Rasterisation

​×V1​V3​

​. The dot product $\mathbf{N} \cdot \mathbf{V}$N⋅V is then calculated, where $\mathbf{V}$V is the view vector from one vertex to the camera position. If this dot product is negative (for counter-clockwise winding convention), the triangle is culled as back-facing. This simple test discards approximately half of the primitives in closed meshes, providing a significant performance gain early in the pipeline.^[23] Triangle setup begins after the projection of a triangle's vertices from 3D world space to 2D screen space, preparing the necessary parameters for efficient fragment generation during traversal. The three vertices, denoted as $V_1(x_1, y_1)$V1​(x1​,y1​), $V_2(x_2, y_2)$V2​(x2​,y2​), and $V_3(x_3, y_3)$V3​(x3​,y3​), are used to compute the edge equations for each side of the triangle. For the edge between $V_1$V1​ and $V_2$V2​, the edge function is defined as $E_{12}(x, y) = (y_1 - y_2)x + (x_2 - x_1)y + (x_1 y_2 - x_2 y_1)$E12​(x,y)=(y1​−y2​)x+(x2​−x1​)y+(x1​y2​−x2​y1​), which evaluates to zero on the line, positive on one side, and negative on the other. Similar functions $E_{23}$E23​ and $E_{31}$E31​ are derived for the other edges, assuming consistent vertex winding order such that interior points yield non-negative values for all three. These equations enable sub-pixel accurate tests and support incremental evaluation for traversal efficiency.^[24] Traversal methods determine which screen-space pixels (fragments) lie within the projected triangle. A common approach is bounding box scan conversion: compute the axis-aligned bounding box from the minimum and maximum x and y coordinates of the vertices, then iterate over all integer pixel centers within this box. For each pixel $(x, y)$(x,y), evaluate the three edge functions; the pixel is inside if $E_{12}(x, y) \geq 0$E12​(x,y)≥0, $E_{23}(x, y) \geq 0$E23​(x,y)≥0, and $E_{31}(x, y) \geq 0$E31​(x,y)≥0 (adjusting for winding). More efficient hierarchical methods, such as edge walking, traverse scanlines incrementally: start from the top vertex, advance active edges per row using updates like $E(x, y+1) = E(x, y) - (x_2 - x_1)$E(x,y+1)=E(x,y)−(x2​−x1​), and span horizontal spans between edges. These techniques minimize tests outside the triangle while enabling parallel processing in hardware.^[24] Barycentric coordinates provide the weights for interpolating attributes across the triangle and are derived from the edge functions. For a pixel $P(x, y)$P(x,y), the coordinates $\alpha, \beta, \gamma$α,β,γ satisfy $P = \alpha V_1 + \beta V_2 + \gamma V_3$P=αV1​+βV2​+γV3​ with $\alpha + \beta + \gamma = 1$α+β+γ=1, and are computed as $\alpha = \frac{E_{23}(P)}{E_{23}(V_1)}$α=E23​(V1​)E23​(P)​, $\beta = \frac{E_{31}(P)}{E_{31}(V_2)}$β=E31​(V2​)E31​(P)​, $\gamma = \frac{E_{12}(P)}{E_{12}(V_3)}$γ=E12​(V3​)E12​(P)​, normalized such that their sum is 1 (noting the denominators are twice the signed area of the triangle). Pixels inside the triangle have $\alpha \geq 0$α≥0, $\beta \geq 0$β≥0, $\gamma \geq 0$γ≥0. These coordinates allow linear interpolation of per-vertex attributes like color in screen space.^[24] In perspective projection, screen-space barycentric interpolation distorts attributes like texture coordinates due to the non-linear w-division. Perspective-correct interpolation addresses this by operating in homogeneous coordinates. For an attribute $f$f, compute vertex values $f_i / w_i$fi​/wi​ and $1 / w_i$1/wi​ (where $w_i$wi​ is the homogeneous depth), then linearly interpolate using screen-space barycentrics: $f' = \frac{\alpha (f_1 / w_1) + \beta (f_2 / w_2) + \gamma (f_3 / w_3)}{\alpha / w_1 + \beta / w_2 + \gamma / w_3}$f′=α/w1​+β/w2​+γ/w3​α(f1​/w1​)+β(f2​/w2​)+γ(f3​/w3​)​, and similarly for the denominator to recover the effective $w$w. This yields values linearly interpolated in 3D eye space, preventing affine warping artifacts in textures or shading.^[25]

Depth and Visibility Resolution

Advanced Techniques

Anti-Aliasing Methods