Graphics Reference
In-Depth Information
Figure 4.17.
Perspective and orthographic
views of a 2
¥
5
¥
3 block.
In engineering drawings one often shows a perspective view along with three
orthographic views - a top, front, and side view, corresponding to looking along the
z-, y-, and x-axis, respectively. See Figure 4.17. For a more detailed taxonomy of pro-
jections see [RogA90].
Finally, in a three-dimensional graphics program one might want to do some 2d
graphics. For example, one might want to let a user define curves in the plane. Rather
than maintaining a separate 2d structure for these planar objects it would be more
convenient to think of them as 3d objects. Using the orthographic projection, one can
simulate a 2d world for the user.
4.10
Homogeneous Coordinates: Pro and Con
The computer graphics pipeline as we have described it made use of homogeneous
coordinates when it came to clipping. The given reason for this was that it avoids a
division by zero problem. How about using homogeneous coordinates and matrices
everywhere? This section looks at some issues related to this question. We shall see
that both mathematical and practical considerations come into play.
Disadvantages of the Homogeneous Coordinate Representation.
The main dis-
advantage has to do with efficiency. First, it takes more space to store 4-tuples and 4
¥ 4 matrices than 3-tuples and 3 ¥ 4 matrices (frames). Second, 4 ¥ 4 matrices need
more multiplications and additions to act on a point than 3 ¥ 4 matrices. Another dis-
advantage is that homogenous coordinates are less easy to understand than Cartesian
coordinates.
Advantages of the Homogeneous Coordinate Representation.
In a word, the
advantage is uniformity. The composite of transformations can be dealt with in a more
uniform way (we simply do matrix multiplication) and certain shape manipulations
become easier using a homogeneous matrix for the shape-to-world coordinate system
