Fragment Shader: Attribute Interpolation

Topic Overview

Vertex Attributes Become Per-Fragment Varyings

Vertex shaders output attributes (color, UVs, normals) per vertex. The rasterizer interpolates these across the triangle so the fragment shader receives smoothly varying values at each pixel. In GLSL these are varyings; in HLSL they are interpolants.

Straight lines in 3D project to straight lines in 2D, but attribute values are not linear in screen space under perspective. Equal screen-space steps correspond to unequal world-space steps due to foreshortening.

Perspective-Correct Interpolation

The GPU interpolates A/w and 1/w, then divides:

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For example, three vertices with colors red, green, blue and w values 1, 2, 1 produce different results at screen position (5,3) depending on whether you interpolate linearly or with perspective correction. Texture coordinates show swimming artifacts without correction; normals produce wrong lighting on angled surfaces.

• Linear: fast but wrong under perspective
• Perspective-correct: fixed-function hardware default since mid-1990s GPUs
• noperspective qualifier: rare GLSL escape hatch for screen-linear interpolation

For This Exercise

You will simulate color interpolation at a fragment position, computing both naive linear and perspective-correct results. This task asks you to print barycentric coordinates, both color values, and explain the difference. Getting this wrong is invisible on flat polygons but immediately visible as texture warping on steep angles.

Your Task

The rasterizer works in screen space, but attributes like texture coordinates vary linearly in eye space. Interpolating them with plain screen-space barycentrics (affine interpolation) produces the warped textures of the original PlayStation. GPUs instead do perspective-correct interpolation: they interpolate a/w and 1/w linearly across the screen, then divide per fragment. Compute both for fragments of a triangle and report the error.

Input

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Rules

• Screen barycentrics: area = (x1-x0)(y2-y0) - (y1-y0)(x2-x0). Then:
  • λ0 = ((x1-px)(y2-py) - (y1-py)(x2-px)) / area;
  • λ1 = ((x2-px)(y0-py) - (y2-py)(x0-px)) / area;
  • λ2 = 1 - λ0 - λ1.
• Outside points: a point with any λ < 0 is marked outside the triangle, but it is still evaluated, as extrapolation.
• Perspective-correct: qᵢ = λᵢ / wᵢ and 1/w = Σqᵢ. The weights are qᵢ / (1/w), and the corrected attribute is Σ qᵢaᵢ / (1/w).
• Affine: Σ λᵢaᵢ. The error is affine - correct.

Output

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Numbers are printed with %.4f, except that a value with |x| < 0.00005 prints as 0.0000.

Errors:

• attrs: N (1-4)
• vertex: I (0-2) X Y W ATTRS...
• vertex I: w must be > 0 (the vertex was clipped)
• vertex I: expected N attribute(s)
• sample: X Y, sample: define vertices 0, 1 and 2 first, sample: degenerate triangle
• unknown command X

A rejected vertex keeps its previous value.

Example

Input:

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Output:

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Requirements

1. Interpolate 1/w and a/w, never w or a. Only quantities divided by w are linear in screen space.
2. Leave the barycentric formula unnormalized until the final division, exactly as above. The expected outputs depend on it.

Hidden tests cover a strongly foreshortened edge, extrapolation outside the triangle, equal w values (where affine is exact), a degenerate triangle, and argument errors.