Why Hidden Surface Removal

Multiple polygons may project onto the same pixel; the closest one to the viewer wins. The z-buffer algorithm (Catmull, 1974) is the most common solution and is built into every modern GPU.

Algorithm

1. Allocate two buffers same size as the framebuffer:

• Color buffer (the framebuffer).
• Depth buffer (z-buffer), initialized to +infinity (or 1.0 in NDC).

1. For each polygon, rasterize. For each pixel (x, y) covered:

• Compute fragment depth z (interpolated barycentrically from vertex z values).
• If z < depthBuf[x, y]:
• colorBuf[x, y] = fragmentColor
• depthBuf[x, y] = z
• Else discard.

1. After all polygons processed, the framebuffer holds the visible image.

Properties

• Independent of polygon order -> can render in any order; trivially parallelizable.
• Constant memory cost = WHdepth_precision (e.g., 1080p at 24-bit = ~6 MB).
• Handles intersecting polygons correctly at fragment level.
• Per-pixel test -> fast on GPU (early-Z, hierarchical Z optimizations).

Z Precision and Z-Fighting

The depth value after the perspective divide is non-linear: z_ndc = (f + n)/(f - n) - 2fn/((f - n)*z_view). Most precision is concentrated near the near plane. Two coplanar surfaces at large z may yield identical z_buf values -> z-fighting (flickering pixels). Mitigations:

• Push the near plane farther.
• Use a 32-bit float depth buffer.
• Reverse-Z: store depth as 1 - z and use float, distributing precision more evenly.
• Polygon offset for decals.

Numerical Example

Three triangles overlap at pixel (300, 200) with fragment z values 0.5, 0.3, 0.7. After processing in any order:

• First write: z = 0.5 -> stored.
• Next: z = 0.3 < 0.5 -> stored.
• Next: z = 0.7 > 0.3 -> discarded.

Final color is from the z = 0.3 fragment, regardless of arrival order.

Memory Cost Table

Pros and Cons

• Pros: Simple, parallel, handles arbitrary scenes including intersections and self-occlusion.
• Cons: Memory; z-fighting at far ranges; transparency requires sorting (or order-independent transparency techniques).

Modern Optimizations

• Hierarchical Z-buffer: store min/max z in a quadtree for fast occlusion culling.
• Early-Z test: hardware skips fragment shader if depth test fails before shading.
• Reverse-Z + float depth: industry-standard precision improvement.