CS 284: CAGD
Lecture #8  Tue 9/18, 2012.
PREVIOUS
<     > CS
284 HOME <     > CURRENT
<     > NEXT
Wrapup: BSplines
Vertex Multiplicities on Uniform BSplines (different from Knot Multiplicities, see below)
 It is OK to place deBoor points on top of one another!
 Effects on parametrization? (none)
 Effects on basis functions? (none)
 Effects on the Bspline curve?
 Experiment with the interactive display panels shown in the book on page 102. {30% down}
Extend a cubic curve by 6 more points and then move the de Boor control points
to study what happens to the Bspline curve when you:
 double up two de Boor points at the end? (curve ends on ctrl poly)
 triple a de Boor endpoint? (curve ends at ctrl point)
 give a de Boor endpoint a vertex multiplicity of four? (overkill?)
 double up two internal de Boor points?
 make a triple internal de Boor point? (make a corner at crtl point)
 give an internal de Boor point a multiplicity of four?
 These are all still uniform Bsplines!
Playing with the Knots: NonUniform BSplines
 Changing Knot Values
 Knots can assume arbitrary, monotonically ordered tvalues.
 Does this affect only the parameterization, or also the shape of the curve? (both)
 Study their influence with applet on p107. {55% down}
 What happens when we double up knots 1 and 2, or knots 3 and 4? (curve ends on ctrl poly)

Effect on Basis Functions
 Study the effect of shifting knots for degree 1 . . . 4 basis functions. (Applet on p110) {60% down}
 How far does the effect of a changed knot value spread? (knot moved +/ degree knots on either side)
 New Knots: Knot Insertion (Curve Refinement)
 Knots can be inserted at will (use Blossoming subdivision)
 Knot Multiplicities (an extreme case of nonuniform knot spacings!)

Gives additional design freedom (applet p107) {55% down}

Effects on the basis functions (BBB p162166) ==> see handout.

Effects on the Bspline (BBB p167172) ==> see handout.

Review
of some facts about behavior of knots
Circle Splines, Spiral Splines, MinimumVariation Curves (MVC)
Goal: To obtain nice round loops with gradual change in curvature.
Basic idea: If circles are your ideal shape, build on circular primitives.
Key difference: These are curves that are not subject to the "LinearPrecision Collapse"
"Fair, G^{2} and C^{2}Continuous Circle Splines," by C. H. Séquin, Kiha Lee, and Jane Yen (2005).
Comparison of different interpolation schemes an the interpolation task in "B+B+B", Chapter 3:
Comparison of those schemes on the more challenging example task used at the beginning of this course:
Going beyond Circle Splines:
"From Spiral to Spline: Optimal Techniques in Interactive Curve Design" Ph.D. thesis by Raph Levien (2009).
A summary of some of the above findings:
Interpolating Splines: Which is the fairest of them all? by Raph Levien and C. H. Séquin (2009)
"Minimum Curvature Variation Curves, Networks, and Surfaces for Fair FreeForm Shape Design" Ph.D. thesis by Henry Moreton (1993).
We will talk about this in the context of optimized surfaces.
Bottom Line: Polynomials are not the only possible primitives !
Homework Assignments:
READ: Rockwood Chapter 8: pp 133151: Surfaces
PREVIOUS
<     > CS
284 HOME <     > CURRENT
<     > NEXT
Page Editor: Carlo H. Séquin