CS465 Fall 2006
Homework 6 Solution
-------------------

1.1

The Y component of f is simply the Y coordinate of the spline curve, y(v).
The X and Z components of f form circles at constant Y values, with radius
equal to the spline's x coordinate x(v), that start at (x(v),0) at u=0 and
proceed counter-clockwise around the positive y axis.

       [x]   [ cos(2*pi*u) * (-3v^2 + 3*v)  ]
   f = [y] = [ -v^3 + 3v^2                  ]
       [z]   [ -sin(2*pi*u) * (-3v^2 + 3*v) ]


1.2

The partial derivatives are calculated from the component functions.

           [ -2*pi*sin(2*pi*u) * (-3v^2 + 3*v) ]                     [ 0      ]
   df/du = [ 0                                 ], at (u,v)=(.5,.5) = [ 0      ]
           [ -2*pi*cos(2*pi*u) * (-3v^2 + 3*v) ]                     [ 3*pi/2 ]

           [ cos(2*pi*u) * (-6v+3)  ]                     [ 0   ]
   df/dv = [ -3v^2 + 6v             ], at (u,v)=(.5,.5) = [ 9/4 ]
           [ -sin(2*pi*u) * (-6v+3) ]                     [ 0   ]

The normal to the surface is found by evaluating df/du and df/dv and taking the 
cross product of the resulting vectors:

   normal = df/du x df/dv = [0 0 1] x [0 1 0] = [-1 0 0]


1.3

Yes, it is well-defined, with the value [0 -1 0]. But df/du = [0 0 0] so
you cannot compute it using the method used in 2.

1.4

No, it is not well-defined, and cannot be computed as in 2.



2.1

  tInd[0]  [0,1,2]
  tInd[1]  [0,2,3]
  tInd[2]  [0,3,1]
  tInd[3]  [1,4,2]
  tInd[4]  [2,4,3]
  tInd[5]  [1,3,4]
  
  For any triangle [i,j,k] above, [j,k,i] and [k,i,j] are equivalently correct.
  However, the other 3 permutations are incorrect, because of the right-handed
  convention.


2.2

  tStrip[0] = [1,2,0,3,1,4,2,3]
  
  Other strips certainly exist.


2.3

   For each edge, there are two possible entries in the table:


   edge v1 v2 fL fR pL sL pR sR          edge v1 v2 fL fR pL sL pR sR

    a   0  1  A  C  b  d  f  c     OR     a   1  0  C  A  f  c  b  d            
    b   0  2  B  A  c  e  d  a     OR     b   2  0  A  B  d  a  c  e          
    c   0  3  C  B  a  f  e  b     OR     c   3  0  B  C  e  b  a  f    
    d   1  2  A  D  a  b  h  g     OR     d   2  1  D  A  h  g  a  b         
    e   2  3  B  E  b  c  i  h     OR     e   3  2  E  B  i  h  b  c     
    f   1  3  F  C  g  i  c  a     OR     f   3  1  C  F  c  a  g  i     
    g   1  4  D  F  d  h  i  f     OR     g   4  1  F  D  i  f  d  h     
    h   2  4  E  D  e  i  g  d     OR     h   4  2  D  E  g  d  e  i 
    i   3  4  F  E  f  g  h  e     OR     i   4  3  E  F  h  e  f  g           


   For each vertex, there are 3 or 4 choices of edge (only 1 should be listed)
	
   vertex edge 
      0   a b c
      1   a d f g   
      2   b d e h
      3   c e f i
      4   g h i

   For each face, there are 3 choices of edge (only 1 should be listed)

   face edge
    A   a b d
    B   b c e
    C   a c f
    D   d g h
    E   e h i
    F   f g i