3D Theory - OpenGL Rendering

Rendering Pipeline

Object coordinates

Points in 3D space are represented by 3 numbers represented by its position in the x, y and z planes.

Normalized coordinates - frustum projection

In order to represent this on a 2D screen we need to define a projection. A frustum projection shows an object, which is closer to the camera as larger than an object which is further away. To do this, the x and y coordinates are scaled by an amount invsely propotional to the position in the z dimention.

Screen coordinates - frustum projection

The viewport transformation scales and shifts the normalised coodinates to fit the size of the screen.

Pipeline

The rendering pipline takes the model data and renders it on the screen. This is done partly by software and partly by hardware, the boundry between these depends on what graphics card and operating system that you are using.

As shown in the diagrams above, in a frustum projection, the x and y coordinates are scaled by an amount invsely propotional to the position in the z dimention. A 3x3 matrix cannot represent such a transformation, so in the pipline an additional variable 'w' is added to each coordinate. This is a scaling value, used to scale the x,y and z values depending on the inverse of its distance from the camera.

To do this, the matricies are increased to 4x4 to include this scaling value.

Projection Matrix

example (width=258 height=440) Aspect=0.629268
FD/aspect=5.93077841512884
FD=3.73205080756888

projection->m00 = FD/aspect;
projection->m01 = 0;
projection->m02 = 0;
projection->m03 = 0;
projection->m10 = 0;
projection->m11 = FD;
projection->m12 = 0;
projection->m13 = 0;
projection->m20 = 0;
projection->m21 = 0;
projection->m22 = (zFar + zNear)/(zFar - zNear);
projection->m23 = -1;
projection->m30 = 0;
projection->m31 = 0;
projection->m32 = (2 * zFar * zNear)/(zFar - zNear);
projection->m33 = 0;

Inverse Projection Matrix

in example above = 0.16861193084690149207499690607832
mi01 = 0
mi02 = 0
mi03 = 0
mi10 = 0
mi11 = - FD/aspect*m23*m32 /det = 1/FD

in example above = 0.26794919243112251215656695782073
mi12 = 0
mi13 = 0
mi20 = 0
mi21 = 0
mi22 = 0
mi23 = - FD/aspect*FD*m23/det = 1/m32 = (zNear-zFar)/(2 * zFar * zNear)

assuming zFar = -20 and zNear=-3 then m23 = 17/(2 * 20 * 3) =0.141666667
mi30 = 0
mi31 = 0
mi32 = m02*m11*m30 - m01*m12*m30 - m02*m10*m31 + m00*m12*m31 + m01*m10*m32 - m00*m11*m32;
= - m00*m11*m32;
= - FD/aspect*FD*m32/det
= 1/m23
= -1
mi33 = + FD/aspect*FD*m22/det = -m22/(m23 * m32) = m22/m32 = (zFar+zNear) / (2*zFar*zNear)

assuming zFar = -20 and zNear=-3 then m33 = -23 / (2*20*3) = -23/120 = 0.19166667

det= m03 * m12 * m21 * m30-m02 * m13 * m21 * m30-m03 * m11 * m22 * m30+m01 * m13 * m22 * m30+
m02 * m11 * m23 * m30-m01 * m12 * m23 * m30-m03 * m12 * m20 * m31+m02 * m13 * m20 * m31+
m03 * m10 * m22 * m31-m00 * m13 * m22 * m31-m02 * m10 * m23 * m31+m00 * m12 * m23 * m31+
m03 * m11 * m20 * m32-m01 * m13 * m20 * m32-m03 * m10 * m21 * m32+m00 * m13 * m21 * m32+
m01 * m10 * m23 * m32-m00 * m11 * m23 * m32-m02 * m11 * m20 * m33+m01 * m12 * m20 * m33+
m02 * m10 * m21 * m33-m00 * m12 * m21 * m33-m01 * m10 * m22 * m33+m00 * m11 * m22 * m33;
= -m00 * m11 * m23 * m32;
=FD/aspect * FD * m23 * m32

[0.168611930846902 0 0 0]
[0 0.267949192431123 0 0]
[0 0 0 0.141666666666667]
[0 0 -1 0.191666666666667]