3D surfaces - Solid representations

The lecture will be a breadth-first search approach of the field, covering main topics to give general ideas of what
can be done when we are talking of surfaces. It should be a first start with key concepts for someone who needs to
deal with surfaces.

Philosophy is :
 

•      Make sure people understand concepts and know about definitions
•      Prepare them for deeper investigation
•      Help them get the ideas with good visualizations of the concepts. If possible give them general toolkits.
•      Have fun, of course, it's Seth's class !

• Definitions of surfaces or volumes - Relation with the way they are constructed :

•  Implicit vs Explicit (introduce next lecture about). 2D, 3D, n-D.

Math def
Manifolds
 
• Parametric surfaces -> NURBS, B-Splines, Bezier, ...
Crash course
Seth NURBS
 

CONSTRUCTIVE SURFACES

 

• --- Procedural, subdivision surfaces -> SIGGRAPH '95 p.180, '94 p.295
 
• --- Volume vs Skin : Volumetric CSG (constructive solid geometry) with trees of binary constructors, etc. -> Roth, CGIP '82, 18(2) p.109-144 + see Justin + OpenGL with stencil buffer.    Brep  (classify the skin, and not every point)
 
 
• Voxel representations... Marching cubes, etc (but the pb is ill-posed...) ? (developed in implicit surfaces lecture)

 Hans demo. Principle too.

http://graphics.lcs.mit.edu/~hkp/volumes.html

 
>>>> yes, just mention it briefly, to be covered more fully in implicit surfs lecture.
 
 

• General surface properties :

• Topological properties : [piecewise] continuity (C0, C1, C2), differentiability, connectedness, completeness (is it the good word I'm not sure ?), maybe compactness "for fun" but I'm not sure everyone will enjoy... Oh you mentionned C-1, too.
 
Insist on huge difference between Continuous vs Discrete
 
• Differential geometry : curvature, geodesics.

>>>> as you wish; please follow your own interests.

• Dynamic Properties

• Non-linear optimization ?  -> Gradient descent, Levenberg-Macquardt

 
• Physics - Finite Elements decomposition. Evolution of surfaces subject to forces. Euler-Lagrange. Hamilton.
If you have some surface to make evolve, Evolver is a convenient tool. see http://www.geom.umn.edu/software/evolver/html/ for documentation and demos. Installed on graphics at /proj2/geomtools/packages/Evolver/
 
 
• Fast Marching Methods and Level Set Methods. 
  3D energy fct to get 2D properties ?  -> snakes. Idea is to preserve continuity
 
 
>>>> these are advanced topics; you should cover them as time allows, or put them in the "discussion" section.