Software Rendering School

On DevMaster.net I found a series of tutorials: Software Rendering School.

I’ve read first two parts – it is really amazing, everything is explained is a very simple language and does not require to remember the University mathematics.

Few nice things from the beginning:

• You may find an explanation what is the difference between the points and vertices – the points are simply random positions set by their coordinates, and vertices may have more data bound to them, vertices are positions bound to primitives like triangle.
• The basic operations you can perform with the points are translation, scaling and rotation.
• Translation is a simple movement:

new.x = point.x + translate.x;

new.y = point.y + translate.y;

new.z = point.z + translate.z

• The scaling is also a kind of a movement, but we multiply the coordinates:

new.x = point.x * scale.x;

new.y = point.y * scale.y;

new.z = point.z * scale.z;

• The most complex operation is the rotation:

//X-Rotation

new.x = v.x

new.y = v.y * cos(a) – v.z * sin(a)

new.z = v.y * sin(a) + v.z * cos(a)

//Y-Rotation

new.x = v.x * cos(a) – v.z * sin(a)

new.y = v.y

new.z = v.x * sin(a) + v.z * cos(a)

//Z-Rotation

new.x = v.x * cos(a) – v.y * sin(a)

new.y = v.y * sin(a) + v.x * cos(a)

new.z = v.z

• Vector, in the simplest way, can be described by its position, direction and magnitude.
• Dot product is cos of the angle between two vectors:

dot  = v1.x * v2.x + v1.y + v2.y + v1.z * v2.z

dot = cos(a)

More about Dot Product: http://mathworld.wolfram.com/DotProduct.html

• Cross product of two vectors is another vector, which is perpendicular to both source vectors:

new.x = v1.y * v2.z – v2.z – v1.z

new.y = v2.x * v1.z – v1.x * v2.z

new.z = v1.x * v2.y – v2.x * v2.y

More about Cross Product: http://mathworld.wolfram.com/CrossProduct.html

And here is a joke about the subject:

Question: What do you get if you will cross an elephant and a grape?

Answer: elephant grape sine-of-theta

•  Matrix is an object with a given number of rows and columns.
• Parallel projection:

x2 = x1 + viewportWidth / 2

y2 = –y1 + viewportHeight / 2

z2 = 0

• Perspective projection is clear when you know that

y2 / y1 = z2 / z1 = d / z

or

x2 / x1 = z2 / z1 = d / z

where d is the distance to the projection plane.