Curves.ppt

2 The use of

The use of curved surfacescurved surfaces allows for a higher level of allows for a higher level of modeling, especially for the construction of highly modeling, especially for the construction of highly

realistic models. realistic models.

There are several approaches to modeling There are several approaches to modeling

curved surfaces: curved surfaces: (1)

(1) Similar to Similar to polyhedral modelspolyhedral models, , we model an object by using small

we model an object by using small curved surface _{curved surface}

patches

patches (instead of polygons) placed next to each (instead of polygons) placed next to each other.

other. (2)

(2) Another approach is Another approach is solid modelingsolid modeling, ,

that constructs a model using elementary solid that constructs a model using elementary solid objects (such as: polyhedra, spheres, cones etc.) as objects (such as: polyhedra, spheres, cones etc.) as

building blocks. building blocks.

Curved Surfaces

There are two ways to construct a model: There are two ways to construct a model:

This is the process of building the model by This is the process of building the model by

assembling many simpler objects. assembling many simpler objects.

For example, creating a (cylindrical) hole in a sphere For example, creating a (cylindrical) hole in a sphere

or a cube. or a cube.

Additive Modeling

Additive Modeling

This is the process of removing pieces from a given This is the process of removing pieces from a given

object to create a new object. object to create a new object.

Subtractive Modeling

Subtractive Modeling

Curved Surfaces

Curved Surfaces

4

Curve Representation

Curve Representation

There are three ways to represent a curve

Explicit

: y = f(x)

y = mx + b y = x2

(–) Must be a single valued function (–) Vertical lines, say x = d?

Implicit

: f(x,y) = 0

x2 + y2 - r2 = 0

(+) y can be multiple valued function of x (–) Vertical lines? (Continuity hard to detect)

Parametric

: (x, y) = (x(t), y(t))

Explicit Representation

Explicit Representation

Curve in 2D: y = f(x)

Curve in 3D: y = f(x), z = g(x)

Surface in 3D: z = f(x,y)

Problems:

How about a vertical line x = c as y = f(x)?

6

Implicit Representation

Implicit Representation

Curve in 2D: f(x,y) = 0

Line: ax + by + c = 0

Circle: x

2

+ y

2

– r

2

= 0

Surface in 3d: f(x,y,z) = 0

Plane: ax + by + cz + d = 0

Sphere: x

2

+ y

2

+ z

2

– r

2

= 0

f(x,y,z) can describe 3D object:

Parametric Form for

Parametric Form for

Curves

Curves

Curves: single parameter t (e.g. time)

x = x(t), y = y(t), z = z(t)

Circle:

x = cos(t), y = sin(t), z = 0

Tangent described by derivative

Magnitude is “velocity”

 

 

 

 

 

 

 

 

     

 

     

 

 

 

 

  

  

dt t dzdt

t dydt

t dx

dt t dp

t z

t y

t x t

8

Parametric Form for

Parametric Form for

Surfaces

Surfaces

Use parameters u and v

x = x(u,v), y = y(u,v), z = z(u,v)

Describes surface as both u and v vary

Partial derivatives describe tangent plane at

each point

p(u,v) = [x(u,v) y(u,v) z(u,v)]

T

Advantages of Parametric

Advantages of Parametric

Form

Form

Parameters often have natural meaning

Easy to define and calculate

Tangent and normal

10

Lagrange Polynomial

Lagrange Polynomial

Problems:

y=f(x), no multiple values

Higher order functions tend to oscillate

No local control (change any (x_i, y_i) changes the whole curve) Computationally expensive due to high degree.

Given n+1 points (x₀, y₀), (x₁, y₁) …….. (x_n, y_n)

Piecewise Linear

Piecewise Linear

Polynomial

Polynomial

To overcome the problems with Lagrange

polynomial

Divide given points into overlap sequences of 4 points construct 3rd degree polynomial that passes through these points, p₀, p₁ , p₂ , p₃ then p₃, p₄ , p₅ , p₆ etc.

Then glue the curves so that they appear sufficiently smooth at joint points.

Questions: Questions:

1.

12 A curve is approximated by a

A curve is approximated by a piecewise polynomialpiecewise polynomial curve.

curve.

Cubic polynomials

Cubic polynomials are most often used are most often used because:

because: (1)

(1) Lower-degree polynomials offer too little flexibility Lower-degree polynomials offer too little flexibility in controlling the shape of the curve.

in controlling the shape of the curve. (2)

(2) Higher-degree polynomials can introduce unwanted Higher-degree polynomials can introduce unwanted wiggles and also require more computation.

wiggles and also require more computation.

