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# Intro to Geometric Modeling (GM)

## 1.

Week 4: Geometric Modeling –Parametric Representation of

Analytic Curves

Spring 2018, AUA

## 2. Intro to Geometric Modeling (GM)

IdeasCAD

Geometric

Geometricmodel

model

The goal of CAD - efficient representation of the unambiguous and

complete info about a design for the subsequent applications:

• mass property calculations

• mechanism analysis

• finite element analysis

• NC programming

Geometric modeling - defining geometric objects using computer

compatible mathematical representation.

Mathematical representation learned in schools will not work.

As well as objects created in Word or Power Point or Photoshop.

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## 3.

Objects of RepresentationCurves

Surfaces

Solids

Standard form vs free-form

Domain of study – Computer Graphics

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## 4.

Types of RepresentationExplicit

Representation

z=ax+by+cz+d

Implicit

Representation

ex+fy+gz+h=0

Parametric

Representation

x=a+bu+cw

y=d+eu+fw

z=g+hu+iw

The question is which one is computer compatible?

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## 5. Advantages of PR

• Get rid of dependency of the coordinates (X, Y, Z) from eachother.

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## 6. Advantages of PR

• Get rid of dependency of the coordinates (X, Y, Z) from each other.• Can be extended to higher objects. (4th parameter )

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## 7. Advantages of PR

• Get rid of dependency of the coordinates (X, Y, Z) from each other.• Can be extended to higher objects. (4th parameter )

• More DOF for controlling curves and surfaces.

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## 8. Advantages of PR

Get rid of dependency of the coordinates (X, Y, Z) from each other.

Can be extended to higher objects. (4th parameter )

More DOF for controlling curves and surfaces.

Transformations (distinct separation between shape and trans.

info).

R=7 circle at 0,0

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R=7 circle at 4,3

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## 9. Advantages of PR

Get rid of dependency of the coordinates (X, Y, Z) from each other.

Can be extended to higher objects. (4th parameter )

More DOF for controlling curves and surfaces.

Transformations (distinct separation between shape and trans. info).

Vector – matrix multiplication (not good for you, good for comp).

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## 10. Advantages of PR

Get rid of dependency of the coordinates (X, Y, Z) from each other.

Can be extended to higher objects. (4th parameter )

More DOF for controlling curves and surfaces.

Transformations (distinct separation between shape and trans. info).

Vector – matrix multiplication (not good for you, good for comp.).

Bounded objects are represented with one-to one relationship.

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## 11. Advantages of PR

Get rid of dependency of the coordinates (X, Y, Z) from each other.

Can be extended to higher objects. (4th parameter )

More DOF for controlling curves and surfaces.

Transformations (distinct separation between shape and trans. info).

Vector – matrix multiplication (not good for you, good for comp.).

Bounded objects are represented with one-to one relationship.

No problems for slope calculation.

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## 12. Advantages of PR

Get rid of dependency of the coordinates (X, Y, Z) from each other.

Can be extended to higher objects. (4th parameter )

More DOF for controlling curves and surfaces.

Transformations (distinct separation between shape and trans. info).

Vector – matrix multiplication (not good for you, good for comp.).

Bounded objects are represented with one-to one relationship.

No problems for slope calculation.

Discretizing entities

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## 13.

Parametric Representation (PR)X = f(t)

Y = g(t)

Z = h(t)

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## 14. PR of 3D Curve

Tangent vectoror

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## 15. PR of Analytic Curves

Analytic curves are defined by analytic equations•Compact form for representation

•Simple computation of properties

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•Little practical use

•No local control

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## 16. Lines: 2 points

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## 17. Lines: point and direction

n0 ≤ L ≤ Lmax

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## 18. Parametric equation from NP Implicit Equation: Example

F(x, y) = 0y

For

x

x2 + y2 - R2 = 0

R

x R cos 2 u, where 0 u 1

y R sin 2 u, where 0 u 1

Parametric equation :

P(u ) [ R cos 2 u, R sin 2 u ]T ,

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0 u 1

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## 19. Circles

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## 20. Ellipses

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## 21. Examples

• Find the equation and endpointsof a line that passes through point

P1, parallel to an existing line,

and is trimmed by point P2.

• Relate the following CAD

commands to their mathematical

foundations:

– The command to measure the

angle between two

intersecting lines.

– The command to find the

distance between a point and

a line.

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P4

y

P5

n1

n1

P3

P2

P1

x

z

P2

P1

P4

P3

P2

P3

D

P4

n1

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