CS 284: CAGD
Lecture #8 -- Th 9/21, 2006.

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Preparation:

Read: Rockwood pp 93-117: B-Spline Curve

Topic: 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: 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

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.

Reading Assignments:

Review again: Rockwood pp. 94-117: B-Splines
Study handout: B+B+B: Effect of Knot Multiplicities
Preview Rockwood Chapter 8: pp 133-151: 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.

E-mail to me your SLF-file and a captures picture (JPG, GIF, PNG)

DUE: Thu. 9/28/2006, 2:10pm.