CS 284: CAGD 
Lecture #7 -- Thu 9/13, 2012.

Preparation:

Read:  Rockwood:  pp 76-83; pp 89-92 (basic labeling scheme for Blossoming)

Read:  Rockwood:  pp 94-103 (basics of B-splines)

Warm-up Exercise: Designing a C2-continuous interpolating curve

Questions about Homework Assignment?

Blossoms (cont.)

A geometrical view of iterated "multi-level" inter-/extra-polation,
which allows us to see the various spline types in a unified manner and to convert easily between them.

• Blossom evaluation as a generalization of the deCasteljau algorithm (p88).
• Summary of key points about Blossom and Splines. (p83).

Topic:  B-Splines

• An approximating spline, controlled by the "deBoor points".
• Relations between Bézier Curves and B-Splines
• B-Spline in Blossom Form (p 94) {0% down}
  
• Watch the de Boor control points fly...
  
• Control Points of the Uniform B-Spline
• Finding de Boor Points Geometrically
• Interpretation: Change to a new set of basis functions -- some linear combination of old ones.

The deBoor Algorithm (= deCasteljau for B-splines)

• Iterated Interpolation to find B-Spline Curve Points
• Graphical Construction (p98)
• Can "t" lie outside the range [2,3] for this example (p99) ?
• Graphical Construction for such an extended point (e.g., t=3.5)
• What is this curve that we are constructing ?
• Finding additional de Boor points for this curve e.g., "456"
• Will using "456" lead to the same curve point for t=3.5 ?

Multi Segmented B-Splines

• Choosing additional de Boor points more freely; add several extra segments:
  
• Joining B-Spline Curves (p 96) {5% down}
  
• Study influence of de Boor control points (p97)
• Concept of limited support
• The valid range for the curve parameter (e.g., 3 4 5   4 5 6   5 6 7   6 7 8 )
• Page 103, bullet 3.
• What do we gain from this restriction ? ( C^degree-1 continuity)
• What do we pay -- if anything ? ( only one new free point per segment)
• Page 103, bullet 4.
•  Periodic (closed) B-Spline Curves (p 105)

B-Spline Basis Functions

• Concentrate on one dimension of a B-Spline curve: e.g., Y(t)= piecewise m-degree polynomial.
• Assemble that basis function from  m+1  m-degree polynomial pieces, joined with C^m-1 continuity.
• How to construct such basis functions: Repeated convolution
• m=1 : triangular hat functions
• m=2 : three parabolic pieces
• m=3 : four cubic pieces

• The limited support of these basis functions
• Comparison with Bezier Basis Functions

The Behavior of B-Splines

• Reviewing the standard seven properties:
• e.g., degrees of continuity ... (Comparison Table)
• Comparing B-splines of degrees 2,3, and 4:
• Study their behavior using the applets on pages 101,102.
• The use of B-splines
• B-spline curves of degree 3  (p 97)
• What can you do with a given number of segments ?
• How many segments does it take to make a 3D knotted space curve ?

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

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

Homework Assignments: