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.
- Show 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.]
- [2000/9/4] (b) A non-rational B-spline has knot vector
[1,2,4,7,8,10,12]. Derive the first of the third order (second
degree) basis functions, N1,3(t), and graph it.
If this knot vector were used to draw a third order B-spline, how many control
points would be required?
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