Orthographic Projection

1. Background Knowledge

In computer vision and graphics, a projection is a mapping from 3D coordinates $(X, Y, Z)$ to 2D image coordinates $(x, y)$. An orthographic (parallel) projection assumes all projection rays are parallel to each other and perpendicular to the image plane, unlike perspective projection where rays converge at a camera center. This is a good approximation when the camera is far away relative to the object size or when exact shape and dimensions (not foreshortening) matter, as in CAD and engineering drawings.

Mathematically, the simplest orthographic projection onto the image (say, the $XY$-plane) just drops the depth coordinate: $(X, Y, Z) \to (X, Y)$. Parallel lines in 3D remain parallel after projection, and relative distances in the projected plane are preserved up to a uniform scale. More general linear orthographic projections can involve rotations and scaling, but the essence in this problem is that depth $Z$ does not affect the 2D location.

2. Algorithm/Approach

For this easy version, the pattern is:

• Treat the input as a 3D point with coordinates $(X, Y, Z)$.
• Apply a fixed, simple linear transformation that ignores the $Z$ component.
• Output the resulting 2D point $(X, Y)$.

Conceptually, you are applying a $2 \times 3$ projection matrix

to the 3D vector $\begin{bmatrix}X \\ Y \\ Z\end{bmatrix}$, but for this problem you can just access the first two coordinates directly.

3. Step-by-Step Strategy

4. Parse input

• Read the three coordinates $X, Y, Z$ from input (often floats or doubles).

2. Apply orthographic projection

• Compute:
• x_{\text{proj}} = X
• y_{\text{proj}} = Y
• Ignore $Z$; it does not appear in the formulas.

3. Output result

• Print or return the 2D point (x_{\text{proj}}, y_{\text{proj}}) in the format required by the platform.

Example in Python-like pseudocode:

def orthographic_project(point3d):
 X, Y, Z = point3d
 x_2d = X
 y_2d = Y
 return (x_2d, y_2d)

4. Common Pitfalls

• Overthinking the math: This problem does not require perspective division, camera intrinsics, or any trigonometry. It’s just dropping the $Z$ coordinate.
• Wrong coordinate order: Make sure you know in what order $X, Y, Z$ are provided, and output only the first two in the correct order.
• Unnecessary type conversion issues: If the platform uses floats/doubles, keep consistent types and formatting (e.g., required number of decimal places).

5. Time & Space Complexity

• Time complexity: $O(1)$ per point, since it’s a constant number of assignments.
• Space complexity: $O(1)$, as you only store a fixed number of scalar values (the input 3D point and output 2D point).