Perspective Division

easyFREE3D to 2D ProjectionsProjectionCV: Image Formation

Implement a perspective division to project a 3D point onto a 2D plane, which is a fundamental concept in 3D to 2D Projections. This process is crucial in understanding how pinhole cameras form images.

The concept of perspective division is based on the idea that the projection of a 3D point $(X, Y, Z)$ onto a 2D plane results in a 2D point $(x, y)$ , where $x$ and $y$ are calculated using the formulas $x = \frac{X}{Z}$ and $y = \frac{Y}{Z}$ .

Start with a 3D point $(X, Y, Z)$ .
Apply the perspective division formula to obtain the 2D coordinates.

x = \frac{X}{Z}, \quad y = \frac{Y}{Z}

This technique is widely used in computer vision applications.

Example

Input

perspective_divide([10, 20, 5])

Output

[2.0, 4.0]

Reasoning

The input point_3d is a 3D point in camera coordinates, $(X, Y, Z) = (10, 20, 5)$ : $X$ is horizontal offset, $Y$ vertical offset, and $Z = 5$ is the depth in front of the pinhole.
Projection scales both lateral coordinates by $\frac{1}{Z}$ , so the point lands on the image plane at unit distance: $x = \frac{10}{5} = 2$ and $y = \frac{20}{5} = 4$ .
The output is the 2D image point [2.0, 4.0]: two numbers instead of three, because depth is consumed by the division. Values are floats rounded to 4 decimal places (here they divide exactly).
Notice the effect of depth: moving the same point twice as far away, $Z = 10$ , gives $(1, 2)$ , so distant objects appear smaller. Any point on the ray through $(2, 4, 1)$ , such as $(20, 40, 10)$ , projects to the same pixel.

Constraints

Z > 0 (point is in front of camera)
Return [x, y] rounded to 4 decimal places

Learn this in

Image FormationCV Study Plan Camera Models & Calibration3D Computer Vision

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Concept primer

Background Knowledge

In a simple pinhole camera model, a 3D scene point $(X, Y, Z)$ in camera coordinates is projected onto a 2D image plane. If the image plane is placed at distance 1 along the camera’s optical axis (the $Z$ -axis), then the geometry of similar triangles gives:

x = \frac{X}{Z}, \quad y = \frac{Y}{Z}.

This operation is called perspective division or perspective projection. It captures the key effect of perspective: points farther away (large $Z$ ) appear closer to the center and smaller in the image.

In computer graphics and vision, we often first transform a point into a “camera” or “clip” space using matrices, then perform homogeneous division: divide by the last coordinate to get back to standard 2D/3D coordinates. Here, you are given a 3D point already in camera coordinates and just need to apply the division by depth $Z$ to obtain normalized 2D coordinates $(x, y)$ .

Algorithm/Approach

The general pattern for this type of problem is:

Take the 3D point components $(X, Y, Z)$ .
Perform component-wise division by depth $Z$ to get the projected 2D point:
$x = X / Z$
$y = Y / Z$
Return $(x, y)$ as the result.

The main concern is handling the special case where $Z = 0$ (or extremely close to 0), which would cause division by zero or numerical instability.

Step-by-Step Strategy
Read input: Obtain the three coordinates $X$ , $Y$ , $Z$ as floating-point numbers.
Check depth:

If $Z = 0$ , decide how the problem expects this to be handled (e.g., undefined, error, or special output). In many contest-style problems, inputs are guaranteed to avoid this case, but still be aware of it.

Compute projection:

x = X / Z
y = Y / Z

Output result:

Print or return the values x and y in the required format (often as floats or doubles, possibly with specified precision).

Example in pseudocode:

def perspective_division(X, Y, Z):
 # assuming Z != 0 per problem constraints
 x = X / Z
 y = Y / Z
 return x, y

Common Pitfalls

Division by zero: If $Z = 0$ , the projection is undefined. Check constraints or add a guard.
Sign of Z: A negative $Z$ means the point is behind the camera in the usual convention; depending on the problem, this may not occur or may be considered invalid.
Integer division: In languages like C++/Java/Python 2, dividing integers does integer division by default. Make sure to use floating-point division (e.g., cast to double/float).
Precision/formatting: Output formatting (number of decimal places) might matter in some problems; follow the problem statement.

Time & Space Complexity

Time complexity: $O(1)$ — the algorithm does a constant number of arithmetic operations.
Space complexity: $O(1)$ — only a few variables are used, independent of input size.

solution.py

Python 3.13

Test results