Bezeir Curves

38
Jan 20, 2022 3D Object Representation 1 3 D Object Representation

Transcript of Bezeir Curves

Page 1: Bezeir Curves

Apr 11, 2023 3D Object Representation 1

3 D Object Representation

Page 2: Bezeir Curves

Apr 11, 2023 3D Object Representation 2

Spline Representations Spline: In drafting terminology

A flexible strip used to produce a smooth curve through a designated set of points.

Several small weights are distributed along the strip. Spline Curve: In computer graphics

Any composite curve formed with (cubic) polynomial sections satisfying specified continuity conditions at the boundary of the pieces.

Spline surface Described with two sets of orthogonal spline curves.

Page 3: Bezeir Curves

Apr 11, 2023 3D Object Representation 3

Interpolation and Approximation Splines Control points

A set of coordinate positions which indicates the general shape of the curve.

A spline curve is defined, modified, and manipulated with operations on the control points.

Interpolation curves: the curve passes through each control points Digitize drawings Specify animation path

Approximation curves: fits to the general control-points path without necessarily passing through any control points

Page 4: Bezeir Curves

Apr 11, 2023 3D Object Representation 4

Interpolation and Approximation Splines

Page 5: Bezeir Curves

Apr 11, 2023 3D Object Representation 5

Convex hull The convex polygon boundary that encloses a set of

control points. Provide a measure for the deviation of a curve or

surface from the region bounding the control points

A convex hull for 2 sets of control points

Page 6: Bezeir Curves

Apr 11, 2023 3D Object Representation 6

Control graph A set of connected line segments connecting the

sequence of control points.

Page 7: Bezeir Curves

Apr 11, 2023 3D Object Representation 7

Parametric Continuity Continuity conditions

To ensure a smooth transition from one section of a piecewise parametric curve to the next.

Zero-order (C0) continuity First-order (C1) continuity Second-order (C2) continuity

Page 8: Bezeir Curves

Apr 11, 2023 3D Object Representation 8

Zero-order (C0) continuity

Simply the curves meet.

Page 9: Bezeir Curves

Apr 11, 2023 3D Object Representation 9

First-order (C1) continuity First parametric derivatives

(tangent lines) of the co ordinate functions of 2 successive curves are equal at their joining point.

Parametric coordinate form x =x(u), y=y(u) z=z(u) u1<=u<=u2

Page 10: Bezeir Curves

Apr 11, 2023 3D Object Representation 10

Second-order (C2) continuity

Both first and second parametric derivatives of the 2 curve sections are the same at intersection.

Page 11: Bezeir Curves

Apr 11, 2023 3D Object Representation 11

Parametric Continuity

Page 12: Bezeir Curves

Apr 11, 2023 3D Object Representation 12

Spline Specifications Three methods for specifying a spline

A set of boundary conditions imposed on the spline.

The matrix that characterizes the spline. The set of blending functions (or basis

functions)

Page 13: Bezeir Curves

Apr 11, 2023 3D Object Representation 13

1.Boundary Conditions

Suppose x(u)=axu3+bxu2+cxu+dx

be the cubic polynomial representation Endpoints coordinates - x(0) and x(1). First derivatives at the endpoints x’(0) and

x’(1). Determine the values of ax, bx, cx, and dx. Form the matrix from the boundary conditions.

Page 14: Bezeir Curves

Apr 11, 2023 3D Object Representation 14

2.Matrix Representation

U - row matrix of parameter u , C – Coefficient column matrix.

C = Mspline . Mgeom

Mgeom – control point coordinate values(4 element column matrix

Mspline – 4x4 matrix – characterization for the spline curve x(u) = U. Mspline . Mgeom

Page 15: Bezeir Curves

Apr 11, 2023 3D Object Representation 15

3.Blending Function

gk- constant parameters such as control points, slope of curve at the control points.

BFk(u) – blending func

Page 16: Bezeir Curves

Apr 11, 2023 3D Object Representation 16

Bezeir Curves and Surfaces

Page 17: Bezeir Curves

Apr 11, 2023 3D Object Representation 17

“Computers can’t draw curves.”

The more points/line segments that are used, the smoother the curve.

Page 18: Bezeir Curves

Apr 11, 2023 3D Object Representation 18

Bezier Curves

An alternative to splines. M.Bezeir a French mathematician who

worked for the Renault motor car company. He invented his curves to allow his firm’s

computers to describe the shape of car bodies.

