CS 284: CAGD
Lecture #8  Th 9/21, 2006.
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Preparation:
Read: Rockwood pp 93117: BSpline Curve
Topic: BSplines

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

Relations between Bezier Curves and BSplines

BSpline in Blossom Form (p 94)

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
 Joining BSpline 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 )

What do we gain from this restriction ? ( C^{n1} 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
 study their influence with applet on p105.

New Knots
Reading Assignments:
Review again: Rockwood pp. 94117: BSplines
Study handout: B+B+B: Effect of Knot Multiplicities
Preview Rockwood Chapter 8: pp 133151: Surfaces (  we will talk about rational curves later)
New Homework Assignment: Construct a Parameterized Bell
Your task is to design a curved thin (metal) surface of finite thickness
that has a chance of emitting a nice sound when struck and used as a bell.
There are many different ways in which such a surface could be defined: Some possible approaches to composing a Bell shape.
For this assignment you should concentrate of using a sweep in some
form and using some spline for either the cross secition or for the
guide curve, or for both.
There should be from 3 to 5 parameters with wich you can change the
shape of the bell interactively without loosing continuity in the
overall surface.
Let yourself be inspired by this collection of
pictures and references,
but keep your shape simple!
Create a SLIDE file with all parameters set to their preferred values.
Capture your design pictorially using the screen saver.
 Email to me your SLFfile and a captures picture (JPG, GIF, PNG)
DUE: Thu. 9/28/2006, 2:10pm.
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