Intro to Geometric Modeling (GM)
Advantages of PR
Advantages of PR
Advantages of PR
Advantages of PR
Advantages of PR
Advantages of PR
Advantages of PR
Advantages of PR
PR of 3D Curve
PR of Analytic Curves
Lines: 2 points
Lines: point and direction
Parametric equation from NP Implicit Equation: Example
Circles
Ellipses
Examples
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Категория: МатематикаМатематика

Intro to Geometric Modeling (GM)

1.

Week 4: Geometric Modeling –
Parametric Representation of
Analytic Curves
Spring 2018, AUA

2. Intro to Geometric Modeling (GM)

Ideas
CAD
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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2

3.

Objects of Representation
Curves
Surfaces
Solids
Standard form vs free-form
Domain of study – Computer Graphics
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3

4.

Types of Representation
Explicit
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 each
other.
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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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6

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
8

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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9

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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10

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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11

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 vector
or
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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
15

16. Lines: 2 points

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17. Lines: point and direction

n
0 ≤ L ≤ Lmax
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18. Parametric equation from NP Implicit Equation: Example

F(x, y) = 0
y
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
18

19. Circles

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20. Ellipses

+B/A
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21. Examples

• Find the equation and endpoints
of 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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