CS 284: CAGD 
Lecture #8 -- Wed 9/23, 2009.

Preparation:
Rockwood:  pp 94-117 (B-splines)

Warm-up: Comparisons of Splines

Topic:  B-Splines (cont.)

Key Properties of B-Splines

• DeBoor algorithm:  Iterated Interpolation to find B-Spline Curve Points
• Graphical Construction (p98) -- compare to deCasteljeau algorithm
  
• Can "t" lie outside the range [2,3] for this example (p99) ?
• Graphical Construction for such an extended point (e.g., t=3.5)
• B-Spline Basis Functions
  
  • Strongly overlapping control domains leads to built-in smoothness.
• Periodic (closed) B-Spline Curves (p105)
• End-around re-use of control points.
• B-splines of different degrees (Applet on p102  {30% down})
• How many control segments does it take to make the first curve segment appear ?
• Note that the quadratic B-spline touches the control polygon -- Why ?
• Use of B-splines:
• What can you do with a given number of segments ? 
• How many segments does it take to make a knotted 3D space curve ?
• Degree elevation of a B-spline segment
• Using Blossoming and detour via Bezier control points.

Vertex Multiplicities (different from Knot Multiplicities, see below)

• It is OK to place deBoor points on top of one another!
• Effects on parametrization ?
• Effects on basis functions ?
• Effects on the B-spline curve ?
• Experiment with the interactive display panels shown in the book on page 102.
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 ?
2. tripple a de Boor end-point ?
3. give a de Boor end-point a multiplicity of four ?
4. double up two internal de Boor points ?
5. make a tripple internal de Boor point ?
6. give an internal de Boor point a multiplicity of four ?
• These are all still uniform B-splines!
  

Non-uniform B-Splines

• Changing the Knot Values
  
• can assume arbitrary, monotonically ordered t-values
• does this affect only the parametrization, or also the shape of the curve ?
• study their influence with applet on p106.{50% down}
  
• Effect on B-Spline Curve (Applet on p107 {55%})
• What is the effect on reducing the knot interval ? 
• What happens when we double-up knots 1 and 2, or knots 3 and 4 (Applet on p 107)
• Effect on Basis Functions (Applet on p110 {60% down})
• Study effect of shifting knots for degree 1 ... 4 basis functions.
• How far does the effect of a changed knot value spread ?
• Knot Insertion (Curve Refinement)
  
• Knots can be inserted at will,
• leading 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
• Effects on the basis functions (BBB p162-166)
• Effects on the B-spline (BBB p167-172)   ==> see handout.
• Review of some facts about behavior of knots

B-Spline Derivatives

• See handout (BBB p 393-398) -- read on your own, if and when you need it.
  

Next Reading Assignments:

DUE:  Mon. 9/28/2006, 10:40am.

New Homework Assignments:

Construct a Parameterized Goblet

Create a SLIDE file with all parameters set to their preferred values.
Capture your design pictorially using the screen saver.

• E-mail to me your SLF-file and a captured picture (in JPG, GIF, or PNG)
  

DUE:  Wed. 9/30/2006, 10:40am.