Introduction

Typically in computer graphics, we wish to interpolate values associated with perspectively-projected vertices of a line segment. The dimension N eye-space vertices and are projected onto a dimension N-1 hyperplane called screen-space. The values associated with each vertex (e.g. color, texture-coords, normals) are typically interpolated along the line segment based on the screen-space projected coordinates of the corresponding vertices. However, the values required at each interpolated screen-space point need to be the values as if interpolated in eye-space and projected onto the interpolated position.

The relationship between interpolation in screen-space and in eye-space is nonlinear. Linearly interpolating the The resulting values from linearly interpolating some fraction t between two screen-space vertices do not match the eye-space interpolated values of the eye-space point projecting on the interpolated screen-space point. In order to obtain the proper interpolated values, we need to delve into the details of the relationship between interpolating in screen-space and in eye-space.

In order to derive a relationship between the two spaces, we will examine a simple case of only two dimensions (N=2). Eye-space points (X,Z) will be represented by upper-case letters, and screen-space points (x,z) will be represented by lower-case. (X,Z)-points represent points Z units away from the eye-plane (Z=0) that will be projected onto the Z=1 plane with respect to the eye (center-of-projection) at the origin (0,0). The viewing direction is in the +Z direction.

The projection of a 2D eye-space point (X,Z) to a 1D screen-space point x (onto z=1 plane) is quite simple. given the eye-space point (X,Z) we can compute x using similar triangles:

x/1 = X/Z
  x = X/Z

Given an eye-space edge from point (X1,Z1) to (X2,Z2), we can interpolate between them in eye-space as follows:

  E(T) = (1-T)*(X1,Z1) + T*(X2,Z2)

[ X' ]   [ (1-T)*X1 + T*X2 ]
[ Z' ] = [ (1-T)*Z1 + T*Z2 ]

S(t) = x' = (1-t)*x1 + t*x2

Now we can project the interpolated point along the edge into screen-space from the interpolation equation E(t) and the perspective equation. Given to edge endpoints and a parametric t value, we obtain an interpolated eye-space point (X',Z'). We can project this point into screen-space as x'=X'/Z'.

Projected E(T) = x' = X'/Z' = ((1-T)*X1 + T*X2) / ((1-T)*Z1 + T*Z2)

We can derive a relationship between interpolation in screen-space and in eye-space by relating their paremetric interpolating values t and T respectively. We will do this by observing their relationship when they are equivalent points; in other words, when the projected E(t) equals S(t).

           S(t) = Projected E(T)
             x' = X'/Z'
(1-t)*x1 + t*x2 = ((1-T)*X1 + T*X2) / ((1-T)*Z1 + T*Z2)

from the projection equations:
    x = X/Z
    X = x*Z
so
    X1 = x1*Z1
    X2 = x2*Z2

by substitution:

(1-t)*x1 + t*x2 = ((1-T)*x1*Z1 + T*x2*Z2) / ((1-T)*Z1 + T*Z2)
                = [ ((1-T)*Z1) / ((1-T)*Z1 + T*Z2) ] * x1 +
                  [ (T*Z2) / ((1-T)*Z1 + T*Z2) ] * x2

we now have an equation of the form:

    a*x1 + b*x2 = A*x1 + B*x2

where a=A and b=B for this equality to hold.

so we will use b=B:

    t = (T*Z2) / ((1-T)*Z1 + T*Z2)
    t*((1-T)*Z1 + T*Z2) = Z2*T
   t*(Z1 - Z1*T + Z2*T) =
 Z1*t - Z1*t*T + Z2*t*T = 
 Z2*t*T - Z1*t*T - Z2*T = -Z1*t
   T*(Z2*t - Z1*t - Z2) =
                      T = -Z1*t / (Z2*t - Z1*t - Z2)
                        = Z1*t / (Z1*t + Z2 - Z2*t)
                      T = Z1*t / (Z1*t + Z2*(1-t))

    t = (T*Z2) / ((1-T)*Z1 + T*Z2)
    T = Z1*t / (Z1*t + Z2*(1-t))

We must now show how the eye-space and screen-space points are related. We will interpolate a value in eye-space and show the corresponding interpolation in screen-space in terms of eye-space coordinates.

Given two eye-space points E1 and E2, we can interpolate as follows:

E(T) = E1*(1-T) + E2*T

E(T) = E1 * [ (1-t)*Z2 / (Z1*t + Z2*(1-t)) ] + 
       E2 * [ Z1*t / (Z1*t + Z2*(1-t)) ]
     = [ E1 * (1-t)*Z2 + E2 * Z1*t ] / (Z1*t + Z2*(1-t))

         = [ (E1/Z1)*(1-t) + (E2/Z2)*t ] / [ (1/Z1)*(1-t) + (1/Z2)*t ]

In other words, when interpolating values in screen-space, in order to compute the appropriate corresponding values in eye-space we must interpolate the projected values and the 1/Z terms and divide for each interpolated sample point.