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

PREVIOUS < - - - - > CS 284 HOME < - - - - > CURRENT < - - - - > NEXT

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.

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 ? ( Cdegree-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 Cm-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,

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

These are curves that are not subject to the "Linear-Precision Collapse"
Motivation: you want to obtain nice round loops with gradual change in curvature

"Fair, G2- and C2-Continuous Circle Splines," by C. H. Séquin, Kiha Lee, and Jane Yen (2005).

"From Spiral to Spline: Optimal Techniques in Interactive Curve Design" Ph.D. thesis by Raph Levien (2009).

"Minimum Curvature Variation Curves, Networks, and Surfaces for Fair Free-Form Shape Design" Ph.D. thesis by Henry Moreton (1993).

Homework Assignments:

Review: Rockwood pp. 94-117: B-Splines .

Study handout: B+B+B: "Effect of Knot Multiplicities" and answer this question:
-- What is the key difference in the result (if any) of piling multiple knot values on top of each other and placing multiple control points on top of each other?

Read: "Fair, G2- and C2-Continuous Circle Splines," by C. H. Séquin, Kiha Lee, and Jane Yen.  Then write a few sentences on the following issues:
-- What are the one or two key lessons learned from this paper, regarding the design/introduction of a new CAD technique (such as a new curve type)?

Send your answers to both questions to me by e-mail by Monday, September 17, 2012,  8pm. -- Thanks.

PREVIOUS < - - - - > CS 284 HOME < - - - - > CURRENT < - - - - > NEXT
Page Editor: Carlo H. Séquin