Fundamentals of Computer Graphics, 3rd Edition

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7

Viewing

In the previous chapter we saw how to use matrix transformations as a tool for

arranging geometric objects in 2D or 3D space. A second important use of geo-

metric transformations is in moving objects between their 3D locations and their

positions in a 2D view of the 3D world. This 3D to 2D mapping is called a viewing

transformation, and it plays an important role in object-order rendering, in which

we need to rapidly find the image-space location of each object in the scene.

When we studied ray tracing in Chapter 4, we covered the different types of

perspective and orthographic views and how to generate viewing rays according

to any given view. This chapter is about the inverse of that process. Here we

explain how to use matrix transformations to express any parallel or perspective

view. The transformations in this chapter project 3D points in the scene (world

space) to 2D points in the image (image space), and they will project any point on

a given pixel’s viewing ray back to that pixel’s position in image space.

If you have not looked at it recently, it is advisable to review the discussion of

perspective and ray generation in Chapter 4 before reading this chapter.

By itself, the ability to project points from the world to the image is only

good for producing wireframe renderings—renderings in which only the edges

of objects are drawn, and closer surfaces do not occlude more distant surfaces

(Figure 7.1). Just as a ray tracer needs to find the closest surface intersection

along each viewing ray, an object-order renderer displaying solid-looking objects

has to work out which of the (possibly many) surfaces drawn at any given point

on the screen is closest and display only that one. In this chapter, we assume we

are drawing a model consisting only of 3D line segments that are specified by

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142 7. Viewing

Figure 7.1. Left: wireframe cube in orthographic projection. Middle: wireframe cube in

perspective projection. Right: perspective projection with hidden lines removed.

the (x, y, z) coordinates of their two end points. Later chapters will discuss the

machinery needed to produce renderings of solid surfaces.

7.1 Viewing Transformations

The viewing transformation has the job of mapping 3D locations, represented

as (x, y, z) coordinates in the canonical coordinate system, to coordinates in the

image, expressed in units of pixels. It is a complicated beast that depends on

Some APIs use “viewing

transformation” for just the

piece of our viewing trans-

formation that we call the

camera transformation.

many different things, including the camera position and orientation, the type

of projection, the field of view, and the resolution of the image. As with all

complicated transformations it is best approached by breaking it up in to a product

of several simpler transformations. Most graphics systems do this by using a

sequence of three transformations:

• A camera transformation or eye transformation, which is a rigid body trans-

formation that places the camera at the origin in a convenient orientation.

It depends only on the position and orientation, or pose, of the camera.

• A projection transformation, which projects points from camera space so

that all visible points fall in the range −1 to 1 in x and y. It depends only

on the type of projection desired.

• A viewport transformation or windowing transformation, which maps this

unit image rectangle to the desired rectangle in pixel coordinates. It de-

pends only on the size and position of the output image.

To make it easy to describe the stages of the process (Figure 7.2), we give names

to the coordinate systems that are the inputs and output of these transformations.

The camera transformation converts points in canonical coordinates (or world