(3)

(3) No lower-degree representation allows a curve No lower-degree representation allows a curve segment to be defined by two given endpoints with segment to be defined by two given endpoints with

given derivative at each endpoints. given derivative at each endpoints. (4)

(4) No lower-degree curves are nonplanar in 3D. No lower-degree curves are nonplanar in 3D.

Why Cubic Curves?

If the

If the directionsdirections (but not necessarily the magnitudes) (but not necessarily the magnitudes) of the two segments’ tangent vectors are equal at a

of the two segments’ tangent vectors are equal at a

join point.

join point.

G

G1 1 GeometricGeometric ContinuityContinuity

If the direction and magnitude of

If the direction and magnitude of QQnn(₍t_t) through the _{) through the}

n

nth derivative are equal at the join point.th derivative are equal at the join point.

C

Cn nParametricParametric ContinuityContinuity

Measure of Smoothness

Measure of Smoothness

If two curve segments join together.

If two curve segments join together.G

G0 0 Geometric_Geometric Continuity_Continuity _ C_C0 0 Parametric_Parametric Continuity_Continuity

C0

If the

If the directions and magnitudes_{directions and magnitudes} of the two of the two

segments’ tangent vectors are equal at a join point.

segments’ tangent vectors are equal at a join point.

C

C1 1 Parametric_Parametric Continuity_Continuity

C1

If the direction and magnitude of

If the direction and magnitude of QQ22(₍t_t) (curvature or _{) (curvature or}

acceleration

acceleration) are equal at the join point.) are equal at the join point.

C

C2 2ParametricParametric ContinuityContinuity

14

Measure of Smoothness

Measure of Smoothness

S

Joint Point

C0

C1

C2

Q₁

Q₂

Q₃ TV₃=2*TV₁ TV₂=TV₁

• QQ₁₁& Q& Q₂₂ are C are C11 and G and G11

continuous continuous

• QQ₁₁& Q& Q₃₃ are G are G11 continuous continuous

only as Tangent vectors only as Tangent vectors

have different magnitude. have different magnitude. • Observe the effect of Observe the effect of

increasing in magnitude increasing in magnitude

of TV of TV

P₀

• By increasing parametric By increasing parametric continuity we can

continuity we can

increase smoothness of

increase smoothness of

the curve.

Desirable Properties of a

Desirable Properties of a

Curve

Curve

Simple control

lines need only two points

curves will need more (but not significantly

more)

Intuitive control

Physically meaningful quantities like position,

tangent, curvature etc.

Global Vs. Local Control

Portion of curve effected by a control point.

General Parameterization

16 Given

Given n _n + 1 points + 1 points P_P00((x_x00, , y_y00), ), P_P11((x_x11, , y_y11), …, ), …, P_Pnn((x_xnn, , y_ynn))

If we require the curve to

If we require the curve to pass throughpass through all the points.

all the points.

Interpolation

Interpolation

we wish to find

we wish to find a curvea curve that, in some sense, that, in some sense, fits the fits the shape outlined by these points

shape outlined by these points..

If we require only that the curve be

If we require only that the curve be nearnear these points.

these points.

Approximation

Approximation

Based on requirements we are faced with Based on requirements we are faced with

two problems: two problems:

Interpolation Vs.

Interpolation Vs.

Approximation

Parametric Representation

Parametric Representation

of Lines

of Lines

Interpolation of two points

In Parametric form:

_P

1 P₂

 

 





_



















































 



























2 1 2 1 1 1 2 1 2 1 1 2 1 1 2 1

1

0

1

1

1

1

1

)

(

)

(

)

(

)

(

)

(

)

(

)

(

x

x

t

t

x

x

t

x

x

x

t

t

x

y

y

t

y

t

y

x

x

t

x

t

x

P

P

t

P

t

P

Parameter T Co-eff C Basis

Matrix Geometry_G

18

 



     



 

 

 













































1

0

,

,

,

2 3 2 3 2 3

t

d

t

c

t

b

t

a

t

z

d

t

c

t

b

t

a

t

y

d

t

c

t

b

t

a

t

x

t

z

t

y

t

x

t

Q

z z z z y y y y x x x x

 

t

T

C

Q







 





d

d

d

c

c

c

b

b

b

a

a

a

t

t

t

t

Q

z y x z y x z y x z y x T

































C

2

1

3

Parametric Cubic Curves

Parametric Cubic Curves

Parametric Cubic Curves

Now co-efficient matrix C can be expressed as a multiple of basis(weight) matrix M and geometry matrix G.

 

t



x

     

t

y

t

z

t



T

C

T

M

G

Q

















vectorgeometry ix

basis matr

G

G

G

G

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

t

t

t

t

Q



















































4 3 2 1

44 43

42 41

34 33

32 31

24 23

22 21

14 13

12 11

2

3

₁

)

(

Each element of geometry vector G has 3 component for x, y and z.

