Modeling Curves and Surfaces

1
Modeling Curves and Surfaces
2
Motivation

• Need representations of smooth real world objects
• Art / drawings using CG need smooth curves
• Animation camera paths
• Character Paths

3
Naïve approach

• Simply use lines polygons to approximate curves
  surfaces(check out Paint!)
• curve piecewise linear function
• lots of storage if accuracy desired
• hard to interactively manipulate the shape
• Instead, well use higher-order functions
• Note difference between stored model and
  rendered shape

4
Approaches

• Explicit Functions y f(x) e.g. y 2x2
• 1. Only one value of y for each x
• 2. Difficult to represent a slope of infinity
• Implicit Equations f(x,y) 0 e.g. x2 y2 -
  r2 0
• 1. Need constraints to model just one part of a
  curve
• 2. Joining curves together smoothly is
  difficult
• Parametric Equations x f(t), y f(t) e.g.
  x t23, y 3t22t1
• 1. Slopes represented as parametric tangent
  vectors
• 2. Easy to join curve segments smoothly
• See http//www.doc.ic.ac.uk/dfg/AndysSplineTutori
  al/

5
Parametric Curves

• Linear x axt bx
• y ayt by
• z azt bz
• Quadratic x axt2 bxt cx
• y ayt2 byt cy
• z azt2 bzt cz
• Cubic x axt3 bxt2 cxt dx
• y ayt3 byt2 cyt dy
• z azt3 bzt2 czt dz

0 t 1
6
Joining Curve Segments Together

• G0 geometric continuity Two curve segments
  join together
• G1 geometric continuity The directions of
  the two segments tangent vectors are equal at
  the join point
• C1 continuity Tangent vectors of the two
  segments are equal in magnitude and direction
• (C1 ? G1 unless tangent vector 0,0,0)
• Cn continuity Direction and magnitude through
  the nth derivative are equal at the join point

7
Joining Examples

 
 
R2
R1
R0
TV3
TV2
8
Notation

• x axt3 bxt2 cxt dx
• y ayt3 byt2 cyt dy Q(t) T C
• z azt3 bzt2 czt dz

9
Alternate Notation

• Q(t) T M G
• T Matrix Basis Matrix
  Geometry Matrix
• Idea Different curves can be specified by
  changing the geometric information in the
  geometry matrix.
• The basis matrix contains information about the
  general family of curve that will be produced.

10
Example Parametric Line

• x(t) axt bx x'(t) ax
• y(t) ayt by y'(t) ay
• z(t) azt bz z'(t) az

11
Example (cont.)

• Since two points, P1 P2 define a straight line,
  then the geometry matrix should be
• G1 P1 G2 P2.
• What is the basis matrix?
• We know that at t 0, Q(0) P1
• Q'(0) P2 - P1
• at t 1, Q(1) P2
• Q'(1) P2 - P1

t gt 1
t 1
t 0
P2
t lt 0
P1
12
Example (cont.)

• Solve these 4 simultaneous equations to find the
  basis matrix Mline
• Q(0) (0, 1) Mline G P1
• Q'(0) (1, 0) Mline G P2 - P1
• Q(1) (1, 1) Mline G P2
• Q'(1) (1, 0) Mline G P2 - P1

13
Example (cont.)

• m21x1 m22x2 x1 (m11 m21)x1 (m12 m22)x2
  x2
• m21y1 m22y2 y1 (m11 m21)y1 (m12 m22)y2
  y2
• m21z1 m22z2 z1 (m11 m21)z1 (m12 m22)z2
  z2
• m11x1 m12x2 x2 - x1 m11 -1 m12 1
• m11y1 m12y2 y2 - y1 m21 1 m22 0
• m11z1 m12z2 z2 - z1

14
Blending functions

• Multiplying the T and M matrices gives you a set
  of functions in t these are called blending
  functions.
• There is one blending function for each of the
  pieces of geometric information in the G matrix.
• The value of a blending function at a certain
  value of t determines the effect of the
  corresponding piece of geometric information at
  that point along the curve.
• Line blending functions

f(t)
P1 fn.
1
P2 fn.
1
0
t
15
Hermite Curves
Defined by two endpoints and tangents at the
endpoints.
R4
R1
P4
P1
16
Hermite Curve Examples
17
Hermite Curve Examples (cont.)
Demo
18
Hermite Matrices

• Q(t) T MH GH

19
How do we calculate MH?
Assume the following.
20
Hermite Derivation Cont.
Converting to matrices we have Q(t)TC Not we
can also split C into a product of two matrices
C MHGH where G is the set of restrictions on
the curve
21
Tangent vectors ?
22
Recall Q(t)t3 t2 t 1MHGH
The two point restrictions yield the following
23
Tangent equations
The tangent restrictions yield the following
equations.
24
Stacking the four equations
25
Finding the inverse we have
Which result in the following blending functions
26
Hermite Blending Functions

• Q(t) T MH GH

27
Piecewise Curve

• We can create a long continuous curve by
  connecting multiple Hermite curves.

What goes on Here?
28
Another approach

• Here we will blend four points instead of two
  points and two vectors.
• The scheme will use an interesting method called
  tweening in order to obtain points in between
  other points.

