Notes for Wednesday March 31 -- splines 2

More about cubic splines

In class we looked at how to define Hermite splines so that they can easily be strung end to end to make one continuous curve.

The trick is to re-use the P1 and R1 of each Hermite curve segment as the P0 and R0 of the next Hermite curve segment.

All of the code for that is in the included hw7.zip.

I also showed an example of the Catmull-Rom interpolating spline, and we looked at how I had used that to create a text font for a project in VR.

The 16 values of the Catmull-Rom matrix are given by: 

   catmullRomMatrix = [ -.5,1,-.5,0, 1.5,-2.5,0,1, -1.5,2,.5,0, .5,-.5,0,0 ]
It is used to transform any four successive key points (A,B,C,D) of a curve to the cubic coefficients (a,b,c,d) of the corresponding cubic curve (at3 + bt2 + ct + d) between points B and C.

Remember from class that for the first segment of the curve, since there is no point to the left of B, we just set A = B.

Similarly, for the last segment of the curve, since there is no point to the right of C, we just set D = C.

World Builder

At the end of class, we watched Bruce Branit's classic 2007 short film World Builder.

Homework -- due before class on Wednesday April 7

The contour of a surface of revolution can be defined by two functions: z(v) and r(v), where both are functions of parameter v, which varies between 0.0 and 1.0.

As u and v both vary between 0.0 and 1.0, the surface at that value of u is at point:

[ r(v) cosθ , r(v) sinθ , z(v) ].
where 0 ≤ θ < 2π.

  1. Starting with the code we implemented in class, which is in hw7.zip, implement a surface of revolution (SOR). That's a mesh which looks like something that might be created on a lathe or a potter's wheel.

    It can be defined by two spline curves S0 and S1 as follows:

    S0 defines a changing value along the z axis,

    S1 defines a varying radius.

    Both splines are indexed by v.

    Here is the general formula, where ε is a very small positive number (eg: 1/1000): 

       z0 = evalSpline(S0, v)
   z1 = evalSpline(S0, v + ε)

   r0 = evalSpline(S1, v)
   r1 = evalSpline(S1, v + ε)

   tilt = atan2(r0 - r1, z1 - z0)

   x = r0 * cos(2 π u)          // POSITION
   y = r0 * sin(2 π u) 
   z = z0

   nx = cos(tilt) * cos(2 π u) // NORMAL
   ny = cos(tilt) * sin(2 π u)
   nz = sin(tilt)

   vertex = [ x,y,z, nx,ny,nz ]
    Hint: You can define a function uvToSOR(u,v,S) The third argument S is an array that contains two elements: an array of keypoints for S0 and an array of keypoints for S1.

    You can use an Hermite, Bezier or Catmul-Rom spline for this assignment, whichever you prefer.

  2. Use instances of your SOR mesh to create an animated creature. You can use the same SOR mesh or different SOR meshes for different body parts (eg: torso, head, arms, legs, etc). Your creature can be a human, an insect, a dragon, or just something from your imagination.