CS488 - Introduction to Computer Graphics - Lecture 31

Modelling

Splines without Tears (or Jerks)

Why do splines exist at all?

• Sometimes we absolutely need curves
• Example: specifiying fonts for SVG

General piece-wise curves

1. One example is line segments: you have been making them all along.

When you put them in a mesh there is an extra requirement

• They must join at the ends: end of one must be the start of the next
• This is continuity.

Linear splines

1. Blending is the key operation.
2. Start with two points P1, P2
3. Get the in-between points using P(t) = P1 + t(P2 - P1) for 0<t<1.

Linear curves necessarily give derivative discontinuities (called C1 continuity)

We can make the discontinuities unimportant by putting the points close enough together

• That is, you give me a curve I can evaluate at any points,
  • I can pick points close enough together that linear interpolation between the points produces negilible derivative discontinuities.
• But how do you specify this curve?
  • algebraic equation
  • OR ???

The usual way is a piecewise continuous non-linear curve, with as much continuitity as you desire at the joins.

Non-linear blending

1. Start with three points P1, P2, P3
2. Blend in pairs
   • P10(t) = P1 + t (P2 - P1)
   • P11(t) = P2 + t (P3 - P2)
3. Blend the blend
   • P20(t) = P10(t) + t (P11(t) - P10(t))
4. The result is a quadratic curve

You can take this to as many levels as you want. What does it give you? Continuity. Which is what?

There are many types of continuity

• Algebraic
  1. C0
     • no break in the curve
     • provided by linear splines
     • Essence is the following
       1. The previous segment P1, P2 is given
       2. You want to add a new segment Q1, Q2
       3. Continuity requires Q1 = P2
       4. User provides Q2
     • Interaction is local
       • Why is locality important?
       • If you change Q1 only two segments change to preserve continuity
       • If you change Q2 only two segments change to preserve continuity
  2. C1
     • Tangent is continuous where segments join.
       • direction
       • length
     • What are the tangents at P3 & Q1?
       1. Linear splines
          • At t=1, dP(t)/dt = P2 - P1
          • At t=0, dQ(t)/dt = Q2 - Q1
          • Condition for tangent continuity obviously too stringent!
          • No local control
       2. Quadratic splines
          • At t=1,
            • dP10(t)/dt = P2 - P1
            • dP11(t)/dt = P3 - P2
            • dP20(t)/dt = P2 - P1 + P11(1) - P10(1) + 1*(P3 - 2P2 + P1) = 2(P3 - P2)
          • At t=0,
            • dQ10(t)/dt = Q2 - Q1
            • dQ11(t)/dt = Q3 - Q2
            • dQ20(t)/dt = Q2 - Q1 + Q11(0) - Q10(0) = 2(Q2 - Q1)
          • Two constraints
            1. Q1 = P3
            2. Q2 = Q1 + P3 - P2 = P3 + (P3 - P2)
          • No local control
       3. Cubic splines
          • blending
          • ...
          • local control
  3. C2
  4. etc.
• Geometric
  1. G0: same as C0
  2. G1: only directions of tangents need to be the same
     • For quadratic splines the two constraints are
       1. Q1 = P3
       2. Q2 - Q1 = a (P3 - P2)

          Q2 = P3 + a (P2 - P3)

     • Local control for quadratic splines

Can be extended to surfaces

• called `spline patches'
• bilinear interpolation on a quadrilateral grid
• isobaric interpolation on a triangular grid

Splines are good for modelling.

• But modelling what?

Animation

Quite simple, really

• Parameters of the model are functions of time.
  • parameters of moving objects
  • parameters of camera, including things like depth of field
• The modeller must specify these functions.
• Constraints.
  • The functions must be physically `realistic'.
  • The functions must be easily specifiable.

`Specifiable': Think splines.

`Realistic': Think continuity.

1. C0 continuity: no teleportation, possible discontinuities in velocity, which violate Newtonian mechanics.
2. C1 continuity: no discontinuities in velocity, possible infinite accelarations.
3. C2 continuity: acceleration continuous, possible infinite jerks.
4. etc.

Most impotant point

Definition of `realistic' varies

1. Between camera and actors
2. As type of actor changes: compare classic Disney animation to live action, which obeys Newtonian dynamics (and other constraints, too)

Return to:

• Bill Cowan's teaching page.
• Bill Cowan's CS488 page for Winter 2009.
• Bill Cowan's CS488 lecture note page for Winter 2009.