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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3.

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