Lect28 Curves+Surfaces2

7/27/2019 Lect28 Curves+Surfaces2

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159.235 Graphics 1

159.235 Graphics & Graphical

Programming

Lecture 28 - Curves & Surfaces II


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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. The following

illustrates an example with control points set P0 P

9:

Knot.

Control point.

m=9 (10 control points)m-1 knots

m-2 knot intervals.


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First segment Q3 is defined by pointP0 throughP3 over the

range t3 = 0 to t4 = 1. So m at least 3 for cubic spline.

Knot.

Control point.P1

P2

P3

P0

Q3m=9 (10 control points)

m-1 knots

m-2 knot intervals.


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Second segment Q4 is defined by pointP1 throughP4 overthe range t4 = 1 to t5= 2.

Knot.

Control point.

Q4

P1

P3

P4

P2

m=9 (10 control points)m-1 knots

m-2 knot intervals.


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Uniform Non-Rational B-Splines

The unweighted spline set will look as follows, 10 control

points, 10 splines, but only 8 knots and 7 knot intervals.

You can see why t3

to t4

is the first interval with a curve

since it is the first with all four B-Spline functions.

t9 to t10 is the last interval, t3 to tm+1 are the knots.

43

t86 mm+1

0


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Cubic B-Splines

We have talked about a bi-infinite set of splines.

What does this look like?

The unweighted B-Splines are shown for clarity. How does a weighted set affect the shape of the

curve?

t4 8 12


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Generating a CurveX(t)

t

t

Opposite we see an

example of a shape to be

generated.

Here we see the curve

again with the weightedB-Splines which

generated the required

shape.


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B-Spline Example

A closed curve is rather like a bi-infinite set

a periodic set ofNknots in which:

for all i, ti+N= ti and ti = i fori = 0..N-1

Consider a closed curve with 5 control

points with 5 knots. N=5, ie. x1 x4, etc.

Control points are: x0 = (2,0); x1 = (1,1);x2 = (-1,1); x3 = (-1,-1) and x4 = (1,-1).


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B-Spline Example

Draw the interval: t = [4..5] , the others are handled the same way.

11

11

11

11

0141

0303

0363

1331

6

11)4()4()4()( 23 ttttX

)0,1211(

)32,32()5()32,32()4(

)4664,664(6

1

))4(4)4(6)4(64,)4(6)4(64(6

1

322

322

midwayand

XandXHence

ttttt

or

ttttt

x0

x1x2

x3 x4

x4

x1


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How Smooth is a B-Spine?

Smoothness increases with orderkin Bi,k Quadratic, k= 3, gives up to C1 order continuity.

Cubic, k= 4 gives up to C2 order continuity.

However, we can lower continuity order too withMultiple Knots, ie.ti = ti+1= ti+2= Knots are coincident and so now we have non-

uniform knot intervals.

A knot with multiplicity m is continuous to the (k-1-m)th

derivative.

A knot with multiplicity khas no continuity at all, ie. the curve isbroken at that knot.


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Non-Uniform Non-Rational B-Splines

Note: as m increases by 1, an extra B-Splinefunction is attached to that knot: qiBi,k

B-Splines have special forms at multiple knots. Obtained from a recursive formula defining how

B-Splines are built, by setting m successive knotsto be equal: tk+1 = tk+2= = tk+m

No point in having m>k Thus for endpoints to be on curve, they must have

a multiplicity ofk, multiplicity of 4 in cubic case.


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B-Splines at Multiple Knots

The multiple-knot B-Splines are as follows:

t

B3,4(t)

B0,4(t)

B1,4(t) B2,4(t)

i=0,1,2,3 4 5 6 7

B0,4 is totally discontinuous at t = t0 , B1,4 is position continuous,

B2,4 is gradient continuous, B3,4 is curvature continuous.


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B-Spline Continuity Example

First knot shown with 4control points, and their

convex hull.

P1

P2P0

P3


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B-Spline Continuity Example

First two curve segmentsshown with their respective

convex hulls.

