CS 184: COMPUTER GRAPHICS

Lecture #24 -- We: 4/21, 1999.

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Preparation:

Skim: Ch 21 "Animation"

Lecture Topics

Review Bezier Curves

Exercises:
How many DOF's in a cubic Bezier segment in R3 ?
Sketch a Bezier curve from its control points.
Approximate a curve with several Bezier segments.

B-Splines

B-Spline are approximating splines,
composed of subsequent pieces of polynomial curves
that share many of their control points
and thus guarantee a higher degree oc continuity at the joints.
The cubic B-spline has C2 continuity at the joints.

Properties of B-Splines

Piecewise cubic polynomial
Does NOT interpolate control points
Convex hull property
Construction by 3-fold linear interpolation
Invariant under affine transformations
Start-to-end symmetry
Twice differentiable (C2) at joints
Infinitely differentiable everywhere else
May have cusps (may not be G2 everywhere)
Good to make smooth closed loops
 

Bicubic Bezier Patches

Same math as for Bezier curves, applied in two directions !
One direction acts as "control rails" for the other direction, and vice versa.
The total patch needs 16 control points, only the corner points are interpolated.
 

Bicubic B-spline Surfaces

One patch of a cubic B-spline also takes 16 control points,
but it specifies a patch only over a domain corresponding roughly to the central 1/9th of the domain.
Other patches sharing a side with this patch also share 12 control points,
thereby guaranteeing C2 continuity across the seams.
(The weaker dots outside the 16 black control points are "knots" that play a role in the mathematical definition of the B-spline. You may learn more about them in CS284.)
 

Use of Splines in your Final Projects

Smooth camera paths.
Rollercoaster tracks.
Snake bodies.
Curved smooth bodies and shells.
 

Finish Midterm Discussion

 

Ideas for Final Project

Some bug/vehicle exploring an interesting surface (Moebius band) under user control.
Some (small) roller-coaster loop.
Some (imaginary) board game.
Some (weird) machinery.
Some (weird) moving creature on simple surface.
Some (small) fly-through world.

More on Final Project: Animation

Enter Time-Dependence
Choreographed, Coordinated, Constrained Motion
(Will be relevant during final project)

Turning wheels. Interlocking gears. Synchronized gear wheels.
Operable joints. Waving tentacles
Piston, rod, crankshaft.
Jitter bug assemblies.
Uniform or non-uniform scaling ==> "breathing" objects
Real deformations ==> "bellows".

More Advanced Tasks:
Terrain following -- cheat: simplify !
Collision detection -- don’t !

Current Homework Assignment:

ASG#9  "Procedural Generation and Rendering"
DUE: Thursday 4/22/99, 11:59pm.
CAN BE DONE WITH YOUR PARTNER OF CHOICE !

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