29
de Casteljau Algorithm (tweening)
Tweening three points to create a parabola
P1
P1
B
A
P
P
P0
P0
P2
P2
P(t) (1-t)A tB
A(t) (1-t)P0 tP1
substituting
B(t) (1-t)P1 tP2
P(t) (1-t)2 P0 2t(1-t)t P1 t2 P2
Blending functions are in green!
30
Tweening four points
P2
B
P1
E
C
P
D
A
t.3
P3
P0
A(t) (1-t)P0 tP1 B(t) (1-t)P1 tP2 C(t)
(1-t)P2 tP3 D(t) (1-t)A tB etc
Applying proper substitutions we
get (do this for homework!)
P(t)P0(1-t)3 P1 3(1-t)2t P2 3(1-t)t2 P3 t3
Blending functions are in green!
31
Bernstein polynomials
The blending functions are known as the Bernstein
polynomials They exists for any number of points
but we will stay with four. They are closely
related to the well known Pascals triangle whose
third row is represented by
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1
(a b)3 a3 3a2b 3ab2 b3 if a (1-t)
and b t we have (a b)3 (1-t)3 3(1-t)2t
3(1-t)t2 t3
32
The Bernstein polynomials of degree 3
1
33
Bézier Curves
34
Bézier Matrices

• Q(t) T MB GB

• P1 P4 endpoints
• R1 3(P2-P1), R4 3(P4-P3)
• constant velocity curves if control points are
  equally spaced

35
Bézier Blending Functions
36
General Bézier Curves

• Let P1 through Pn1 be points that define the
  curve. Then
• where
• Bi,n(t) C(n,i) ti (1-t)n-i Blending
  functions
• and
• C(n,i) n!/(i!(n-i)!) Binomial
  coefficient

37
General Bézier Curves (cont.)

• For two points, n 1
• QB(t) (1-t)P1 tP2
• For three points, n 2
• QB(t) (1-t)2P1 2t(1-t)P2 t2P3
• For four points, n 3
• QB(t) (1-t)3P1 3t(1-t)2P2 3t2(1-t)P3
  t3P4

38
Bézier Curve Characteristics

• 1. The functions interpolate the first and last
  vertex points.
• 2. The tangent at P0 is given by P1 - P0, and the
  tangent at Pn by Pn - Pn-1.
• 3. The blending functions are symmetric with
  respect to t and (1-t). This means we can
  reverse the sequence of vertex points defining
  the curve without changing the shape of the
  curve.
• 4. The curve, Q(t), lies within the convex hull
  defined by the control points.
• 5. If the first and last vertices coincide, then
  we produce a closed curve.
• 6. If complicated curves are to be generated,
  they can be formed by piecing together several
  Bézier sections.
• 7. By specifying multiple coincident points at a
  vertex, we pull the curve in closer and closer to
  that vertex.

39
B-Spline curves

• Bezier drawbacks
• Moving control points will affect the entire
  Bezier curves.
• A Bezier curve cannot use a cubic curve to
  approximate or represent n points without the
  inconvenience of using multiple curve segments
  (or by increasing the degree of the curve.
• Basically, Bezier curves do not provide enough
  flexibility in curve design

40
B-Splines

• A B-spline is a complete piecewise cubic
  polynomial consisting of any number of curve
  segments.
• A B-spline curve is a series of m-2 curve
  segments that we label Q3,Q4,Qm ,defined by m1
  control points P0,P1,,Pm, mgt3.

41
B-spline formulation

• These curve segments are defined as

Where Qi is the i-th B-spline segment and Pi is
the set of four points in a sequence of control
points
42
In other words
Where Bi-3k represents the blending functions.
i is the segment number and k is the local
control point index. The value of u over a
single curve segment is between 0 and 1. Each
segment is defined by four control points and
each control point influences four curve
segments.
43
Types of B-Spline curves

• Uniform
• Here the blending functions are all the same.
• Nice for designing closed curves
• Non-Uniform
• The blending functions are different at each end
  of the curve
• Not closed

44
Uniform B-Spline Demonstrator
45
Cubic B-spline Blending
B1
B2
B0
B3
Q3
0 1 2 3 4 5 6 7
knots
46
Approximate Gaussian Curve
Bi(u)
ui ui1 ui2 ui3 ui4
47
Closed Uniform B-Splines

• Probably the best application for Uniform
  B-Splines is when the curve is closed.
• This is automatically created when the last
  control point is continued on to the first
  control point.

48
Non-Uniform B-Splines

• In this case the distances between the knots are
  allowed to vary. Recall in the case with uniform
  this distance is set at one.
• This is of course represented by the knot vector.
  In the uniform case it was something like lt0 1 2
  3 4 5 6 7gt
• In the Non-Uniform case it may be something like
  lt0 0 0 0 1 2 3 3 3 gt
• Repeating the knot values changes the blending
  functions so they are not all the same.

49
Non-Uniform B-Splines segments
Demo
curve
Control pts
Blending functions
50
What happens with 4 points
Note that the non- uniform cubic B-spline reduces
to the usual Bezier curves that we have already
studied
Bezier blending functions