20









x x x x x

g

g

g

g

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

t

t

t

G

M

T

t

x

4 3 2 1 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 2 3

1

)

(

 













_x

function blending a x x x

g

m

tm

m

t

m

t

g

m

tm

m

t

m

t

g

m

tm

m

t

m

t

g

m

tm

m

t

m

t

t

x

4 44 34 24 2 14 3 3 43 33 23 2 13 3 2 42 32 22 2 12 3 1 41 31 21 2 11 3

























































































Parametric Cubic Curves

Parametric Cubic Curves

Multiplying out only the x-component we get

The curve is a weighted sum of the elements of geometry matrix

Derivative of

Derivative of

Q(t)

_Q(t)

Derivative of Q(t) is the parametric tangent vector of the curve.

 

 

 

 

 





 

t



a_xt b_xt c_x a_yt b_yt c_y a_zt b_zt c_z



Q

C t

t C

T dt

d t

Q

t z dt

d t

y dt

d t

x dt

d t

Q dt

t dQ

 

 

 

 

 

 



  

  



2 3

2 3

2 3

0 1

2 3

) (

2 2

2

2

22 A curve segment

A curve segment Q_Q((t_t) is defined by ) is defined by constraintsconstraints on: on:

Each cubic polynomial of

Each cubic polynomial of Q_Q((t_t) has ) has 4 coefficients4 coefficients,, (1)

(1) endpoints endpoints (2)

(2) tangent vectors tangent vectors and

and (3) (3) continuity between segments continuity between segments

so

so 4 constraints4 constraints will be needed, will be needed, allowing us

allowing us to formulate

to formulate 4 equations in the 4 unknowns4 equations in the 4 unknowns,, then solving for the unknowns.

then solving for the unknowns.

Curve Design :

Curve Design :

Determining

A cubic

A cubic Hermite curveHermite curve segment interpolating the segment interpolating the endpoints

endpoints P_P11 and and P_{P4 4} is determined by constraints onis determined by constraints on

the endpoints

the endpoints P_P11 and and P_P44 and

and

tangent vectors at the endpoints

tangent vectors at the endpoints R_R11 and and R_R44

P

P₁₁

P

P₄₄

R

R₁₁ RR44

Hermit Curves

24 The Hermite Geometry Vector:

The Hermite Geometry Vector:

             4 1 4 1 R R P P G_H

 

x x H H H H x x x x x G M t t t G M T C T d t c t b t a t x                1 2 3 2 3

The constraints on

The constraints on x_x(0) and (0) and x_x(1):(1):

 





 

t t t M _H G_Hx x      

 3 2 2 1 0

Hence the tangent-vector constraints:

Hence the tangent-vector constraints:

 





 

R _xx





M_HH G_HH_xx x G M R x         0 1 2 3 1 0 1 0 0 0 4 1

The 4 constraints can be written as:

The 4 constraints can be written as:

x

x H H

26                                0 0 0 1 0 1 0 0 1 2 3 3 1 1 2 2 0 1 2 3 0 1 0 0 1 1 1 1 1 0 0 0 1 H M

 



 

   





 





 



4

2 3 1 2 3 4 2 3 1 2 3 2 3 2 1 3 2 R t t R t t t P t t P t t G B G M T t z t y t x t

Q _{H H H H}

1

1

1

1 tt

f

f((tt))

P

P₁₁ PP₄₄

R

R₁₁

R

R₄₄

The Hermite Blending Functions

The Hermite Blending Functions

Hermit Curves

28

Hermit Curves

Hermit Curves

   

 

   

 

    

 

   

 

7 4 7 4

4 1 4 1

R kR

P P

R R P P

7 6 5 4 3 2

1, P , P , P , P , P , P P

1 1

Indirectly specifies the endpoint tangent vectors by

Indirectly specifies the endpoint tangent vectors by

specifying two intermediate points that are not on

specifying two intermediate points that are not on

the curve.

the curve.

 





 

1 2



2 1



1 Q 0 PP 3 P P

R

 

 



 

 



P

P₁₁

P

P₄₄

P

P₂₂

P

P₃₃

B

30 The B

The Bézier ézier Geometry Vector:Geometry Vector:

             4 3 2 1 P P P P G_B B HB

H M G

P P P P R R P P

G  

                                        4 3 2 1 4 1 4 1 3 3 0 0 0 0 3 3 1 0 0 0 0 0 0 1

 







MH_H MH_HB



G_B H T MHB_B G_BB T G M M T G M T t Q              

B

0 0

0 1

0 0

3 3

0 3

6 3

1 3

3 1

   

 

   

 





 

 

 _{H HB}

B M M

M

 













3 ₄ 3

2 2

2 1

3 _{3 1 3 1}

1 t P t t P t t P t P

G M

T t

Q _{B B}

 

 

 



 



The 4 polynomials in

The 4 polynomials in BB_{B B} = = T . MT . M_{B B} are called the are called the Bernstein polynomials

Bernstein polynomials..

