CS 284: CAGD 
Lecture #8 -- Tue 9/18, 2012.

Preparation: Read: "Fair, G²- and C²-Continuous Circle Splines," by C. H. Séquin, Kiha Lee, and Jane Yen. 

Warm-up Exercise: Comparison of Different Splines

Wrap-up:  B-Splines

Vertex Multiplicities on Uniform B-Splines (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 B-spline 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 B-spline curve when you:
1. double up two de Boor points at the end?  (curve ends on ctrl poly)
   
2. triple a de Boor end-point?  (curve ends at ctrl point)
   
3. give a de Boor end-point a vertex multiplicity of four?  (overkill?)
   
4. double up two internal de Boor points?
5. make a triple internal de Boor point?  (make a corner at crtl point)
   
6. give an internal de Boor point a multiplicity of four?
• These are all still uniform B-splines! 

Playing with the Knots: Non-Uniform B-Splines

• Changing Knot Values
• Knots can assume arbitrary, monotonically ordered t-values.
• 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)
  
• This leads to a finer subdivision of the B-spline into individual segments.
  
  
• Knot Multiplicities (an extreme case of non-uniform knot spacings!)
  
• Gives additional design freedom  (applet p107)  {55% down}
  
• Effects on the basis functions (BBB p162-166)   ==> see handout.
• Effects on the B-spline (BBB p167-172)   ==> see handout.
• Review of some facts about behavior of knots

Circle Splines, Spiral Splines, Minimum-Variation Curves (MVC)

• SIGGRAPH'03 Presentation
• Kiha's demos: use Explorer!

• Cubic Hermite interpolation [see B+B+B, Chapter 3]:
• Continuity =?; DoF = ?; Extent of support = ?.
• Non-uniform Lagrange interpolation:
  
• Continuity = ?; DoF = ?; Extent of support = ?.
• Linear positional circle interpolation [Wenz]:
• Continuity = ?; DoF = ?; Extent of support = ?.
• Trigonometric positional circle interpolation [Szilvasi-Nagi]:
• Continuity = ?; DoF = ?; Extent of support = ?.
• Linear angle-interpolated circle splines [Sequin et al]:
• Continuity = ?; DoF = ?; Extent of support = ?.
• Trigonometric angle-interpolated circle splines, reparameterized for C2-continuity [Sequin et al]:
• Continuity = ?; DoF = ?; Extent of support = ?.

• Trigonometric angle-interpolated circle splines, reparameterized for C²-continuity [Sequin et al]
• Linear angle-interpolated circle splines [Sequin et al]
• Trigonometric positional circle interpolation [Szilvasi-Nagi]
• Linear positional circle interpolation [Wenz]
• Non-uniform Lagrange interpolation.

Bottom Line:  Polynomials are not the only possible primitives !

Homework Assignments: