CS 284: CAGD
Lecture #7  Thu 9/13, 2012.
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Preparation:
Read: Rockwood: pp 7683; pp 8992 (basic labeling scheme for Blossoming)
Read: Rockwood: pp 94103 (basics of Bsplines)
Questions about Homework Assignment?
Blossoms (cont.)
A geometrical view of iterated "multilevel" inter/extrapolation,
which allows us to see the various spline types in a unified manner and to convert easily between them.
Topic: BSplines

An approximating spline, controlled by the "deBoor points".

Relations between Bézier Curves and BSplines

BSpline in Blossom Form (p 94) {0% down}

Watch the de Boor control points fly...

Control Points of the Uniform BSpline

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 Bsplines)

Iterated Interpolation to find BSpline 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 BSplines

Choosing additional de Boor points more freely; add several extra segments:
 Joining BSpline 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 )

What do we gain from this restriction ? ( C^{degree1} continuity)

What do we pay  if anything ? ( only one new free point per segment)

Periodic
(closed) BSpline Curves (p 105)

Concentrate on one dimension of a BSpline curve: e.g., Y(t)= piecewise mdegree
polynomial.
 Assemble that basis function from m+1 mdegree polynomial pieces, joined with C^{m1} 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 BSplines

Reviewing the standard seven properties:
 e.g., degrees of continuity ... (Comparison Table)

Comparing Bsplines of degrees 2,3, and 4:
 Study their behavior using the applets on pages 101,102.

The use of Bsplines
 Bspline 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 ?
Nonuniform BSplines

Changing the Knot Values
 can assume arbitrary, monotonically ordered tvalues
 does this affect only the parametrization, or also the shape of
the curve ?
 study their influence with applet on p106.{50% down}

Effect on BSpline Curve (Applet on p107 {55%})
 What is the effect on reducing the knot interval ?
 What happens when we doubleup 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,
Knot Multiplicities (an extreme case of nonuniform knot spacings!)
Circle Splines, Spiral Splines, MinimumVariation Curves (MVC)
These are curves that are not subject to the "LinearPrecision Collapse"
Motivation: you want to obtain nice round loops with gradual change in curvature
"Fair, G^{2} and C^{2}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 FreeForm Shape Design" Ph.D. thesis by Henry Moreton (1993).
Homework Assignments:
Review: Rockwood pp. 94117: BSplines
.
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, G^{2} and C^{2}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 email by Monday, September 17, 2012, 8pm.  Thanks.
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Page Editor: Carlo H. Séquin