CS 284: CAGD
Lecture #6 -- Mo 9/18, 2000.

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Preparation:

Rockwood: pp 85-104

Main Lecture Topics

Blossoming; B-Splines; 3D Curves

Using the Blossoming Machinery

What can we do with this machinery ?
- - Review: Extension of a quadratic Bezier curve.
- - Extension of a cubic Bezier curve.
- - Subdivision of a cubic Bezier curve (p82).
Summary of key points about Blossom and Splines.
Blossom evaluation as a generalization of the deCasteljau algorithm (p88).
Illustration of the Cubic Blossom (p89).
- - Find out how the point moves around in the (whole ?) plane.

B-Splines

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

The de Boor Algorithm

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 legal range for a parameter
- - Page 103, bullet 3.
What do we gain from this restriction ?
What do we pay -- if anything ?
- - Page 103, bullet 4.

Relation to the Current Homework Assignment

On-line demo of the capabilities of the SLIDE program pa2.slf

The Behavior of B-Splines

Reviewing the standard seven properties
Degrees of Continuity
Comparing B-Splines of different degrees
- - Study their behavior using the panes on pages 101,102.
The Use of B-Splines
- - B-Spline Curves of degree 3 (p 97)
- - Periodic (closed) B-Spline Curves (p 105)
- - What can you do with a given number of segments ?
- - How many does it take to make a knot ?

Current Homework Assignment: (to be done individually)

Smooth Sweeps along Complex 3D Curves

Implement smooth curve interpolation in 3D.
- - Start from the file " http://www.cs.berkeley.edu/~sequin/CS284/CODE/pa2.slf ".
- - Write the corresponding formulae for the z-components of the inner Bezier control points.
- - Add the necessary z-expressions in all places where I have not already done so.
- - I have added a "Z-scale" slider that multiplies all the new z-coordinates with a slider-controlled constant,
so that you can readily go back and forth between 2D (multiplier=0.0) and 3D (multiplier=1.0).
- - Lift the data points out of the x-y-plane, and convince yourself that everything still "works smoothly."
Use the SLIDE "sweep" construct to form a small ribbon going along a closed B-spline curve approximating the original 15 control points.
- - This can be done by sweeping a "star" type cross section, and giving the star 2 skinny spokes.
- - I have already put in most of the needed code; if you turn on "drawSweep" in the "slf_swp" slider menu, you should see such a ribbon in red, displayed edge-on.
- - However, the ribbon seems to have 10 pinched-off, twisted points in its loop; WHY is this ? -- Write down your thoughts.
- - Under what conditions do you get a nice "prismatic" sweep with no funny points ?
- - How do you have to set parameters such as : "minimizeTorsion" "closed (curve)" "symmetry" "azimuth" ... in order to achieve a nice result ?
- - Now lift the original control points out of the x-y-plane by increasing the "Z-scale" slider.
- - How is the behavior of the sweep different now ?
- - Pay particular attention to the end condition where the loop closes; under what circumstances do you get a smooth, seamless closure ?
- - Answer the same questions also for a 5-pointed star (set "Spokes" to 5).
Now try to make such a nice sweep along your own smooth, composite, interpolating curve.
- - It is somewhat tedious, since you have to specify a separate sweep construct for each of the 15 Bezier segments.
- - But again, the code is already in place, and you can see such a ribbon in green, if you turn on "drawSweep" in the "slf_bez" menu.
- - You may now want to turn off the B-spline at this time, so that you can focus on the Bezier segments.
- - For a 2D curve, as long as the ribbon is perpendicular to the curve's plane, everything should work fairly nicely.
- - But there may be problems in 3D. -- WHY ? What can you do about them ?
Now use all the originally given points, as well as your own calculated control points, to make a closed polyline with 45 joints, and then sweep a 5-pointed star along that polyline.
- - The code for this is right next to the B-spline "backbone" code (near line 467 in pa2.slf).
- - Under what conditions do you get nice prismatic beams with nicely mitered joints ?
- - How do you have to set parameters such as : "minimize Torsion" "closed (curve)" "symmetry" "azimuth" ... in order to achieve a nice result ?
- - Can the "twist" parameter be of any use in achieving nice, properly mitred joints ? HOW ?

DUE: WED 9/20/00, 9:10am.
On line:

Set the initial values for the sliders to the proper values, so that when we execute your program, we will see a nice "prismatic" sweep of a 3-pointed star along your best composite Bezier curve. (You may want to change some of the parameters from the values that you gave them in pa1.) Capture a hardcopy print-out of this sweep.

Put your SLIDE file in the proper place {see instructional page} (same as with your last assignment);

and also mail a copy of your final modified pa2.slf file to j-yen@cs.berkeley.edu

Hand in at the beginning of class:

A hardcopy print-out of your sweep;

and a page or two of text, answering the questions raised above.

Next Reading Assignment:

Rockwood: pp 105-117 (Uniform B-splines)