CS 285: SOLID MODELING

Lecture #6 -- Thu, Sep. 13, 2007.

Self-Test

What does it mean for an object to be "symmetric"?

What are the possible symmetry families for a 3D object of finite size ?

If an object has a 6-fold rotation axis and no mirror or glide planes, what symmetry families can it possibly belong to ?

If an object has a 5-fold rotation axis and a 3-fold axis, what symmetry families can it possibly belong to ?

What class does a bilaterally symmetrical object (like a mouse) belong to ?

What kind of symmetry operations are there in 2D, -- and in 3D, -- and in 4D ?

Brain teaser: -- Why does an ordinary wall mirror reverse left and right, but not up and down ?

Visualization of Symmetry Groups Using Shape Generator Programs

Understanding Chart I:  with "Sculpture Generator I"  {The program for you to experiment with}

Understanding Chart II:  with "Escher Sphere Editor" {The program for you to experiment with}

What are the symmetries of the objects you encounter in your daily life ?

• How to determine the symmetry group of an object:
  Find a maximal-valence rotation axis, make it the z-axis, go to chart 1,  look for C₂ axes perpendicular to it, also for mirror planes, ...
  if you find more than one rotation axis with valence >= 3, go to chart 2;  5-fold axes ==> icosa/dodeca;  4-fold axes at right angles ==> cube/octa, ...
  

From a Graphical Design to a Realizable 3D Solid Part

• Need to distill out an individual tile;
• Need to give the tile some thickness,
• and possibly some relief in the form of a height field;
• Need to provide a (possibly simpler) back-surface;
• Need to connect these two surfaces into a coherent B-rep with side walls;
• Also may have to figure out a coloring pattern for the various tiles.
  

Discussion of Assignment #2 -- Generating Symmetrical Venn Digrams

• How do know whether your solution is satisfactory ?
• How many regions to you expect in your diagram ?
• Pascal's Triangle
• Can you make completely symmetrical Venn diagrams with n variables, for n=4, 6, 7, ... ?
• Some solutions: with ellipses, isoscelese triangles, equilateral triangles, rectangles
  
• 7-fold symmetric solution
  
• Higher order solutions
• THIS YEAR'S SOLUTION
  

New Homework Assignment#3: (Due 9/18/07: 

Pick your FIVE favorite papers for in-class presentation from the handed-out collection.
(This is your opportunity to influence the contents and focus of this course offering.)

New Homework Assignment#4: (Due 9/20/07:  08 am, electronically)

Use the sweep facility in SLIDE to generate your monogram.

Some Basic Concepts Needed:

• Finding the Tangent
• Normal Plane
• Osculating Plane
• Normal vector
• Binormal
• Difference between Normal and second derivative
• The three coordinate planes and their relations to the 3 vectors.

• 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" parameters.

• How to gracefully join the two ends so that prism edges line up
• How well does SLIDE do this ?  Test cases: Hamiltnian cycles on cube, dodeca, face-diagonals of dodeca, ...
  

• Any smooth curve in computer graphics becomes piecewise linear at some point.

•  see sweep description
• explore these parameters in the new homework assignment, study monogram.slf and monomesh.slf