Advanced Graphics,
Dr Neil Dodgson,
University of Cambridge
Computer Laboratory
Part II course, 2000
Part 4: Other 3D modelling mechanisms
A: Generative models
B: Converting swept objects to polygons
C: Constructive Solid Geometry
D: Implicit surfaces, voxels and marching cubes
E: Subdivision surfaces
...back to part 3  on to part 5...
4A) Generative models
Sweeps
These are three dimensional objects generated by sweeping a
two dimensional shape along a path in 3D. Two special cases of the
sweep are surfaces of revolution, where the path is
a circle; and extrusions, where the path is a straight line.
Surfaces of revolution are covered in R&A section
62. Sweeps are covered in R&A section
63 and FvDFH section 12.4.
Extrusion
On the left (above) is an example extrusion. On the right is its generating polygon (the red star), with the generated 3D object shown in semitransparent cyan.
Surface of revolution
On the left
(above) is an example surface of revolution. On the right (above) is
its generating quadrilateral (the red polygon), with the generated 3D
object shown in semitransparent cyan. Below that is another view of
the same surface of revolution.
Revolution or extrusion?
Some objects can be generated in more than one way. The hollow
cylinder shown above (left) could be generated as either a surface of
revolution (above right) or as an extrusion (immediately right).
General sweeps
If we push the idea of a sweep to its limit we can think of many
things which could be modified to produce a three dimensional swept
shape:

Cross section
Some two dimensional shape that is to be swept along the sweep
path. It does not have to be circular. At right are two swept objects,
one with a circular crosssection, one with a polygonal crosssection.

Sweep path
The path along which the two dimensional cross section is swept to
produce the three dimensional shape. It may be any curve. At right we
see two views of the same swept object: a polygonal
crosssection is swept along a convoluted path.

Twist
How the cross section twists (rotates) as it moves along the sweep
path. The default would be to have no twist at all. At right is a
swept object with and without some twist.

Scale
How the cross section scales (changes size) as it moves along the
sweep path. The default would be to have it stay the same size along
the whole path. Above are a cylinder, and the same cylinder with
different scales along its length.

Normal vector direction
The normal vector of the 2D cross section will usually
point along the sweep path at each point. Changing this will change the
nature of the swept object. See R&A Figure 617
for an example.
At right is a swept object with a circular crosssection,
semicircular path, and varying scale.
You may be able to think of parameters, other than those in the list
above, which could be modify.
Exercises
 [1998/7/12] Show how the following object can be represented as a
swept object.
 Use the following different methods of specifying a geometrical
model for this picture (assuming it's a three dimensional model and not
a line drawing). Come as close as you can to the original for any of the
methods, and describe the difficulties in using a particular method for
this model.
 Extrusions
 Surfaces of revolution
 General sweeps
 For each of the following categories list five realworld objects
which could be represented by the primitives in the category.
 The raytracing primitives in Part 2A
 Extrusions
 Surfaces of revolution
 General sweeps
 A flume (water tunnel) at a swimming complex is modeled as a
circle swept along a particular path. The designers also want to model
the volume swept out by a person traveling down the flume. (We can
approximate the crosssection of a person with something roughly
elliptical and we'll assume the `virtual' person doesn't move legs or
arms while hurtling along.) Explain which parameters in the list
would be need to be modified to specify the shape of the flume and
which would need to be modified to model the volume swept out by a
person traveling down the flume (alternatively, specify which
parameters would be held constant, in each case, for the entire length
of the sweep).

Part 4: Other 3D modelling mechanisms
A: Generative models
B: Converting swept objects to polygons
C: Constructive Solid Geometry
D: Implicit surfaces, voxels and marching cubes
E: Subdivision surfaces
...back to part 3  on to part 5...
Neil Dodgson 
Advanced Graphics 
Computer Laboratory
Source file: p4a.html
Page last updated on Thu Sep 14 16:58:06 BST 2000
by Neil Dodgson
(nad@cl.cam.ac.uk)