sd 1.5.2 geometric modeling

7/25/2019 SD 1.5.2 Geometric Modeling

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Introduction to Geometric Modeling

Manuel Ventura

Ship Design I

MSc in Marine Engineering and Naval Architecture

M.Ventura Introduction to GeometricModeling

2

Summary

1. Parametric Curves

2. Parametric Surfaces


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Parametric Curves1. Mathematical Formulations

Cubic Splines

Bzier

B-Spline

Beta-Spline

NURBS

2. Interpolation and approximation of curves

3. Analysis of curves


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Cubic Spline (1)

Considering the wooden spline(virote) a thin elastic beam, andfor small deflections, theEuler law relates the deflectionof the beam axis y(x) with the

bending moment M(x) by theexpression:

( ) =y x M x

EI

( )

where:

E Modulus of Young

I Moment of inertia of the beam section


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Cubic Spline (2) Assuming that the beam is simply supported on the weights,

then the bending moment varies linearly between them, i.e.,M(x) = Ax + B. Replacing in the expression and integratingresults

( )y x M x

EIdx Ax B dx Ax Bx Cx D= + + + +

( )( )=

1

EI= 3 2

( )P t At Bt Ct D= + + +3 2

( )( )( )( )

=

=

=

=

1

0

1

0

1

0

1

0

TP

TP

pP

pP

In each segment, the curve can be defined as a function ofthe parameter t normalized for the interval [0,1]

The constants can be obtained from thefollowing boundary conditions:


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Cubic Spline (3)

Finally the curve can be represented in the matrix form as

[ ]H =

+ + +

+

+

+

2 2 1 1

3 3 2 1

0 0 1 0

1 0 0 0

[ ]G

p

p

T

T

i

i

i

i

=

+

+

1

1

( ) [ ][ ] [ ]GHttttP 123=

where


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Bzier Curves (1) The curves generally known as Bzier resulted from

separate research from Casteljau (Citroen) and PierreBzier (Renault) in the beginning of the 1960s.

A Bzier curve is defined by:

P t C B t for t i

n i

i

n

( ) ( ),= =

0

0 1

( )

( )

B n

i n i

t t

n

it t i n

n i

n i i

n i i

,

!

!( )!

=

=

=

1

1 0 1 for , , . . . ,

where Bn,j are the Bernestein base functions, of degree n


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Bzier Curves (2)

The Bzier curve is tangent to the first and last segmentsof the control polygon

The curve order is equal to the number of vertices of thecontrol polygon.

The curve is entirely contained in the convex hull of the

control points.


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B-Spline Curves (1) They were studied by N. Lobatchevsky in the XIX century Their use for curve fitting to experimental data began in

1946 with Schoenberg They were first introduced in CAD systems by J. Ferguson

(Boeing) in 1963.

=

=n

i

kii tNPtCk0

, )()(

)()()(

0


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B-Spline Curves (3)The B-Spline curves have the following properties:

Linear precision

Convex hull, in k consecutive control points

Variation diminishing

Are invariant when submitted to affine transformations

When the order of the B-Spline is equal to the number ofcontrol points, the knot vector consists only in the values ofthe extremities with the multiplicity equal to the order

{ 0, 0, 0, 0, 1, 1, 1, 1 }and the B-Spline base functions are equivalent to Bernesteinfunctions and the curve degenerates into a Bzier curve.


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Beta-Splines (1)

Os cubic Beta-splines were introduced on 1981 by Barsky

They are a generalization of the B-Splines based in notionsof geometric continuity and in the mathematical modeling oftension

The requirements of parametric continuity of the 2 order

(C2) between the B-Splines segments is replaced by therequirements of geometric continuity of 2 order (G2) of theunit tangent vector and of the curvature vector

This originates discontinuities of the 1st and 2nd parametricderivative, that are expressed as functions of theparameters 1 and 2, designated by bias and tension,respectively.


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Beta-Splines (2) A Beta-spline curve is defined by:

( ) 1


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Representation of Conic Shapes (1) A NURBS curve of the 2nd degree, with 3 points represents

a conic shape if the conic form factor, kc, defined by:

k w w

wc=

1 3

2

24

.

.

hiperbolekparabolak

elipsek

c

c

c

>=

0 Surface with single curvature(concave or convex)

Gauss Curvature distribution

sd 1.5.2 geometric modeling

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