Equations for Geomwtric Modelling

8/8/2019 Equations for Geomwtric Modelling

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CCU ME ComputerCU ME Computer-Aided Machine Designided Machine Design

Com put er-Aided Mac hine Design

Chapter 4 Curves

I. Foundation of Curves


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Desc r ip t ion of Curves

Ways to mathematically describe curves for geometricmodeling including

Intrinsic equation

Explicit equation

Implicit equation

Parameter equation


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In t r insic Equat ion

An intrinsic property is one that depends on only thefigure in question, and not its relation to a coordinate

system or other external frame of reference

Example

Rectangle has four equal angles : intrinsic

property Rectangle particular has two vertical side: not

intrinsic property


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In t r ins ic Curves Equat ion

)(or)(1

sgsf ==

s

dsd

d


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Transfer Int r insic t o ex t r insic

)()(

0)(cos)(

0)(sin)(

sincos

)()(

1

2

2

2

2

sysxSolve

ds

dxs

ds

yd

ds

d

ds

dy

ds

dds

dys

ds

xd

ds

d

ds

dx

ds

d

ds

dy

ds

dx

syysxx

ds

d

==

=+=

==

==

==


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Ex p l i c i t & Im p l i c i t Equa t ion

Standard explicit equation of a curve in the plane y=f(x)

Not represent closed or multiple-valued curve

Implicit equation of curves f(x, y) = 0

Parameter- based modeling

Intersection & point classification

Both explicit and implicit forms are axis dependent

The choice of the coordinate system affects the easeof modeling the curves and calculating their

properties


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Example

Lines Explicit: y=ax+b

Implicit: Ax+By+c=0

Conic curves

Explicit: y=ax2+bx+c (parabola)

Implicit: Ax2+2Bxy+cy2+Dx+Ey+F=0


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Sym m et r ic Proper t y o f Imp l i c i t f unc t ion

Symmetric about x=-yf(x,y)= f(-y,-x)xy=-k2

Symmetric about x=yf(x,y)= f(y,x)xy=k2

Symmetric about y axisf(x,y)= f(-x,y)y=x2

Symmetric about x axisf(x,y)= f(x,-y)x=y2

Symmetric about originf(x,y)=f(-x,-y)x2+y2=r2


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Param et r ic Equat ion

Explicit form not represent most shapes used in geometricmodeling

The shapes of most objects are intrinsically independent of any

coordinate system. Most modeling applications require that thechoice of a coordinate system should not affect the shapes

It is the relationship between the points themselvesthat determines the resulting shape of a curve of

surface fit through a set of points, not the relationshipbetween there points and some arbitrary coordinatesystem


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Tangent lines or planes parallel to the principal axes or planes ofany chosen coordinate system results in values of infinity ofother ill-defined mathematical properties

Curves and surfaces are often non-planar and bounded, and arenot easily represented by an ordinary non-parametric function

Not easy to program & computability

Best describe the way curves are drawn by a plotter or somecomputer graphics display screens.

x(t), y(t) -> servo motor or electron beam


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[ ]

[ ]

+=

+=

+=

=

==

+=

+==

=

=

linefor

helixforsincos

32

13

:example

surfaceafor),(),(),(),(

curvespaceafor)()()()(

3

2

nucz

muby

luax

buz

uayuax

uz

uuyux

wuzwuywuxwuP

uzuyuxuP


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Curve Segm ent

If the parameter of equation u is defined by u[0,1], thencurve is bounded, i.e., curve segment

We call the process that makes the parameter under the

range u[0,1] normalization

We can change the parameter space of curve equation

arbitrarily It is called reparameterization


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Transfer ex (im )p l ic i t t oparam et r ic equat ion

y=f(x) Let x =u, y=f(u) => got the parametric equation

f(x,y,z)=0

Set x=x and find y=y(x), z=z(x)

Let x=u, y=y(u), z=z(u) => got the parametric equation

Let u=(x-x0)/(x1-x0) u=0, x=x0; u=1 x=x1 x=x(u)=x0+(x1-x0)u, y=y(x(u)), z=z(x(u))

normalization

Every parametric form has a corresponding implicit form, but

there are implicit forms that have no known parametric

representation


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Types of Param et r ic Equat ion

P(u)=( x(u) y(u) z(u) ) x(u), y(u), z(u) : a fixed polynomial form

Aun+Bun-1+Cun-2+Dun-3+

x(u), y(u), z(u) : a basis fucntions Hermite, Bezier, B-spline


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Som e t erm s in Param et r icEquat ion

Model Space 3D space defined by the Cartesian coordinates

x, y, z

Parameter Space 3D space defined by (x, u), (y, u), (z, u)

Cross Plots

Plots of a curve in terms of its parameter spacecomponents


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Tangent Vec t or on aParam et r ic Equat ion

[ ]

[ ]

du

dx

dudz

du

dz

du

dy

du

dx

du

dy

uuuu

dx

dz

dz

dy

dx

dy

du

udz

du

udy

du

udx

zyxdu

udpup

uzuyuxup

===

=

==

=

:spacemodel

)()()(

)()(

:spaceparameter)()()()(

y

x

z p(u)

pu(u)

pu0

pu1

p0 u=0

p1 u=1


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Advant ages of Param et r icEquat ion

Separation of variables and direct computation of point coordinate

Easy to express parametric equations as vectors

Each variable is treated alike

More degrees of freedom to control curve shape

Transformation may be performed direct on them

Accommodate all slopes without computational breakdown

Extension or contraction into higher or lower dimensions is direct and easy

without affecting the initial representation Curves bounded when the parameter is constrained

Be presented by many parameterization

Equations for Geomwtric Modelling

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