lecture 4 [compatibility mode]

8/2/2019 Lecture 4 [Compatibility Mode]

1/26

Mathematical Representation of

Curves

ME C382: COMPUTER AIDED DESIGN

1


2/26

Disadvantages of Non-parametric Representations

Explicit non-parametric representation can not be used forclosed curves (like circles) and multi-valued curves (likeparabolas). Because it is a one-to-one relationship.

The above problem is overcome by implicit representation butthe latter is laborious.

Implicit representation requires that two surface equations besolved for y and z for a given value of x

Infinite slope situations can not be dealt with in computerprogram

Shapes of most objects are coordinate system independent.

2


3/26

Parametric Representation: Advantages

It overcomes all difficulties of the non-

parametric representations

It allows multi-valued and closed functions tobe easily defined

tangent vectors

The equations are polynomials and thus

computationally more suitable than equationsinvolving roots

3


4/26

PARAMETRIC CURVES

PARAMETRIC CUBIC CURVE

HERMITE CURVE

4


5/26

PARAMETRIC CURVES

Classification

Plane Curves & Space Curves

Eg Circle v/s Helix

Circle v/s Bezier Curve

Interpolation curve v/s Approximation curves

Hermite Curves v/s Bezier curve

5


6/26

Parametric Form of a Curve

In parametric form, each point on a curve isexpressed as a function of a parameter u.

The parameter acts as a local coordinate forpoints on the curve

The parametric equation for a 3-D curve in space

P(u) = [ x y z]T = [ x(u) y(u) z(u)]T,

umin u umax

Coordinates of a point on the curve are thecomponents of its position vector.

6


7/26

PARAMETRIC REPRESENTATION OF LINES

121

121

121

10)(

)(

toequivalentisThis

10)PP(PP:1

+=

+=

+=

uyyuyy

xxuxx

uuMethod

1

1

121

PP

bygivenisandparameterthelikeactwillitself

nPP

:2)(

=

+=

+=

L

L

LL

Methodzzuzz

7


8/26

Circle

The parametric equation of a circle (in the x-y plane)is

x = xc + R cos(u)

y = yc + R sin(u) 0 u 2

z = zc

Thus, determination of the radius (R) and center of the

circle (xc, yc) is necessary (and is sufficient) forparametric representation of a circle.

8


9/26

u=0

u=2P = x z

P=(x, y, z)

Pn=(xn, yn, zn)Pn+1=(xn+1, yn+1, zn+1)

u=

u

9

Pc

u=3/2


10/26

Computation Of Parametric Circle For Computer Display

x = xc + R cos(u)y = yc + R sin(u) 0 u 2

z = zc

xn+1 = xc + R cos(u+u)yn+1 = yc + R cos(u+u)

zn+1 = zn

Expanding the trigonometric terms and simplifyingxn+1 = xc + (xn xc) cos(u) - (yn yc) sin(u)

yn+1 = yc + (yn yc) cos(u) + (xn xc) sin(u)

zn+1 = znTrigonometric terms have to be calculated only once for

a given u.

10


11/26

Ellipse Two cases of ellipses

we will consider:Basic Ellipse, = 0

,

11


12/26

Basic Ellipse

x = xc + A cos(u)

y = yc + B sin(u) 0 u 2

z = zc

12

u

P


13/26

Ellipse

x = xc + A cos(u)y = yc + B sin(u) 0 u 2

z = zc

P=(x, y, z)

Pn=(xn, yn, zn)

u

Pn+1=(xn+1, yn+1, zn+1)

13

u=0

u=2Pc=(xc, yc, zc)

Pc

u=3/2

u=


14/26

Computation Of Parametric Basic Ellipse For Computer Display

xn+1 = xc + (xn xc) cos(u) (A/B)(yn yc) sin(u)

yn+1 = yc + (yn yc) cos(u) + (A/B)(xn xc) sin(u)

zn+1 = zn

14


15/26

PARAMETRIC REPRESENTATION OF SYNTHETICCURVES

15


16/26

What are synthetic curves?

Synthetic curves represent a curve fitting problem toconstruct a smooth curve that passes through given datapoints. Polynomials are the typical forms.

Synthetic curves take up where the analytic curves leave the latter are not that efficient at geometric design ofmechanical parts

Some examples of complex geometric design are:

Car bodies

Ship hulls Airplane fuselage and wings

Propeller blades

Shoe insoles

Aesthetically designed bottles

16

They are to be made by free-form curves and surfaces.


