To create perspective projection in standard computer graphics, you fundamentally simulate how objects appear smaller as they move further away from the viewer, mimicking human vision.
In computer graphics, this is achieved through a specific process:
The Core Method: Transformation and Projection
Based on standard techniques, to obtain perspective projection, we project the results of perspective transformation on to any of the orthographic projection planes, say, z=0 plane. This two-step process is crucial.
- Perspective Transformation: First, the 3D coordinates of objects are transformed. This transformation is designed to map points in 3D space to a new 3D space in such a way that objects further away appear "compressed" towards the center of the view. A common way this is conceptually achieved is by mapping each point $(x, y, z)$ to $(x', y', z')$ where the $x'$ and $y'$ values are inversely proportional to the original $z$ value (or a related depth value). This is typically done using a perspective projection matrix.
- Orthographic Projection: After the perspective transformation, the resulting 3D points are then projected orthographically onto a 2D plane. As the reference states, this is often the z=0 plane in a normalized device coordinate space. This step essentially discards the depth information (the modified z' value), leaving only the 2D coordinates $(x', y')$ that represent the point's position on the screen.
Why This Works
The reference highlights a key property: The original line AB and transformed line A'B' intersect the z=0 plane at the same point on it. This is the principle that ensures lines and shapes are correctly projected. The perspective transformation warps the geometry such that when you discard the depth (by projecting onto the plane), the points land exactly where they should in the 2D view to create the perspective effect.
Imagine a point in 3D space. The perspective transformation repositions this point in a way that its new X and Y coordinates, when projected onto the viewing plane, accurately represent its position based on its distance from the camera. Points far away will have their transformed coordinates closer to the center of the projection plane compared to points that are closer.
Key Steps in the Graphics Pipeline
Creating perspective projection involves stages within the typical 3D graphics pipeline:
- Model-View Transformation: Objects are positioned and oriented relative to the camera (viewer).
- Perspective Transformation: The 3D scene is transformed to create the perspective effect (using a perspective projection matrix).
- Projection: The transformed 3D points are projected onto a 2D plane (e.g., by dividing x, y, and z by z and then dropping z, or effectively projecting onto the z=0 plane as per the reference).
- Viewport Transformation: The 2D projected coordinates are mapped to the final screen coordinates.
| Step | Description | Outcome |
|---|---|---|
| Perspective Transformation | Applies matrix to 3D points, encoding depth into coordinates. | Transformed 3D points reflecting perspective. |
| Projection onto Ortho Plane | Discards transformed depth (e.g., onto z=0 plane). | 2D coordinates on the projection plane. |
| Viewport Transformation | Maps 2D coordinates to screen pixels. | Final 2D position on the display. |
Practical Insights
- The perspective projection matrix is a standard tool in libraries like OpenGL, Vulkan, and DirectX. You typically specify the field of view (how wide the camera sees), the aspect ratio (width/height of the view), and the near and far clipping planes (minimum and maximum distances for objects to be rendered).
- Objects between the near and far planes are transformed and projected. Objects outside this range are typically clipped (removed).
- The z=0 plane mentioned in the reference is a common convention in normalized device coordinates (NDC), where the transformed and projected coordinates are typically mapped to a range like [-1, 1] for X and Y, and [0, 1] or [-1, 1] for Z (representing depth). Projecting onto the z=0 plane conceptually means taking the X and Y values from this normalized space.
In essence, perspective projection in computer graphics is a mathematical process that warps 3D space based on viewer position and field of view, followed by flattening this warped space onto a 2D plane to create the illusion of depth on a flat screen. The crucial element is the perspective transformation that ensures lines converge correctly when projected.
