(Smooth) CURVES and SURFACES
Guest Lecture #1 for CS 184
Monday 3/28, 2005  2:40pm4:00pm,  306 Soda Hall
( NEXT: Spline Lecture #2)
Introduction to Splines and
Parametric Curve Representations
Even 200 years ago engineers used smooth shapes:
 in ship building for fast, smooth hulls,
 in watch making for the spiral spring that powers the watch ...
Motivational task: How to construct a "smooth" spiral ?
 piecewise linear  yields connectivity;
 composed of circular arcs  yields G1 (tangent) continuity;
 composed of elliptical arcs  yield G2 (curvature) continuity.
How do you represent straight lines, planes ?
How do you generalize that to curves, curved surfaces ?
 parametric representations of a spiral.
 other examples of parametric curves.
What are "splines" ?  Wood strips used in ship building that yield
nice smooth curves,
because nature is trying to minimize overall bending energy (= arclength
integral of curvature squared).
Depending on what fixtures are being used one can apply different constraints
to those strips
 Position Constraints are implemented with pegs, and
 Tangent Constriants can be realized with clamps.
Mathematical "splines" are linearized approximations to the above difficult
optimization problem.
When making smooth shapes by piecing together smooth curves, consider
the
degrees
of smoothness at the joints:
 Parametric Continuity: differentiability of the parametric
representation (C0, C1, C2, ...)
 Geometric Continuity: smoothness of the resulting displayed
shape (G0=C0, G1=tangentcont., G2=curvaturecont. )
Distinguish
between:
 Interpolating splines (pass through all the data points;
example Hermite splines), and
 Approximating splines (only come close to data points; example
BSplines).
Cubic Bezier Curves
These very handy curves are a mixture of the above two "pure" schemes.
Cubic Bezier Curve is defined by:
 2 interpolated endpoints, and
 2 approximated intermediary control points that define the tangent
directions at the endpoints.
Cubic polynomials: ==> allow to make inflection points and true space
curves in 3D.
Basic
underlying math;
(Cubic) Polynomial: infinitely differentiable > Continuity = Cinfinity
Cubic Hermite Spline: 2 endpoints, 2 tangent directions.
Cubic Bezier Curve: 2 endpoints, 2 approximated intermediary control
points.
 (P1,P2) = 1/3 of starting velocity vector
 (P3,P4) = 1/3 of ending velocity vector
Influence of parameters (qualitative view).
DOF's (Degrees of Freedom).
Properties of Bezier Curves
Cubic polynomial, ==> true space curves;
Interpolates endpoints;
Approximation of other two control points;
End tangents defined by pairs of control points;
Convex hull property;
Invariant under affine transformations;
Starttoend symmetry;
Infinitely differentiable;
May have loops, cusps.
Construction of Bezier Curves
DeCasteljau
construction algorithm: (Construction by three iterations of linear
interpolation)
How to draw a Bezier curve from its control points:
 the geometrical construction.
 the polynomial resulting from the repeated interpolations.
Subdivision of a Bezier curve into two pieces
that together are identical to the original one.
Higherorder Bezier Curves and Patches
It
is hard to make "interesting" curves with highorder Bezier curves.
 points in the middle of the curve get influenced by every control point!
But 5th degree (quintic) Bezier curves are very useful if you also want to get control over
curvature at the ends.
There are better ways to make complicated curves with many undulations ...
Stitching Bezier Curves Together
Aiming for more control over complicated curves.
Making Interpolating Curves from several Bezier segments.
Tangent
Continuity Conditions at the joints.
Curvature
Continuity Conditions at the joints.
Counting Degrees of Freedom (DoF)
 Each control point in 2D has 2 DoF.
 Each constraint imposed removes one DoF.
 How many DoF are remaining ?
How to use these DoF's ?
> Optimize the curve !
Global Optimization (approximation)
 Define a cost function:
 E.g., Strain Energy = arclength integral of square of curvature.
How
to approximate a given smooth curve with Bezier segments.
A better way to make extensive smooth curves:
BSplines (an approximating spline)
An easy way to make a smooth C2continuous space curve, e.g., for a path of
an airplane or for a camera path,
is to sprinkle a few control points through
space that define the coarse topology and geometry of the curve
and then
let them be approximated by a
cubic Bspline.
Assuming for the moment a closed curve with N=6 control
points, such a curve would be described as
a sequence
of N=6 cubic curve segments, each one of which is controlled by four
consecutive control points,
but producing a curve segment that is only
about as long as the distance between the two middle control points.
Because
a subsequent curve segment reuses three of the control points of the previous
segment,
they blend together in a very smooth (C2continuous) way.
The
interpolation function is given in all relevant text books:
One segment depending on the 4 control points A,B,C,D,
is given by the polynomial:
Q = A(1 3t +3t^{2} t^{3})/6 + B(4 6t^{2}
+3t^{3})/6 + C(1 +3t +3t^{2} 3t^{3})/6
+ D(t^{3})/6
thus the point at t=0 is given by A/6 + 4B/6 + C/6,
and the point at t=1 is given by B/6 + 4C/6 + D/6.
Properties of Cubic BSplines

Piecewise cubic polynomial;

Does NOT interpolate control points;

Convex hull property;

Construction by 3fold linear interpolation

Invariant under affine transformations;

Starttoend symmetry;

Twice differentiable (C2) at joints;

Infinitely differentiable everywhere else;

May have cusps, thus may not be G2 everywhere;

Good to make
smooth closed loops.
Use of Splines in your Final Projects
Smooth camera paths.
Rollercoaster tracks.
Snake bodies (sweep along a spline curve).
Curved smooth bodies and shells.
REFERENCES  TO LEARN MORE:
A good introductory booklet on Splines with interactive demos:
"Interactive Curves and Surfaces," (with Multimedia Tutorial on CAGD),
A. Rockwood and P. Chambers, Morgan Kaufman Publishers, Inc.
Slides prepared by Prof. O'Brien.
Page Editor: Carlo H. Séquin