Computer Graphics Inf4/MSc Computer Graphics Lecture 10 Curves and Surfaces I

Computer Graphics Inf4/MSc

Computer Graphics

Lecture 10

Curves and Surfaces

I


23/10/2007 Lecture 10 2

Types of Curves

• Explicit: y = mx + b r = Arx + Bry + Cr

• Implicit: Ax + By + C = 0 (x – x0)2 + (y – y0)2 – r2 = 0

• Parametric:

x = x0 + (x1 – x0)t x = x0 + rcos

y = y0 + (y1 – y0)t y = y0 + rsin


23/10/2007 Lecture 10 3

Why parametric?

• Parametric curves are very flexible.• They are not required to be functions

– Curves can be multi-valued with respect to any coordinate system.

• Parameter count generally

gives the object’s dimension.

(x(u,v), y(u,v), z(u,v))• Coord functions independent.


23/10/2007 Lecture 10 4

Specifying curves• Control Points:

– A set of points that influence the curve’s shape.

• Knots:– Control points that lie on the curve.

• Interpolating spline:– Curve passes through the control

points.

• Approximating spline:– Control points merely influence

shape.


23/10/2007 Lecture 10 5

Piecewise curve segments

We can represent an arbitrary length curve as a series of curves pieced together.

But we will want to control how these curves fit together …


23/10/2007 Lecture 10 6

Parametric Cubic Curves

• In order to assure C2 continuity our functions must be of at least degree 3. This is also the lowest order to describe a non-planar curve.

• Cubic has 4 degrees of freedom and can control 4 things.

• Use polynomials: x(t) of degree n is a function of t. - y(t) and z(t) are similar and each is handled independently (why parametric is easier to handle than implicit form).

• That is: i

n

ii xatx

0

)(


23/10/2007 Lecture 10 7

An Illustrative Example

• The whole story of parametric splines is in deriving their coefficients.

• How we do that, by satisfying constraints set by the knots and continuity conditions, is what classifies the spline system and distinguishes the different systems.

• Example:

Cubic Hermite Splines


23/10/2007 Lecture 10 8

Hermite splines

• 4 degrees of freedom, 2 at each end to control C0 and C1 continuity at each end.

• Polynomial can be specified by the position of, and gradient at, each endpoint of curve.

• Determine: x = X(t) in terms of x0, x0/, x1, x1

/

Now:

X(t) = a3t3 + a2t2 + a1t + a0

and X/(t) = 3a3t2 + 2a2t + a1


23/10/2007 Lecture 10 9

Finding Hermite coefficientsSubstituting for t at each endpoint:

x0 = X(0) = a0 x0/ = X/(0) = a1

x1 = X(1) = a3 + a2 + a1 + a0 x1/ = X/(1) = 3a3 + 2a2+ a1

And the solution is:

a0 = x0 a1 = x0/

a2 = -3x0 – 2x0/ + 3x1 – x1

/ a3 = 2x0 + x0/ - 2x1 + x1

/


23/10/2007 Lecture 10 10

The Hermite matrix: MH

The resultant polynomial can be expressed in matrix form:

X(t) = tTMHq ( q is the control vector)

/1

1

/0

0

23

0001

0010

1323

1212

1)(

x

x

x

x

ttttX

We can now define a parametric polynomial for each coordinate required independently, ie. X(t), Y(t) and Z(t)


23/10/2007 Lecture 10 11

Hermite Basis (Blending) Functions

x0 x1

x0/

x1/

The graph shows the shape of the four basis functions – often called blending functions.

They are labelled with the elements of the control vector that they weight.

Note that at each end only position is non-zero, so the curve must touch the endpoints


23/10/2007 Lecture 10 12

Family of Hermite curves.

x(t)

y(t)

Note : Start points on left.


23/10/2007 Lecture 10 13

Displaying Hermite curves.

• Simple : – Loop through t – select step size to suit.

– Plug x values into the geometry matrix.

– Evaluate P(t) x value for current position.

– Repeat for y & z independently.

– Draw line segment between current and previous point.

• Joining curves:– Coincident endpoints for C0 continuity

– Align tangent vectors for C1 continuity

.


23/10/2007 Lecture 10 14

Bézier Curves

• Hermite cubic curves are difficult to model – need to specify point and gradient.

• More intuitive to only specify points.• Pierre Bézier specified 2 endpoints and 2

additional control points to specify the gradient at the endpoints.

• Can be derived from Hermite matrix:– Two end control points specify tangent


23/10/2007 Lecture 10 15

Bézier Curves

P2

P1

P4

P3

P3

P4

P2

P1

Note the Convex Hull has been shown as a dashed line – use as a bounding extent for intersection purposes.