17/26

Need for synthetic curves?

Need for synethetic curves arises intwo occassions:

When a curve is represented by acollection of measured data points, and

When an existing curve must change tomeet new desi n re uirements the

designer needs a curve representationthat is directly related to the datapoints and is flexible enough to bend,twist or change the shape by changing

one or more data points.

17


18/26

Most commonly used Synthetic Curves Hermite Cubic Spline

Each curve segment is defined for only 2 control points; It passes

through the control points and therefore it is an interpolant

It has only upto C1 continuity

Bezier Curve

oes no pass roug e con ro po n s u on y approx ma es

the trend

It also has only upto C1 continuity

B-Spline Curve

It is also most generally an approximator; an interpolating B-Spline

is also sometimes possible

It has upto C2 continuity

18


19/26

Hermite Cubic Spline Curve Segment

They are used to interpolate the given data butnot to design free-form curves. (Cubic) Splines derive their name from French

curves or splines

Hermite cubic spline is one type of generalparametric cubic spline with degree equal to 3and being determined by two data points and

. Hermite Cubic Spline can be a 3-D planar curve or

3-D twisted curve. Algebaric Form

Geometric Form Four Point Form

19


20/26

Parametric Equation of Hermite Cubic Spline Segment

10,)(3

0

==i

ii uuCuP

tscoefficeinalgebraic)(orPolynomial

parameter

=

=

iC

u

20


21/26

)(

formvectorexpandedIn

)(

)(

)(

aswrittenisequationthisformscalarIn

01

2

2

3

3

01

2

2

3

3

01

2

2

3

3

01

2

2

3

3

+++=

+++=

+++=

+++=

CuCuCuCuP

CuCuCuCuz

CuCuCuCuy

CuCuCuCux

zzzz

yyyy

xxxx

[ ] [ ]vectortsCoefficien][

][and1][

where

][][)(formmatrixIn

0123

23

=

==

=

C

CCCCCuuuU

CUuP

TT

T

21


22/26

:giving,1at,and0at,,conditionsboundarytheApplying

.andendpointstwoth thesegment wicurvesplinecubicheConsider t

10,)(

ispointanyatcurvetheector totangent vThe

1231

1

11

1

3

0

1

CCCCP

CPCP

uPPuPP

PP

uiuCuP

o

o

oo

oo

o

i

i

i

+++=

=

=

==

=

=

( ) ( )( )

( ) ( ) ( ) ( )ts.coefficiengeometricascalledareand,,

u-uu2u-u3u2u-13u-2u)(

grearranginandequationparametricin thengSubstituti

2

23

tscoefficienfor theuslysimultaneoequationsfourtheSolving

11

1

2323

1

2323

113

1012

1

1231

PPPP

PPPPuP

PPPPC

PPPPC

PC

PC

oo

oo

oo

o

o

oo

++++++=

++=

=

=

=

=

22


23/26

( ) ( ) ( ) ( )[ ]

functions.blendingcalledareThese

u-u(u)F

u2u-u(u)F3u-2u(u)F

13u-2u(u)F

vectorconditionboundaryort vectorcoefficiengeometric][u-uu2u-u3u2u-13u-2u)(

23

4

23

3

23

2

23

1

11

1

2323

1

2323

=

+=

+=

+=

==

++++++=

PPPPVPPPPuP

oo

oo

matrixHermite][M

10][V],[M[U](u)

P(u)FP(u)FP(u)FP(u)F(u)P

H

H

T

14031201

=

=

+++=

uP

23


24/26

0001

0100

1233

1122

][matrixHermite MH

==

24

]][[Cobtainwe

],][[][)(and][][)(Comparing

VM

VMUuPCUuP

H

HTT

=

==


25/26

[ ]

[ ]

+++++=

=

=

231436666

aswrittenisectortangent vtheSimilarly

0123

0100

11111000

][][

2222

1

1

H

H

PuuPuuPuuPuuuP

M

CMV

25

=

=

0100

2466

3366

0000

][

where

][][][

u

H

uH

T

oo

M

VMU


26/26

Example 1 Find the equation of a Hermite Cubic Spline

that passes through the points (1,2) and (3,4)and whose tangent vectors are two lines

, .

26

lecture 4 [compatibility mode]

Documents