Advanced Graphics,
Dr Neil Dodgson,
University of Cambridge
Computer Laboratory
Part II course, 1999
Part 3: Splines
A: Bezier curves
B: B-splines
C: NURBS
...back to part 2 | on to part 4...
3C) NURBS
NURBS are covered in SMAG section
4 and in some detail in R&A Section 5-13.
Non-uniform rational B-splines are the curves that are currently used
in any graphics application that requires curves and surfaces with
more functionality than Bezier curves can offer. In addition to the
features listed in Part 3B, NURBS are
invarient with respect to perspective transforms.
NURBS curves incorporate -- as special cases -- uniform B-splines,
non-rational B-splines, Bezier curves, lines, and conics. NURBS
surfaces incorporate planes, quadrics, and tori.
Exercises
- Review from IB: What are homogeneous
coordinates and what are they used for in computer graphics?
- Explain how to use homegeneous coordinates to get rational
B-splines given that you know how to produce non-rational B-splines.
- Convince your supervisor that you understand why NURBS includes
Uniform B-splines, Non-Rational B-splines, Beziers, lines, conics,
quadrics, and tori.
- When would you use Bezier curves and when would you use B-splines?
(i.e. why have B-splines, in general, replaced Bezier curves in CAD?)
- [1998/7/12] Consider the design of a user interface for a NURBS
drawing system. Users should have access to the full expressive power
of the NURBS representation. What things should users be able to
modify to give them such access and what effect does each have on the
resulting shape?
- For each of the items (in the previous question) that the user can
edit: (i) Give sensible default values; (ii) Explain how they would be
constrained if a `demo' version of the software was to be limited to
cubic Uniform Non-rational B-Splines.
- [1999/7/11] (c) Show how to construct a circle using non-uniform
rational B-splines (NURBS). (d) Show how the circle definition from
the previous part can be used to define a NURBS torus. [You need
explain only the general principle and the location of the torus'
control points.]
|
Part 3: Splines
A: Bezier curves
B: B-splines
C: NURBS
...back to part 2 | on to part 4...
Neil Dodgson |
Advanced Graphics |
Computer Laboratory
Source file: p3c.html
Page last updated on Thu Oct 7 17:03:00 BST 1999
by Neil Dodgson
(nad@cl.cam.ac.uk)