CS 284: CAGD 
Lecture #8 -- Mo 9/22, 2003.

Preparation:

Review:

• Blossoms are a multi-parameter representations for polynomials.
• There are N components, where N >= degree of polynomial.
• Blossoms are totally symmetrical in all their components.
• Each component displays an affine distance relationship.
• Blossom labels indicate these interpolated values for all components.
• When all component values are the same, then we get the value of the polynomial for that value.
• When blossoms are drawn graphically in the plane, 2 polynomials (x & y) are being expressed simultaneously.
• Graphical blossoms would work just as well in 3D, if we had the proper tools, 3D graphics, haptics, VR ...

Using the Blossom Machinery:

• Last time: Extensions of a quadratic or cubic Bezier curve.
• Generalized extension of a cubic Bezier curve: applet on page 91.
• Blossom as a visualization of the deCasteljau algorithm:
• Evaluation and subdivision of a cubic Bezier curve (p82).
• Blossom evaluation as a generalization of the deCasteljau algorithm (p88):
• Illustration of the Cubic Blossom (p89).
• Using the applet (p89), find out how the point moves around in the (whole ?) plane.

B-Splines

• An approximating spline, controlled by the "deBoor points".
• Relations between Bezier Curves and B-Splines
• B-Spline in Blossom Form (p 94)
• 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

- - Joining B-Spline Curves (p94)
- - 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^n-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: Y(t)= piecewise m-degree polynomial.

Assemble that function from 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 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
  - - study their influence with applet on p105.
• New Knots can be inserted at will
  - - leading to a finer subdivision of the B-spline into individual segments.

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Current Homework Assignment: (to be done individually)

Program deCasteljau

DUE: Mon. 9/29/2003, 9:10am.

Reading Assignment: