#8: Curves and Curved Surfaces CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006.

#8: Curves and Curved Surfaces

CSE167: Computer GraphicsInstructor: Ronen Barzel

UCSD, Winter 2006

2

Outline for today

Summary of Bézier curves Piecewise-cubic curves, B-splines Surface Patches

3

Curves: Summary

Use a few control points to describe a curve in space

Construct a function x(t) moves a point from start to end of curve as t goes from 0 to 1

tangent to the curve is given by derivative x’(t)

We looked at: Linear -- trivial case, just to get oriented Bézier curves -- in particular, cubic

p0

p1

p0

p1p2

p0

p1

p2

p3

Linear Quadratic Cubic

4

Linear Interpolation: Summary

Given two points p0 and p1

“Curve” is line segment between them

p0

p1

t=1.

. 0<t<1t=0

• Three ways of writing expression:

x(t) = (1 − t)p0 + (t)p1 K Weighted average of the control points

x(t) = (p1 − p0 )t + p0 K Polynomial in t

x(t) = p0 p1[ ]−1 1

1 0

⎡

⎣⎢

⎤

⎦⎥t

1

⎡

⎣⎢⎤

⎦⎥ K Matrix form

5

Cubic Bézier Curve: Summary

Given four points p0, p1, p2, p3

curve interpolates the endpoints intermediate points adjust shape: “tangent handles”

Recursive geometric construction de Casteljau algorithm

Three ways to express curve Weighted average of the control points: Bernstein polynomials Polynomial in t Matrix form

x•

p0

p1

p2

p3

6

Notice: Weights always add to 1 B0 and B3 go to 1 -- interpolating the endpoints

x(t) =B0 t( )p0 + B1 t( )p1 + B2 t( )p2 + B3 t( )p3

The cubic Bernstein polynomials: B0 t( ) =−t3 + 3t2 −3t+1

B1 t( ) =3t3 −6t2 + 3t

B2 t( ) =−3t3 + 3t2

B3 t( ) =t3

Bi (t) =1∑

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

B0 (t) B1(t) B2 (t) B3(t)

Cubic Bézier: Bernstein Polynomials

7

Cubic Bézier: Polynomial, Matrix, Notation

• Polynomial in t :

x(t) =rat 3 +

rbt 2 +

rct + d

ra = −p0 + 3p1 − 3p2 + p3( )rb = 3p0 − 6p1 + 3p2( )rc = −3p0 + 3p1( )

d = p0( )

• Matrix form:

x(t) = p0 p1 p2 p3[ ]

GBez

1 24 44 34 4 4

−1 3 −3 1

3 −6 3 0

−3 3 0 0

1 0 0 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

BBez1 24 44 34 4 4

t 3

t 2

t

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

T{

• (New) Notation:

Bez(t,p0 ,p1,p2 ,p3) = x(t) for the given control points

8

Tangent to Cubic Bézier

• Polynomial in t :

′x (t) = 3rat 2 + 2

rbt +

rc

ra = −p0 + 3p1 − 3p2 + p3( )rb = 3p0 − 6p1 + 3p2( )rc = −3p0 + 3p1( )

d = p0( ) ← d not used in tangent

• Matrix form:

′x (t) = p0 p1 p2 p3[ ]

GBez

1 24 44 34 4 4

−1 3 −3 1

3 −6 3 0

−3 3 0 0

1 0 0 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

BBez1 24 44 34 4 4

3t 2

2t

1

0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

′T{

• Notation:

Be ′z (t,p0 ,p1,p2 ,p3) = ′x (t) for the given control points

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nth-order Bézier curve







x t( ) = Bin t( )pi

i=0

n

∑

Bin t( ) =

ni

⎛⎝⎜

⎞⎠⎟1−t( )n−i t( )

B01 t( ) =−t+1 B0

2 t( ) =t2 −2t+1 B03 t( ) =−t3 + 3t2 −3t+1

B11 t( ) =t B1

2 t( ) =−2t2 + 2t B13 t( ) =3t3 −6t2 + 3t

B22 t( ) =t2 B2

3 t( ) =−3t3 + 3t2

B33 t( ) =t3

10

Evaluate/draw curves by:

Sampling uniformly in t

Adaptive/Recursive Subdivision

x(0.0)

x(0.25)

x(0.5)

x(0.75)

x(t)

x(t)

x(1.0)

11

Outline for today


12

More control points

Cubic Bézier curve limited to 4 control points Cubic curve can only have one inflection Need more control points for more complex curves

With k points, could define a k-1 order Bézier

But it’s hard to control and hard to work with The intermediate points don’t have obvious effect on

shape Changing any control point can change the whole curve

• Want local support: each control point only influences nearby portion of curve





13

Piecewise Curves With a large number of points…

Construct a curve that is a sequence of simple (low-order) curves end-to-end

Known as a piecewise polynomial curve

E.g., a sequence of line segments: a piecewise linear curve

E.g., a sequence of cubic curve segments: a piecewise cubic curve In this case, piecewise Bézier

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Continuity For the whole curve to look smooth, we need to consider

continuity: C0 continuity: no gaps. The segments must match at the endpoints C1 continuity: no corners. The tangents must match at the

endpoints C2 continuity: tangents vary smoothly. (makes smoother curves.)

Note also: G1, G2, etc. continuity• Looks at geometric continuity without considering parametric

continuity• Roughly, means that the tangent directions must match, but not the

magnitudes• Gets around “bad” parametrizations• Often it’s what we really want, but it’s harder to compute







15

Constructing a single curve from many

Given N curve segments: x0(t), x1(t), …, xN-1(t) Each is parameterized for t from 0 to 1

Define a new curve, with u from 0 to N:

Alternate: u also goes from 0 to 1

x(u) =

x0 (u), 0 ≤u≤1x1(u−1), 1≤u≤2M M

xN−1(u− N −1( )), N −1≤u≤N

⎧

⎨⎪⎪

⎩⎪⎪

x(u) =xi (u−i), where i = u⎢⎣ ⎥⎦ (and x(N) = xN−1(1))

x(u) =xi (Nu−i), where i = Nu⎢⎣ ⎥⎦

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Given N+1 points p0, p1, …, pN

Define curve:

N+1 points define N linear segments x(i)=pi

C0 continuous by construction G1 at pi when pi-1, pi, pi+1 are collinear C1 at pi when pi-pi-1 = pi+1-pi

Piecewise-Linear curve

x(u) =Lerp(u−i,pi ,pi+1), i ≤u≤i +1

=(1−u+ i)pi + (u−i)pi+1, i = u⎢⎣ ⎥⎦

p0

p1

p2

p3

p4

p5

p6

x(1.5)

x(5.25)

x(2.9)

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Piecewise Bézier curve: segments

• Given 3N +1 points p0 ,p1,K ,p3N

• Define N Bézier segments :

x0 (t) = B0 (t)p0 + B1(t)p1 + B2 (t)p2 + B3(t)p3

x1(t) = B0 (t)p3 + B1(t)p4 + B2 (t)p5 + B3(t)p6

M

xN −1(t) = B0 (t)p3N −3 + B1(t)p3N −2 + B2 (t)p3N −1 + B3(t)p3N

x0(t)

x1(t)

x2(t)

x3(t)

p0

p1p2

p3

p4p5

p6

p7 p8

p9

p10 p11

p12

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Piecewise Bézier curve

x(u) =

x0 (13 u), 0 ≤u≤3

x1(13 u−1), 3≤u≤6

M M

xN−1(13 u−(N −1)), 3N −3≤u≤3N

⎧

⎨⎪⎪

⎩⎪⎪

x(u) =xi13 u−i( ), where i = 1

3u⎢⎣ ⎥⎦

x0(t)

x1(t)

x2(t)

x3(t)

x(3.5)

x(8.75)

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Piecewise Bézier curve

3N+1 points define N Bézier segments x(i)=p3i

C0 continuous by construction G1 continuous at p3i when p3i-1, p3i, p3i+1 are collinear

C1 continuous at p3i when p3i-p3i-1 = p3i+1-p3i

C2 is harder to get

p0

p0

p1

p2

P3

P3

p2

p1

p4

p5

p6

p6

p5

p4

C1 continuousC1 discontinuous

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Piecewise Bézier curves

Used often in 2D drawing programs Inconveniences:

Must have 4 or 7 or 10 or 13 or … (1 plus a multiple of 3) points

Not all points are the same• UI inconvenience: Interpolate, approximate, approximate, interpolate, …

• Math inconvenience: Different weight functions for each point

UI solution: “Bézier Handles”

Math solution: B-spline

21

UI for Bézier Curve Handles

Bézier segment points: Interpolating points presented normally as curve control

points Approximating points presented as “handles” on the curve

points UI features:

All curve control points are the same Can have option to enforce C1 continuity



(www.blender.org)

22

Blending Functions

Evaluate using “Sliding window”

Local support: Window spans 4 points Window contains 4 different Bernstein polynomials Window moves forward by 3 units when u passes to next Bézier

segment Evaluate matrix in window:

QuickTime™ and aTIFF (LZW) decompressor


x(u) = t3 t2 t 1⎡⎣ ⎤⎦

T1 244 34 4

−1 3 −3 13 −6 3 0−3 3 0 01 0 0 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

BBez

1 24 4 34 4

p3i

p3i+1

p3i+2

p3i+3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

GBez

1 2 3

where t = 13u− i and i = 1

3u⎢⎣ ⎥⎦

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B-spline blending functions

Still a sliding “window”, but use same weight function for every point Weight function goes to 0 outside the 4-point neighborhood local support -- each spot on curve only affected by

neighbors Shift “window” by 1, not by 3

No discrete segments: every point is the same





x(u) = t3 t2 t 1⎡⎣ ⎤⎦

T1 244 34 4

16

−1 3 −3 13 −6 3 0−3 0 3 01 4 1 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

BB−spline

1 24 44 34 4 4

pi

pi+1

pi+2

pi+3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

GB−spline

1 2 3

where t = u − i and i = u⎢⎣ ⎥⎦

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B-Spline

Widely used for surface modeling Intuitive behavior Local support: curve only affected by nearby control

points Techniques for inserting new points, joining curves,

etc. Doesn’t interpolate endpoints

Not a problem for closed curves Sometimes use Bézier blending at the ends

But wait, there’s more! Rational B-Splines Non-uniform Rational B-Splines

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Big drawback of all cubic curves: can’t make circles! Nor ellipses, nor arcs. I.e. can’t make conic sections

Rational B-spline: Add a weight to each point Homogeneous point: use w Weight causes point to “pull” more (or less) With proper points & weights, can do circles

Rational B-splines widely used for surface modeling Need UI to adjust the weight Often hand-drawn curves are unweighted, but

automatically-generated curves for circles etc. have weights

Rational Curves

pull more

pull less

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Non-Uniform Rational B-Spline (NURBS)

Don’t assume that control points are equidistant in u Introduce knot vector that describes the separation of the points

Features… Allows tighter bends or corners Allows corners (C1 discontinuity) Certain knot values turn out to give a Bézier segment Allows mixing interpolating (e.g. at endpoints) and approximating

Math is messier Read about it in Buss

Very widely used for surface modeling Can interactively create as if uniform Techniques for cutting, inserting, merging, filleting, revolving, etc…

27

Outline for today


28

Curved Surfaces

Remember the overview of curves: Described by a series of control points A function x(t) Segments joined together to form a longer curve

Same for surfaces, but now two dimensions Described by a mesh of control points A function x(u,v) Patches joined together to form a bigger surface

29

x(u,v) describes a point in space for any given (u,v) pair u,v each range from 0 to 1 Rectangular topology

Parametric curves: For fixed u0 , have a v curve x(u0,v) For fixed v0 , have a u curve x(u,t0) For any point on the surface, there are a pair

of parametric curves that go through point

Parametric Surface Patch

0 1

1

u

v

x

y

z

x(0.8,0.7)

u

v

x(0.4,v)

x(u,0.25)

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Parametric Surface Tangents

The tangent to a parametric curve is also tangent to the surface

For any point on the surface, there are a pair of (parametric) tangent vectors Note: not necessarily perpendicular to each other

u

v

• Notation:

• The tangent a long a u curve, AKA the tangent in the u direction, is written as :

∂x

∂u(u,v) or ∂

∂u x(u,v) or xu (u,v)

• The tangent a long a v curve, AKA the tangent in the v direction, is written as :

∂x

∂v(u,v) or ∂

∂v x(u,v) or xv (u,v)

• Note that each of these is a vector-valued function:

• At each point x(u,v) on the surface, we have tangent vectors ∂∂u x(u,v) and ∂

∂v x(u,v)

∂x∂u

∂x∂v

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Parametric Surface Normal

Get the normal to a surface at any point by taking the cross product of the two parametric tangent vectors at that point. Order matters!

rn(u,v) =

∂x∂u

(u,v)×∂x∂v

(u,v)

Typically we are interested in the unit normal, so we need to normalize

rn* (u,v) =

∂x

∂u(u,v) ×

∂x

∂v(u,v)

rn(u,v) =

rn* (u,v)rn* (u,v)

∂x∂u

∂x∂v

rn

32

Polynomial Surface Patches x(s,t) is typically polynomial in both s and t

Bilinear:

Bicubic:

x(s, t) =ast+bs+ ct+d

x(s,t) =(at+b)s+ (ct+d) -- hold t constant ⇒ linear in s

x(s,t) = (as+ c)t + (bs+ d) -- hold s constant ⇒ linear in t

x(s, t) =as3t3 +bs3t2 + cs3t2 +ds3 + es2t3 + fs2t2 +gs2t+hs2

+ ist3 + jst2 + kst+ ls+mt3 +nt2 + ot+p

x(s,t) =(at3 +bt2 + ct+d)s3 + (et3 + ft2 +gt+h)s2 -- hold t constant ⇒ cubic in s

+(it 3 + jt 2 + kt + l)s+ (mt 3 + nt 2 + ot + p)

x(s,t) = (as3 + es2 + is+m)t 3 + (bs3 + fs2 + js+ n)t 2 -- hold s constant ⇒ cubic in t

+(cs3 + gs2 + ks+ o)t + (ds3 + hs2 + ls+ p)

33

Bilinear Patch (control mesh)

A bilinear patch is defined by a control mesh with four points p0, p1, p2, p3

AKA a (possibly-non-planar) quadrilateral Compute x(u,v) using a two-step construction

p0 p1

p2

p3

u

v

34

Bilinear Patch (step 1)

For a given value of u, evaluate the linear curves on the two u-direction edges

Use the same value u for both: q0=Lerp(u,p0,p1)

q1=Lerp(u,p0,p1)

p0 p1

p2

p3

u

v

q0

q1

35

Bilinear Patch (step 2)

Consider that q0, q1 define a line segment. Evaluate it using v to get x

p0 p1

p2

p3

u

v

q0

q1

x

x =Lerp(v,q0 ,q1)

36

Bilinear Patch (full) Combining the steps, we get the full formula

p0 p1

p2

p3

u

v

q0

q1

x

x(u,v) =Lerp(v,Lerp(u,p0 ,p1),Lerp(u,p2 ,p3))

37

Bilinear Patch (other order)

Try the other order: evaluate first in the v direction

r0 =Lerp(v,p0 ,p2 ) r1 =Lerp(v,p1,p3)

p0 p1

p2

p3

u

vr0

r1

38

Bilinear Patch (other order, step 2)

Consider that r0, r1 define a line segment. Evaluate it using u to get x

x =Lerp(u,r0 ,r1)

p0 p1

p2

p3

u

vr0

r1

x

39

Bilinear Patch (other order, full)

The full formula for the v direction first:

x(u,v) =Lerp(u,Lerp(v,p0 ,p2 ),Lerp(v,p1,p3))

p0 p1

p2

p3

u

vr0

r1

x

40

Bilinear Patch (either order)

It works out the same either way!

x(u,v) =Lerp(v,Lerp(u,p0 ,p1),Lerp(u,p2 ,p3))x(u,v) =Lerp(u,Lerp(v,p0 ,p2 ),Lerp(v,p1,p3))

p0 p1

p2

p3

u

v

q0

q1

r0

r1

x

41

Bilinear Patch



p0

p1

p2

p3



p0

p1

p2

p3

42

Bilinear patch formulations

• As a weighted average of control points:

x(u,v) = (1− u)(1 − v)p0 + u(1− v)p1 + (1 − u)vp2 + uvp3

• In polynomial form:

x(u,v) = (p0 − p1 − p2 + p3)uv + (p1 − p0 )u + (p2 − p0 )v + p0

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• We have this matrix form for linear curves:

x(t) = p0 p1[ ]−1 1

1 0

⎡

⎣⎢

⎤

⎦⎥t

1

⎡

⎣⎢⎤

⎦⎥

• Can derive this for bilinear patches :

xx (u,v) = v 1[ ]−1 1

1 0

⎡

⎣⎢

⎤

⎦⎥p0 x p1x

p2 x p3x

⎡

⎣⎢

⎤

⎦⎥

−1 1

1 0

⎡

⎣⎢

⎤

⎦⎥u

1

⎡

⎣⎢⎤

⎦⎥

xy (u,v) = v 1[ ]−1 1

1 0

⎡

⎣⎢

⎤

⎦⎥p0 y p1y

p2 y p3y

⎡

⎣⎢

⎤

⎦⎥

−1 1

1 0

⎡

⎣⎢

⎤

⎦⎥u

1

⎡

⎣⎢⎤

⎦⎥

xz (u,v) = v 1[ ]

VT{

−1 1

1 0

⎡

⎣⎢

⎤

⎦⎥

BLinT1 24 34

p0z p1z

p2z p3z

⎡

⎣⎢

⎤

⎦⎥

Gx,y,z

1 24 34

−1 1

1 0

⎡

⎣⎢

⎤

⎦⎥

BLin1 24 34

u

1

⎡

⎣⎢⎤

⎦⎥

U{

• Compactly, we have:

x(u,v) =

VTCxU

VTCyU

VTCzU

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

where

Cx = BLinT GxBLin

Cy = BLinT GyBLin

Cz = BLinT GzBLin

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

are cons tants

Bilinear patch: Matrix form

44

Bilinear Patch Properties of the bilinear patch:

Interpolates the control points The boundaries are straight line segments connecting the control points If the all 4 points of the control mesh are co-planar, the patch is flat If the points are not coplanar, get a curved surface

• saddle shape, AKA hyperbolic paraboloid The parametric curves are all straight line segments!

• a (doubly) ruled surface: has (two) straight lines through every point

Kinda nifty, but not terribly useful as a modeling primitive



p0

p1

p2

p3



p0

p1

p2

p3

45

Bézier Control Mesh

A bicubic patch has a grid of 4x4 control points, p0 through p15

Defines four Bézier curves along u: p0,1,2,3; p4,5,6,7; p8,9,10,11; p12,13,14,15

Defines four Bézier curves along v: p0,4,8,12; p1,6,9,13; p2,6,10,14; p3,7,11,15

Evaluate using same approach as bilinear…

p0 p1

p2

p3

p4 p5

p6

p7

p8 p9

p10

p11

p12 p13

p14 p15

u

v

46

Bézier patch (step 1) Evaluate all four of the u-direction Bézier curves at u, to get

points q0 … q3

p0 p1

p2

p3

p4 p5

p6

p7

p8 p9

p10

p11

p12 p13

p14 p15

u

v

q0

q1

q2

q3

q0 =Bez(u,p0 ,p1,p2 ,p3)q1 =Bez(u,p4 ,p5 ,p6 ,p7 )q2 =Bez(u,p8 ,p9 ,p10 ,p11)q3 =Bez(u,p12 ,p13,p14 ,p15 )

47

Bézier patch (step 2)

Consider that points q0 … q3 define a Bézier curve; evaluate it at v

p0 p1

p2

p3

p4 p5

p6

p7

p8 p9

p10

p11

p12 p13

p14 p15

u

v

q0

q1

q2

q3

x

x(u,v) =Bez(v,q0 ,q1,q2 ,q3)

48

Bézier patch (either order)

Sure enough, you get the same result in either order.

p0 p1

p2

p3

p4 p5

p6

p7

p8 p9

p10

p11

p12 p13

p14 p15

u

v

r0 r1

r2 r3

x


x(u,v) =Bez(v,q0 ,q1,q2 ,q3)

⇔

r0 =Bez(v,p0 ,p4 ,p8 ,p12 )r1 =Bez(v,p1,p5 ,p9 ,p13)r2 =Bez(v,p2 ,p6 ,p10 ,p14 )r3 =Bez(v,p3,p7 ,p11,p15 )

x(u,v) =Bez(u,r0 ,r1,r2 ,r3)

q0

q1

q2

q3

49

Bézier patch Properties:

Convex hull: any point on the surface will fall within the convex hull of the control points

Interpolates 4 corner points. Approximates other 12 points, which act as “handles” The boundaries of the patch are the Bézier curves defined by the

points on the mesh edges The parametric curves are all Bézier curves





50

Evaluating a Bézier patch

Most straightforward: follow the 2-step construction Lets you evaluate patch using existing methods for

evaluting curves

Won’t write out the weighted average or polynomial forms

But can do matrix form…

51

U =

u3

u2

u1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

V =

v3

v2

v1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

BBez =

−1 3 −3 13 −6 3 0−3 3 0 01 0 0 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=BBezT

Cx =BBez

T GxBBez

Cy =BBezT GyBBez

Cz =BBezT GzBBez

Gx =

p0x p1x p2x p3x

p4x p5x p6x p7x

p8x p9x p10x p11x

p12x p13x p14x p15x

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

, Gy= L , Gz= L

x u,v( ) =VTCxU

VTCyU

VTCzU

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Bézier patch, matrix form

52

Bézier patch, matrix form

Cx stores the coefficients of the bicubic equation for x Cy stores the coefficients of the bicubic equation for y Cz stores the coefficients of the bicubic equation for z Gx stores the geometry (x components of the control points) Gy stores the geometry (y components of the control points) Gz stores the geometry (z components of the control points) BBez is the basis matrix (Bézier basis) U and V are the vectors formed from the powers of u and v

Compact notation Leads to efficient method of computation Can take advantage of hardware support for 4x4 matrix

arithmetic

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Tangents of Bézier patch

The “q” and “r” curves we constructed are actually the parametric curves at x(u,v) Remember, tangents to surface = tangents to parametric curves So we can compute the tangents using the same construction curves Notice that the tangent in the u direction is for the curve that

varies in u, i.e. for the “r” curve; Similarly, the tangent in the v direction is for the “q” curve.

(Get the normal as usual by taking cross product of the tangents)

p0 p1

p2

p3

p4 p5

p6

p7

p8 p9

p10

p11

p12p13

p14

p15

u

v

r0 r1

r2 r3

x

q0

q1

q2

q3

∂x∂u

∂x∂v

54

Tangents of Bézier patch, construction


∂x∂v

(u,v) =Be ′z (v,q0 ,q1,q2 ,q3)

r0 =Bez(v,p0 ,p4 ,p8 ,p12 )r1 =Bez(v,p1,p5 ,p9 ,p13)r2 =Bez(v,p2 ,p6 ,p10 ,p14 )r3 =Bez(v,p3,p7 ,p11,p15 )

∂x∂u

(u,v) =Be ′z (u,r0 ,r1,r2 ,r3)

p0 p1

p2

p3

p4 p5

p6

p7

p8 p9

p10

p11

p12p13

p14

p15

u

v

r0 r1

r2 r3

x

q0

q1

q2

q3

∂x∂u

∂x∂v

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Tangents of Bézier patch, matrix form

The matrix form makes it easy to evaluate the tangents directly:

U =

u3

u2

u1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

′U =

3u2

2u10

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

V =

v3

v2

v1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

′V =

3v2

2v10

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

x(u,v) =VTCxU

VTCyU

VTCzU

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

∂x∂u

(u,v) =VTCx ′U

VTCy ′U

VTCz ′U

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

∂x∂v

(u,v) =

VT′CxU

VT′CyU

VT′CzU

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

56

Tessellating a Bézier patch

Uniform tessellation is most straightforward Evaluate points on a grid Compute tangents at each point, take cross product to get

per-vertex normal Draw triangle strips (several choices of direction)

Adaptive tessellation/recursive subdivision Potential for “cracks” if patches on opposite sides of an

edge divide differently Tricky to get right, but can be done.

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Building a surface from Bézier patches

Lay out grid of adjacent meshes For C0 continuity, must share points on the edge

Each edge of a Bézier patch is a Bézier curve based only on the edge mesh points

So if adjacent meshes share edge points, the patches will line up exactly

But we have a crease…

58

C1 continuity across Bézier edges

We want the parametric curves that cross each edge to have C1 continuity So the handles must be equal-and-opposite across the edge:

http://www.spiritone.com/~english/cyclopedia/patches.html

59

Modeling with Bézier patches

Original Utah teapot specified as Bézier Patches



60

Modeling Headaches

Original Teapot isn’t “watertight” spout & handle intersect with body no bottom hole in spout gap between lid and body

Rectangular topology makes it hard to: join or abut curved pieces build surfaces with holes build surfaces with awkward topology or structure

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Advanced surface modeling

B-spline patches For the same reason as using B-spline curves More uniform behavior Better mathematical properties Doesn’t interpolate any control points

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NURBS surfaces. Can take on more shapes

• conic sections• kinks

Can blend, merge, fillet, … Still has rectangular topology, though





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Trim curves: cut away part of surface Implement as part of tessellation/rendering And/or construct new geometry







64

Subdivision Surfaces

Arbitrary mesh, not rectangular topology Not parametric No u,v parameters

Can make surfaces with arbitrary topology or connectivity

Work by recursively subdividing mesh faces Per-vertex annotation for weights,

corners, creases Used in particular for character

animation: One surface rather than collection

of patches Can deform geometry without

creating cracks







65

Done

Next class: Scan Conversion Midterm review

Class after that: Midterm!

#8: Curves and Curved Surfaces CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006.

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Transcript of #8: Curves and Curved Surfaces CSE167: Computer Graphics Instructor: Ronen Barzel UCSD, Winter 2006.