Page 19: Bezeir Curves

Apr 11, 2023 3D Object Representation 19

? and provide to How 41 R R

1P4P

1R 4R

mation Approxi

and

!!! points control additional two provide , and providing of Instead

32

41

PP

RR

1P4P

2P 3P

Bezier Approximation

Page 20: Bezeir Curves

Apr 11, 2023 3D Object Representation 20

Bezier Curves Contd

Typically, cubic polynomials Need four points

Two at the ends of a segment Two control tangent vectors

Page 21: Bezeir Curves

Apr 11, 2023 3D Object Representation 21

Bezier Curves Contd

Control polygon: control points connected to each other

Easy to generalize to higher order Insert more control points

Page 22: Bezeir Curves

Apr 11, 2023 3D Object Representation 22

De Casteljau Algorithm

Can compute any point on the curve in a few iterations.

No polynomials, pure geometry. Repeated linear interpolation

Repeated order of the curve times The algorithm can be used as definition of the

curve.

Page 23: Bezeir Curves

Apr 11, 2023 3D Object Representation 23

De Casteljau Algorithm Contd

Page 24: Bezeir Curves

Apr 11, 2023 3D Object Representation 24

De Casteljau Algorithm Contd

Page 25: Bezeir Curves

Apr 11, 2023 3D Object Representation 25

De Casteljau Algorithm Contd

Page 26: Bezeir Curves

Apr 11, 2023 3D Object Representation 26

De Casteljau Algorithm Contd

Page 27: Bezeir Curves

Apr 11, 2023 3D Object Representation 27

De Casteljau Algorithm Contd

3P

1P 2P

t

t1

t1 t1

t1t1

t

tt

t

0P0P

1P

2P

3P

t1

t

t1

t

t1

t

10)1( tPPt

21)1( tPPt

32)1( tPPt

t1

t

t1

t

22

102 )1(2)1(

Pt

tPtPt

32

212 )1(2)1(

Pt

tPtPt

t1

t

33

22

12

03

)1(3

)1(3

)1(

Pt

Ptt

tPt

Pt

ii

i

tti

33

0

)1(3

Page 28: Bezeir Curves

Apr 11, 2023 3D Object Representation 28

0

1

1

3)1( t2)1(3 -tt

3t

)1(3 2 -tt

10)1(3

)1(3)1(3)1()(

0001

0033

0363

1331

1)(

33

0

43

32

22

13

4

3

2

1

23

t, Ptt

i

PtP-ttP-ttPttQ

GMT

P

P

P

P

ttttQ

iii

i

bb

3))1(( tt

Cubic Bezeir Curve -Matrix Representation

Page 29: Bezeir Curves

Apr 11, 2023 3D Object Representation 29

Bezier Polynomial Function

iniin tt

ini

ntB

)1.(.

)!(!

!)(

Page 30: Bezeir Curves

Apr 11, 2023 3D Object Representation 30

Parametric Equations for x(t),y(t)

)(.)(0

tBxtx in

n

ii

)(.)(0

tByty in

n

ii

Page 31: Bezeir Curves

Apr 11, 2023 3D Object Representation 31

Cubic Bezeir func in Matrix Form

In cubic matrix form

Normal matrix form

Page 32: Bezeir Curves

Apr 11, 2023 3D Object Representation 32

Cubic Bezier Curves Contd Bezeir matrix is

Page 33: Bezeir Curves

Apr 11, 2023 3D Object Representation 33

Design Techniques Using Bezier Curves

Closed Bezier curves Specifying the first and last control points

at the same position.

Page 34: Bezeir Curves

Apr 11, 2023 3D Object Representation 34

Design Techniques Using Bezier Curves Contd

Multiple control points Specify multiple control points at a single

co ordinate position gives more weight to that position.

Page 35: Bezeir Curves

Apr 11, 2023 3D Object Representation 35

Design Techniques Using Bezier Curves Contd

When complicated curves are to be generated

Formed by piecing several bezeir sections of lower degree together.

Page 36: Bezeir Curves

Apr 11, 2023 3D Object Representation 36

Bezier Surfaces Two sets of orthogonal Bezeir curves can be used to design an

object surface by specifying an input mesh of control points. The parametric vector function is formed as the Cartesian

product of Bezier blending functions

pi,k specifying the location Each curve of constant u is plotted by varying v over the interval

0 to 1, with u fixed at one of the values in this unit interval.

)()(),( ,0 0

,, uBEZvBEZpvuP nk

m

j

n

kmjkj

Page 37: Bezeir Curves

Apr 11, 2023 3D Object Representation 37

Bezier Surfaces Contd Control points connected by dashed lines Solid lines show curves of constant u and constant v

Page 38: Bezeir Curves

Apr 11, 2023 3D Object Representation 38

Thank You…