CS 284: CAGD  Lecture #7 -- Tu 9/19, 2006.

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Preparation:

Study:  SLIDE Manual Pages about Sweeps

Topic: Understanding Sweeps

Differential Geometry of 3D Curves  (Handout from Ch5, M. E. Mortenson, "Geometric Modeling", John Wiley & Sons, NYC, 1985)

Analysis of Curves (as compared to "Design" or "Data Fitting")

• Intrinsic Curve Properties (defined at each local point on the curve)
• - - Vectors: Tangent, Normal, Binormal;
• - - Planes: Osculating, Normal, Rectifying;
• - - Scalars: Curvature, Torsion.
• Extrinsic Curve Properties (overall, global values)
• - - Arc-length;
• - - Planarity, Linearity;
• - - Closedness, Turning Number (in 2D), Knot Type (in 3D);
• - - Highest polynomial degree, lowest continuity type, # of inflection points.

Construction of the Frenet Frame (Ref. frame for intrinsic properties)

• Finding the Tangent -- velocity -- p'(t) -- normalize ...
• Normal Plane -- from acceleration = change of velocity -- p''(t) -- twp components:
• in line with tangent = true acceleration.
• perpendicular to tangent = turning. ==> Defines principal normal -- points in the direction that curve bends.
• Osculating Plane -- spanned by tangent and principal normal vectors;  = best-fitting plane.
• Osculating Circle -- best-fitting circle in that plane; -- defines curvature.
• Binormal -- perpendicular on osculating plane; -- axis around which tangent vector turns
• Difference between principal normal and second derivative.
• The three coordinate planes and their relations to the 3 vectors. points in direction that curve bends

Serret - Frenet Relations

• What is Curvature ? -- rate of rotation around B (1/r of the osculating circle)
• What is Torsion ? -- rate of rotation of osculating plane (around tangent)
• Inflection points -- where curvature = 0.
• Which unit vectors serve as local rotation axis ?  (==> see above)
• Which unit vectors could best serve as graphical indicators (along curve) ?
• for curvature: use -n; {n would crowd the inside of the bend} ==> "hedgehog" curves.
• for torsion: use B; { allows both signs; t would crowd the tangent vector}.

Application to Current Programming Assignment

DUE DATE postponed till  THURSDAY, 9/21/06.
• Sweeps along piecewise linear paths.
• How to properly join cylidrical tubes at an angle.
• How to join prismatic tubes, so edges line up.
• "Azimuth" and "axturn".
• How to calculate the torsion minimizing azimuths.
• The "twist" parameter.
• How to gracefull join ends of a closed prismatic loop.
• Application of all this to sweeps along smooth curves.
• The sweep parameters in SLIDE.

Topic: Blossoms:

A geometrical view of iterated linear inter- (and extra-)polation.

• Basic Notation and Machinery
• Create N-digit labels for system with N iterative steps.
• Write down interpolation value used in one step as one digit.
• To capture a polynomial of degree D, N>=D.
• Geometric interpretation leads to algorithms for subdivision, knot-insertion, degree elevation, deCasteljeau, Aitken, conversion from Bezier to B-Splines, evv, ...
• General Properties of Blossoms
• Diagonal agreement,
• Symmetry,
• Multiaffinity property.
• Bezier Curves in Blossom Form
• Control point labelling (p82).
• Generalization
• Arbitrary blossom parametrization ==> EvalBlossomProgram (p88).

Using the Blossoming Machinery

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

ExtendedHomework Assignment:  Smooth Sweeps along Complex 3D Curves

1. 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."

2. 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 two 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 ?
(It is advisable to consult the manual pages on SLIDE ...)
- - Now lift the original control points out of the x-y-plane by increasing the "Z-scale" slider at the bottom of the "slf_bez" slider menu..
- - 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).

ATTENTION: -- the "closed" flag does not seem to work when set dynamically:
closed {expr \$oSweep(closed)}
but will work when set explicitly in the SLIDE file, either as:
closed 0
or:
closed 1

3. 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 ?
>>> Just discuss the issues you encounter in the 3D case. Find out what "hooks" are or are not available in the SLIDE sweep.
Briefly summarize what would have to be done to make nice prismatic sweeps along a composite Bezier curve in 3D,
and how the necessary information may be computed.
No need to actually do all the programming -- unless you want to.

4. 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 ?
>>> Implement the sweep for the polyline case. The SLIDE machinery should handle all that.

DUE: Thu. 9/21/2006, 2:10pm.
Set the initial values for the sliders to the proper values, so that when your program is run,
one 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...)
>>> Without the implementation of (3), this will not work well in 3D.
But the 2D case is much simpler. You should be able to get nice results for the 2D case.

• E-mail me a copy of your final modified pa2.slf SLIDE file.

Capture a hardcopy print-out of this sweep.

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.

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