[In this reprinted #altdevblogaday opinion piece, game programmer Simon Yeung takes readers through the steps to linear interpolate the vertex in screen space.]
Continuing with the previous post, after filling the triangle with scan line or half-space algorithm, we also need to interpolate the vertex attributes across the triangle so that we can have texture coordinates or depth on every pixel.
However we cannot directly interpolate those attributes in screen space because projection transform after perspective division is not an affine transformation (i.e. after transformation, the mid-point of the line segment is no longer the mid-point). This will result in some distortion and this artifact is even more noticeable when the triangle is large:
interpolate in screen space
perspective correct interpolation
Condition for linear interpolation
When interpolating the attributes in a linear way, we are saying that given a set of vertices, vi (where i is any integer>=0) with a set of attributes ai (such as texture coordinates), we have a function mapping a vertex to the corresponding attributes, i.e.
substitute equation 1 and 2 into the plane equation of the triangle lies on:
So, 1/zview is an affine function of xndc and yndc, which can be interpolated linearly across the screen space (the transform from NDC space to screen space is a linear transform).
Attributes interpolation
In the last section, we know how to interpolate the depth of a pixel linearly in screen space. The next problem is to interpolate the vertex attributes (e.g. texture coordinates). In view space, we know that those attributes can be interpolated linearly, so those attributes can be calculated by an affine function with the vertex position as parameters, e.g.
Similar to interpolate depth, substitute equation 1 and 2 into the above equation:
Therefore, u/zview is an another affine function of xndc and yndc, which can be interpolated linearly across the screen space. Hence we can interpolate u linearly by first interpolatin 1/zview and u/zview across screen space, and then divide them per pixel.
The last problem…
Now, we know that we can interpolate the view space depth and vertex attributes linearly across screen space. But during the rasterization state, we only have vertices in homogenous coordinates (vertices are transformed by the projection matrix already). How can we get the zview to do the perspective correct interpolation?
Consider the projection matrix (I use D3D one, but the same for openGL):
interpolate in screen space
perspective correct interpolationf(vi)= ai
Say, to interpolate a vertex inside a triangle in a linear way, the function f needs to have the following properties:
f(t0 *v0 + t1 *v1 + t2 *v2 ) = t0 * f(v0) + t1 * f(v1) + t2 * f(v2)
, for any t0, t1, t2 where t0 + t1 + t2=1
which means that we can calculate the interpolated attributes using the same weight ti used for interpolating vertex position. For functions having the above properties, those functions will be an affine function with the following form:
f(x)= Ax + b
, where A is a matrix, x and b are vector
Depth interpolation
When a vertex is projected from view space to normalized device coordinates(NDC), we will have the following relation (ratio of the triangles) between the view space and NDC space:
i.e. w-homogenous = zview
And look at the matrix again and consider the transformed z-coordinates. It will be in the form of:
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| Screen shot of the demo |