Game Engine Architecture

438 10. The Rendering Engine

Division by Z

Perspective projection results in each vertex’s x- and y-coordinates being di-

vided by its z-coordinate. This is what produces perspective foreshortening.

To understand why this happens, consider multiplying a view-space point

pV expressed in four-element homogeneous coordinates by the OpenGL per-

spective projection matrix:

2

0 0 0

2

0 0 0

[1].

1

2

00 0

H V VH

Vx Vy Vz

n

rl

n

tb

ppp rl tb fn

rl tb f n

nf

fn

= →

⎡⎤⎛⎞

⎢⎥⎜⎟⎝⎠

⎢⎥−

⎢⎥⎛⎞

⎢⎥⎜⎟⎝⎠

⎢⎥−

= ⎢⎥⎛++⎞⎛ ⎞⎛⎞+

⎢⎥⎜ ⎟⎜ ⎟⎜⎟−−

⎢⎥⎝−−⎠⎝ ⎠⎝⎠−

⎢⎥⎛⎞

⎢⎥⎜⎟−

⎢⎥⎣⎦⎝⎠−

p pM

The result of this multiplication takes the form

pHV=−⎣⎦⎡⎤abc pz. (10.1)

When we convert any homogeneous vector into three dimensional coor-

dinates, the x-, y-, and z-components are divided by the w-component:

.

xzy

xyzw

www

⎡⎤≡⎡⎤

⎣⎦⎢⎥⎣⎦^

So, aft er dividing Equation (10.1) by the homogeneous w-component, which is

really just the negative view-space z-coordinate −pVz, we have:

[].

H

Vz Vz Vz

Hx Hy Hz

abc

ppp

ppp

⎡⎤

=⎢⎥

⎣⎦−−−

=

p

Thus the homogeneous clip space coordinates have been divided by the view-

space z-coordinate, which is what causes perspective foreshortening.

Perspective-Correct Vertex Attribute Interpolation

In Section 10.1.2.4, we learned that vertex att ributes are interpolated in order to

determine appropriate values for them within the interior of a triangle. Att ri-

bute interpolation is performed in screen space. We iterate over each pixel of the

screen and att empt to determine the value of each att ribute at the correspond-

ing location on the surface of the triangle. When rendering a scene with a perspec-