Chapter 2: The Graphics Pipeline


The purpose of this chapter is discuss a way to draw shapes/geometry to 2D screen. The 2D screen is the monitor that is attached to your computer. The shapes and geometry are what we want to draw. We are going to focus on triangles in this part, though points and lines are other shapes that you could draw. Triangles are the main type of shape, or “primitive”, that is rasterized by most rasterization pipelines. Triangles are 3D shapes with some useful properties, and most other shapes can be created using multiple triangles.

The Graphics Pipeline is the process by which we convert a description of a 3D scene into a 2D image that can be viewed. This involves both transformations between spaces and conversions from one type of data to another.


What is a space? Consider if you and I are standing in a room, 6 feet apart and facing the same direction. Exactly half way inbetween us is a table, and I am on the left of the table. If I were to answer “where is the table”, I would likely say the table is 3 feet to my right. But you would say the table is 3 feet to your left. Neither of us is wrong, we’re just describing relationships in space using a difference reference point. We might say that I am describing the position of the table in “my coordinates” while you are describing it in “your coordinates”.

In computer graphics, we sometimes call these two different coordinate systems spaces. A lot (seriously, a lot) of computer graphics has to do with transforming geometry between spaces. Usually, the spaces are different in more complicated ways (consider if you turned yourself 45 degrees, and I flipped over and did a headstand - our descriptions of the table’s position would now be more complicated and different).

It’s nice to have a common frame of reference for these things. Usually, we describe the positions of most things in our scene in terms of “world coordinates”. This “world space” has a common reference point, or origin, from which the positions of all other objects are described. The world coordinate system also has a sense of directionality, e.g. which way is up, but we’ll discuss this later.

In our rasterizers, our task will be to take triangles with coordinates stored in world space, and transform them into pixels that we can color on the screen.

coordinate frame

Types of Geometry

As discussed, the elements of our scenes will usually be primitive geometry, specifically either points, lines, or triangles. The geometry can be expressed in 2D coordinates, but usually we’ll be talking about 3D geometry.

The output of our rasterizer will be an image that can be viewed. This image will be a two-dimensional array of color values. These color values are called pixels, or “picture elements”.

We’ll usually talk about these pixels as a x and y coordinate on the screen. These coordinates will be integers, and will range from 0 to the width or height of the screen.

pixel coordinates

We will need to find a way to transform from world coordinates to these pixel coordinates, but we’ll cover that after we first introduce the main stages of the graphics pipeline.

The Graphics Pipeline

The Graphics Pipeline exists in a variety of forms depending on the type of computer graphics and rendering you are doing.

What we’ll introduce now is the rasterization pipeline used by OpenGL, which will be more or less what implement for our software rasterizer.

graphics pipeline

input specification: what are the vertices?

vertex shader: move to camera’s perspective

rasterization: which pixels are inside triangle?

fragment shader: what color is each pixel?

testing and blending: which pixels are visible?

Software Rasterizer

Our implementation of a software rasterizer will have four major stages.

  1. Read in triangles
    • We’ll have some way for the user of our software to specify what geometry should be rendered. Initially, you may want to use simple command-line arguments, though eventually we’ll want a way to import triangle meshes from files.
  2. Convert triangels to windows coordinates
    • We need to transform the 3D coordinates of each triangle vertex from world space to window coordinates or pixel coordinates, so that they can be rasterized.
  3. Rasterize each triangle.
    • Rasterization determines which pixels a given triangle covers in pixel space.
    • We’ll use barycentric coordinates to test whether a given point is inside the triangle, and also to interpolate colors across the triangle (if different vertices have different colors)
  4. Write interpolated color values per pixel
    • (using a z-buffer test to resolve depth comparisons)