Centre Knot must lie in the

intersection of the 2 convexhulls.

P1

P2P0P4

P3


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Repeated Control Point

First two curve segments

shown with their respectiveconvex hulls.

Knot is forced to lie on the

line that joins the 2 convex

hulls.

Curve is only C1 continuous

P1=P2

P0P3

P4


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Triple Control Point

First two curve segments

shown with their respective

convex hulls.

Both convex hulls collapseto straight linesall the

curve must lie on these lines.

Curve is only C0 continuous

(Curiously it is actually C2

continuous because tangent

vector magnitude falls to

zero at join. )P1=P2=P3

P0P4


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Non-Uniform Non-Rational B-Splines

Parametric interval between knots does not have to beequal.

Blending functions no longer the same for each interval.

Multiple knots may have to be spaced apart.

Advantages

Continuity at selected control points can be reduced toC1 or lowerallows us to interpolate a control point

without side-effects. Can interpolate start and end points.

Easy to add extra knots and control points.


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Summary of B-Splines Functions that can interpolate a series of control points with C

2

continuity and local control.

Dont pass through their control points, although can be forced.

note that if an orderkB-Spline has kpoints locally colinear then a

straight-line section will result midway in the set, and will touch

(tangentially) the convex hull for (k1) colinear control points.

Uniform

Knots are equally spaced in t.

Non-Uniform

Knots are unequally spaced

Allows addition of extra control points anywhere in the set.


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B-Spline Summary Continued

For interactive curve modelling B-Splines are very good.

To interpolate a number of positions

Catmull-Rom splines are best.

To interpolate with tangent control Hemite or Bzier forms are useful and most often used.

To draw a spline.

Brute force method : Evaluate matrix to get parametric

expressions for coordinates. Then for small increments oft, join with line segments.

Better : use method of forward differences to expressspline in t form.


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Surfacesa Simple Extension

Easy to generalise from cubic curves to bicubicsurfaces.

Surfaces defined by parametric equations of twovariables,s and t.

ie. a surface is approximated by a series ofcrossing parametric cubic curves

Result is a polygon mesh and decreasing step sizeins and twill give a mesh of small near-planarquadrilateral patches and more accuracy.

1010 tands


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Example Bzier surface


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Control of Surface Shape

Control is now a 2D array of control points.

The two parameter surface function, forming the

tensor product with the blending functions is:

Use appropriate blending functions for Bzier and

B-Spline surface functions. Convex Hull property is preserved since bicubic is

still a weighted sum (1).

),(),(

)()(),(

tsZandtsYforsimilarly

qtfsftsX ijjij

i


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Bzier Example

Matrix formulation is as follows:

coordszofarrayisq

tMqMstsz

coordsyofarrayisq

tMqMstsy

coordsxofarrayisq

tMqMstsx

z

TBzB

T

y

T

ByB

T

x

T

BxB

T

44

....),(

44

....),(

44

....),(

Substitute suitable values for

s and t (20 in the above ex.)


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B-Spline Surfaces

Break surface into 4-sided patches choosing suitablevalues fors and t.

Points on any external edges must be multiple knots ofmultiplicity k.

Lot more work than Bzier.

There are other types of spline systems and NURBSmodelling packages are available to make the work mucheasier.

Use polygon packages for display, hidden-surface removaland rendering. (Bzier too)


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Displaying Bicubic Patches

Can calculate surface normals to bicubic surfaces

by vector cross product of the 2 tangent vectors.

Normal is expensive to compute Formulation of normal is a biquintic (two-variable,fifth-

degree) polynomial.

Display.

Can use brute-force methodvery expensive !

Forward differencing method very attractive.


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Curves & Surfaces II - Summary

Bezier Curves

Bicubic patches

Acknowledgments - thanks to Eric McKenzie, Edinburgh, from whose

Graphics Course some of these slides were adapted.

Lect28 Curves+Surfaces2

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