23/10/2007 Lecture 10 16

Bézier MatrixWe must first define the blending functions.

For a polynomial of order n (degree n-1), we have n control points with terms up to tn-1, as follows:

fr is a blending function

the coefficient of tr in the binomial expansion of (t+(1-t))n-1

r

n

rrqftX

1

0

)(

)!1(!

)!1()1( 11

11

rnr

nCwherettCf r

nrnrrn

nr


23/10/2007 Lecture 10 17

Bézier Matrix

• The cubic form is the most popular X(t) = tTMBq (MB is the Bézier matrix)

• With n=4 and r=0,1,2,3 we get:

• Similarly for Y(t) and Z(t)

3

2

1

0

23

0001

0033

0363

1331

1)(

q

q

q

q

ttttX


23/10/2007 Lecture 10 18

Bézier blending functionsThis is how they look –

Again the Convex Hull property holds, ie. The functions sum to 1 at any point along the curve.

Endpoints have full weight

The weights of each function is clear and the labels show the control points being weighted.

q0 q3

q1 q2


23/10/2007 Lecture 10 19

Example Interpolation Spline.

i

i

i

i

BCRi

P

P

P

P

T

GMTtP

1

2

3

0020

0101

1452

1331

2

1

)(

Catmull-Rom spline interpolates control points. The gradient at each control point is the vector between adjacent control points.


23/10/2007 Lecture 10 20

Uniform Nonrational B-Splines

• Cubic Bézier curves are joined together in the same way as Hermite curves for arbitrary length by joining short 4-point pieces with C0 and C1 order continuity.

• However cubic B-Splines can have any number of control points and any length of curve with C0, C1 and C2 order continuity.

• Segments of the B-Spline, joined at knots, depend on a few adjacent control points which are shared between segments for continuity.


23/10/2007 Lecture 10 21

What are B-Splines?

• Consider a bi-infinite set of knots {ti}

• B-Splines are a set of local functions which together form a basis for certain classes of piecewise polynomial functions with these knots.

• Any piecewise polynomial of order k, degree k-1, with continuous (k-2)th derivative is a weighted sum of functions Bik from the set:

)()( tBqtg iki

i


23/10/2007 Lecture 10 22

Parametric B-Spline curvesB-Spline functions are used as blending functions:

fi(t) = Bi,k(t)

Given control points: x1 … xn

And B-Splines: Bi,k where i = 1 … n

Then:

Support of function X is tn < t < tn+k

ie. the range over which X is non-zero.

ii

ki xtBtX )()( ,


23/10/2007 Lecture 10 23

Parametric B-Splines

0141

0303

0363

1331

6

1

:4

),...,(:

)(

3)(

1)(

M

kfor

xxQwhere

tttforMQttX

iii

iiiT

Also works for closed curve, ie. wrapped-round set of n knots.


23/10/2007 Lecture 10 24

What does a B-Spline look like?

t0 t1 t2 t3 t4

1

They look rather like the shape above and there is one starting at each t. The one above is a cubic B-Spline with a support of 4 knot intervals. Thus within one interval there are portions of 4 splines which have influence.


23/10/2007 Lecture 10 25

B-Spline blending functions for one interval.

t1

1/6

4/6

f(t) Here again the functions hold the Convex Hull property.

They are non-negative.

There are four of them and therefore another 4x4 matrix as we’ve seen.


23/10/2007 Lecture 10 26

Uniform Non-rational B-Splines.

• For each i 4 , there is a knot between Qi-1 and Qi at t = ti.

• Initial points at t3 and tm+1 are also knots, so we have a total of m-1 knots.

Knot.Control point.

m = 9 (P0 ..P9)m-1 knots

m+1 control points m-2 curve segments


23/10/2007 Lecture 10 27


• First segment Q3 is defined by point P0 through P3 over the range t3 = 0 to t4 = 1.

Knot.Control point.P1

P2

P3

P0

Q3


23/10/2007 Lecture 10 28


• Second segment Q4 is defined by point P1 through P4 over the range t4 = 1 to t5 = 2.

Knot.Control point.

Q4

P1

P3

P4

P2


23/10/2007 Lecture 10 29

Reading for this lecture

• Foley et al. Chapter 11, section 11.2 up to and including 11.2.3

• Introductory text Chapter 9, section 9.2 up to and including section 9.2.4

Documents

Computer Graphics Inf4/MSc Computer Graphics Lecture 10 Curves and Surfaces I