CS 284: CAGD
Lecture #8  Wed 9/23, 2009.
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Preparation:
Rockwood: pp 94117 (Bsplines)
Topic: BSplines (cont.)
Key Properties of BSplines
 DeBoor algorithm: Iterated Interpolation to find BSpline Curve Points

Graphical
Construction (p98)  compare to deCasteljeau algorithm

Can "t" lie outside the range [2,3] for this example (p99) ?

Graphical Construction for such an extended point (e.g., t=3.5)
 BSpline
Basis Functions
 Strongly overlapping control domains leads to builtin smoothness.

Periodic
(closed) BSpline Curves (p105)
 Endaround reuse of control points.

Bsplines of different degrees (Applet on p102 {30% down})
 How many control segments does it take to make the first curve
segment appear ?
 Note that the quadratic Bspline touches the control polygon  Why ?

Use of Bsplines:
 What can you do with a given number of segments ?
 How many segments does it take to make a knotted 3D space curve
?

Degree
elevation of a Bspline segment
 Using Blossoming and detour via Bezier
control points.
Vertex Multiplicities (different from Knot Multiplicities, see below)
 It is OK to place deBoor points on top of one another!

Effects on parametrization ?

Effects on basis functions ?

Effects on the Bspline curve ?

Experiment with the interactive display panels shown in the book
on page 102.
Extend a cubic curve by 6 more points and then move the de Boor
control points
to study what happens to the Bspline curve when you:

double up two de Boor points at the end ?

tripple a de Boor endpoint ?

give a de Boor endpoint a multiplicity of four ?

double up two internal de Boor points ?

make a tripple internal de Boor point ?

give an internal de Boor point a multiplicity of four ?
 These are all still uniform Bsplines!
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!)
BSpline Derivatives

See handout (BBB p 393398)  read on your own, if and when you need it.
Next Reading Assignments:
Study Rockwood Chapter 8: pp 133151: Surfaces (  we will talk about rational curves later)
Look at the Paper Selection List  Pick your top five candidates 
FIll in: PAPERSELECTION FORM (handout)
DUE: Mon. 9/28/2006, 10:40am.
New Homework Assignments:
Construct a Parameterized Goblet
A goblet is:
1. A drinking vessel, such as a glass, that has a stem and base.
2. Archaic A drinking bowl without handles.
Your task is to design a curved thin (metal) surface of finite thickness
that could serve as a goblet.
There are many different ways in which such a surface could be defined: Some possible approaches to composing a Bell shape also apply to forming a goblet.
For this assignment you should concentrate on 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 goblet interactively without loosing continuity in the
overall surface.
Let yourself be inspired by this collection of images that you get when you do a Google image search for "goblet",  but keep your shape simple!
You may start from this SLIDE file: BellShape.slf
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 captured picture (in JPG, GIF, or PNG)
DUE: Wed. 9/30/2006, 10:40am.
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