B

32 The Bernstein Polynomials

The Bernstein Polynomials

1

1

1

1

t

t

f

f((tt))

1

B

B

2

B

B

3

B

B

4

B

B

B

B

ézier Curves

_{ézier Curves}

What's a Spline?

What's a Spline?

Smooth curve defined by some control

points

34

Splines

Other Curves

Other Curves

Natural Cubic Spline

C0, C1 and C2 continuous cubic polynomial

Smoother than previous curves which don’t have C2

continuity.

Interpolates all of the control points

B-Splines

C0, C1 and C2 continuous cubic polynomial

Don’t interpolate the control points Varieties:

Uniform Vs Nonuniform Rational Vs Non-rational

36

Q(0)

Q(1)

Catmull-Rom Splines

Catmull-Rom Splines

Interpolation splines i.e. Passes through all control points. Tangent at any control point is parallel to the line joining the control points adjacent to that point.

Curve Rendering

Curve Rendering

Brute-Force method

Forward differencing

Recursive sub-division

Brute-Force method

t = 0;

for (i=0; i <= 100; i++) { x(t) = a_xt3+ b

xt2+ cxt+ dx

y(t) = a_yt3+ b

yt2+ cyt+ dy

x(t) = a_zt3+ b

zt2+ czt+ dz

Plot3d( x(t), y(t), z(t) ); t += 0.01;

}

38

Curve Rendering

Curve Rendering

Curve Rendering

Curve Rendering

Forward differencing method

 _{= 1 / n;}

f =d; _{f = a}3+ b3+ c; f 2= 6a3+ 2b2; f 3= 6a3; Plot(f.x, f.y, f.z );

for (i=0; i <= n; i++) {

f += _{f ;} _{f +=} _f 2; f 2+= f 3; [for each var] Plot(f.x, f.y, f.z );

40

Curve Rendering

Curve Rendering

Recursive sub-division

Void DrawCurveRecSub(curve,  ₎

{

if (Straight(curve,  ₎

DrawLine(curve); else {

SubdivideCurve(curve, leftCurve, rightCurve);

DrawCurveRecSub(leftCurve,  _);

DrawCurveRecSub(RightCurve,  _);

} }

P₁

P₄ P₂

P₃ d₂

d₃

Sub-Division of B

Sub-Division of B

ézier

_ézier

Curves

Curves

                            3 1 Bs L PP PP G D i 1 3 3

1 4 2 0

24 4 0 0

0 0 0 8 81 Bs GL 2 4                         3 1 Bs R P PP G D i 1 3 3 1 4 4 0

00 2 4 2

81

Bs

G

Curved Surfaces

Planer Surfaces

Planer Surfaces

Parametric Bicubic

Parametric Bicubic

Surfaces

Surfaces

Generalization of parametric cubic curves

The elements of geometry matrix are curves

themselves instead of constants.









































)

(

)

(

)

(

)

(

1

)

(

)

,

(

4 3 2 1 2 3

t

G

t

G

t

G

t

G

M

s

s

s

t

G

M

S

t

s

Q

_{s s}

46

Parametric Bicubic

Parametric Bicubic

Surfaces

Surfaces









T T T T

48

Hermite Surfaces

Hermite Surfaces

Hermite Surfaces are completely defined by a 4X4 geometry Matrix G_H

The surface is a

stacking of (s) Curves Each (s) curve is a Hermite Curve

With end points and end tangents as

Hermite Surface

50

Hermite Surface

Hermite Surface

P(0,0) P(0,1) P(1,1) P(1,0) t ) 0 , 0 ( P t   ) 1 , 0 ( P t 

 P(1,1)

Connecting Hermite

Connecting Hermite

Patches

Patches

To join two patches in

s

direction:

Patch 1 Patch 2

t = 1

t = 0

s = 1 s = 0

s = 0 _{s = 1} t = 1

t = 0 t s                                         44 43 42 41 24 23 22 21 44 43 42 41 24 23 22 21 2 1 kg kg kg kg g g g g g g g g g g g g Patch Patch

52

B

B

ézier Surface

_{ézier Surface}

Bézier geometry matrix

G

consists of 16 control

points.

Bézier bicubic formulation can be derived in

exactly the same way as the Hermite cubic:

Ref: FV Chapter 11

11.2, 11.2.1, 11.2.3(as far as covered in class) 11.2.6, 11.2.7, 11.2.9

11.3-11.3.2