output

[1, 2, 3]

Reasoning

Inputs: R is the $3 \times 3$ identity matrix $I$ (ones on the diagonal, zeros elsewhere), so the camera axes line up exactly with the world axes. t is $(1, 2, 3)$ , and point is the world origin $(0, 0, 0)$ .
Rotate: each camera coordinate is one row of $R$ dotted with the point, e.g. the first is $1 \cdot 0 + 0 \cdot 0 + 0 \cdot 0 = 0$ . Every row gives $0$ , so $R \cdot X_{world} = (0, 0, 0)$ . Rotating the origin never moves it, whatever $R$ is.
Translate: add $t$ component by component: $(0 + 1,\ 0 + 2,\ 0 + 3) = (1, 2, 3)$ .
Output: [1, 2, 3], a list of three numbers $(X', Y', Z')$ . Since the inputs are integers, the entries stay integers.
Notice: $t$ is exactly where the world origin lands in the camera frame. Swap the roles (point $(1, 2, 3)$ , $t = 0$ ) and the identity rotation hands the point back unchanged, also giving $(1, 2, 3)$ .

Constraints

R: 3x3 rotation matrix
t: translation vector [tx, ty, tz]
point: [X, Y, Z] in world coordinates
Return transformed point [X', Y', Z'] in camera coordinates

Learn this in

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Background Knowledge

Coordinate Systems and Rigid Transformations

In computer vision, objects exist in multiple coordinate systems. The world coordinate system is a fixed reference frame in the scene (often defined by the problem or calibration markers), while the camera coordinate system is centered at the camera's optical center with axes aligned to the camera's orientation. A rigid transformation—consisting of rotation and translation—maps points between these frames without distorting distances or angles. This is fundamental to 3D reconstruction, pose estimation, and any multi-view geometry problem where you need to understand how cameras observe the same scene from different positions.

Rotation Matrices and Translation Vectors

The rotation matrix $R$ is a 3×3 orthogonal matrix (where $R^T R = I$ ) that encodes the camera's orientation relative to the world frame. Each column of $R$ represents where the world's basis vectors point in the camera frame. The translation vector $t$ (3×1) represents the camera's position in world coordinates, but in the transformation equation, it's applied after rotation to shift points into the camera's local frame. Understanding that $t$ is not simply the camera's world position (that would be $-R^T t$ ) is crucial—the formula given applies the rotation first, then translates.

Camera Coordinate Convention

Cameras typically use a convention where the Z-axis points forward (into the scene), X points right, and Y points down. This differs from some mathematical conventions and affects how you interpret the resulting coordinates. Points with positive Z are in front of the camera and will project onto the image plane; negative Z points are behind the camera.

Algorithm/Approach

The transformation is a straightforward affine transformation applied in sequence:

Rotate the world point using the rotation matrix
Translate the rotated point by adding the translation vector

This can be expressed as a single matrix operation if you use homogeneous coordinates, but the basic approach is linear algebra: matrix-vector multiplication followed by vector addition.

Step-by-Step Strategy

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