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# Geometric Modeling - Parametric Representation of Synthetic Curves

## 1.

Week 7: Geometric Modeling Parametric Representation ofSynthetic Curves

Spring 2018, AUA

Zeid, I., Mastering CAD/CAM, Chapter 6

## 2. Planar vs. Space

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## 3. Analytic (known form) vs. Synthetic (free form)

We can create simplisticobjects such as the forklift

given below by using known

equations.

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Creating these curves by using known

analytic curve equations is not

reasonable all the time. Sometimes –

impossible.

3

## 4. Interpolation vs. Approximation

The curve passing through given data (control) points - interpolation curve.The curve not necessarily passing but controlled by data points approximation curve

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## 5. Continuity

The smoothness of the connection of two curves orsurfaces at the connection points or edges.

•C0: simple connection of two curves

•C1: the geometric slopes at the joint must be same

•C2: curvature continuity that not only the gradients but also

the center of curvature is the same

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## 6. Cubic Curves

Parametric equation of a cubic spline segment:3

P (u ) a i u i ,

where

0 u 1

i 0

In an expanded vector form:

P(u ) a 0 a1 u a 2 u 2 a 3 u 3

The tangent vector:

3

P (u ) a i i u i 1

'

i 0

In an expanded vector form:

P' (u ) a1 2 a 2 u 3 a 3 u 2

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## 7. Hermite Cubic Splines

Charles Hermite(1822 - 1901)

Hermite form of a general

cubic spline is defined by

positions and tangent

vectors at two data points.

P’(1)

P(1) (u=1)

P’(0)

tu

P(0) (u=0)

Applying the boundary conditions at u = 0 and u = 1 and performing the necessary

substitutions,

P(u ) P(0) (2 u 3 3 u 2 1) P(1) ( 2 u 3 3 u 2 )

P ' (0) (u 3 2 u 2 u ) P ' (1) (u 3 u 2 )

P ' (u ) P(0) (6 u 2 6 u ) P(1) ( 6 u 2 6 u )

P ' (0) (3 u 2 4 u 1) P ' (1) (3 u 2 2 u )

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## 8. Hermite Cubic Splines

P(u ) P(0) (2 u 3 u 1) P(1) ( 2 u 3 u ) P (0) (u 2 u u ) P (1) (u u )3

2

3

2

'

3

2

'

3

2

X (u ) X (0) (2 u 3 3 u 2 1) X (1) ( 2 u 3 3 u 2 ) X ' (0) (u 3 2 u 2 u ) X ' (1) (u 3 u 2 )

Y (u ) Y (0) (2 u 3 u 1) Y (1) ( 2 u 3 u ) Y (0) (u 2 u u ) Y (1) (u u )

3

2

3

2

'

3

2

'

3

2

Vector

form

Scalar

form

Z (u ) Z (0) (2 u 3 3 u 2 1) Z (1) ( 2 u 3 3 u 2 ) Z ' (0) (u 3 2 u 2 u ) Z ' (1) (u 3 u 2 )

Vector

form

Scalar

form

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## 9. Hermite Cubic Spline – Tangent Vector

P ' (u ) P(0) (6 u 2 6 u ) P(1) ( 6 u 2 6 u ) P ' (0) (3 u 2 4 u 1) P ' (1) (3 u 2 2 u )X ' (u ) X (0) (6 u 2 6 u ) X (1) ( 6 u 2 6 u ) X ' (0) (3 u 2 4 u 1) X ' (1) (3 u 2 2 u )

Y (u ) Y (0) (6 u 6 u ) Y (1) ( 6 u 6 u ) Y (0) (3 u 4 u 1) Y (1) (3 u 2 u )

'

2

2

'

2

'

2

Z ' (u ) Z (0) (6 u 2 6 u ) Z (1) ( 6 u 2 6 u ) Z ' (0) (3 u 2 4 u 1) Z ' (1) (3 u 2 2 u )

Vector

form

Scalar

form

Vector

form

Scalar

form

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## 10. Hermite Cubic Splines - example

YThe Hermite curve fits

the points:

P0 = [1,1]T,

P1 = [3,5]T

and the tangent vectors:

P0’ = [0,4]T,

P1’ = [4,0]T.

Calculate

a) the parametric midpoint of the curve,

b) the tangent vector on

that point.

c) Sketch the curve on

the grid

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X

## 11. Bezier Curves - sl. 1

Parametric equation of Bezier curven

P(u ) Pi Bi ,n (u ),

0 u 1

i 0

Pierre Bezier

(1910-1999)

Renault

where P(u) is the position vector of a point on the curve, Pi

are control points, and Bi,n are the Bernstein polynomials

Paul de Casteljau

(blending functions for the curve).

Bi ,n (u) C(n, i) u (1 u)

i

n i

and C(n,i) are the binomial coefficients: C (n, i )

n!

i! (n i )!

(1930)

Citroën

In an expanded form:

P(u) P0 (1 u) n P1 C (n,1) u (1 u) n 1 P2 C (n,2) u 2 (1 u ) n 2

Pn 1 C (n, n 1) u n 1 (1 u) Pn u n

P(u ) P0 (1 u ) 3 P1 3 u (1 u ) 2 P2 3 u 2 (1 u ) P3 u n 3

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## 12. Bezier Curves - sl. 3

For n = 3:P(u ) P0 (1 3 u 3 u 2 u 3 ) 3 P1 (0 u 2 u 2 u 3 ) 3 P2 (0 0 u 2 u 3 ) P3 (0 0 0 u 3 )

Or, in matrix form:

P (u ) P0

P1

P2

1

3

P3

3

1

Bezier geometric matrix

GB

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3

3

6

3

3

0

0

0

1 u 3

0 u 2

1

0 u

0 1

Bezier basis matrix

MB

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## 13. Bezier Curves - sl. 2

P1P2

P0

t=0

Pk

General Characteristics

•The Bezier curve is defined by n+1 points

•Only P0 and Pn+1 lie on the curve

•The curve is tangent to the first and last polygon

segments

•The curve shape tends to follow the polygon shape.

•Convex hull property.

•The sum of Bi,n functions is always equal to unity.

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P3

t=1

Bezier vs. Hermite Cubic Spline

•The Bezier curve is controlled by data

points. No derivatives

•The order is variable: n+1 points define

nth order curve . -> higher order

continuity

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## 14. Bezier Curves - sl. 5 Practice

• The coordinates of 4 control points are given:P0 = [2,2]T, P1 = [2,3]T, P3 = [3,3]T, P4 = [3,2]T

• Find the equation of the resulting Bezier curve,

• Find the points on the curve for u = 0, ¼, ½, ¾, 1,

• Sketch the curve.

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## 15. Bezier Curves - sl. 4

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## 16. B-spline Curves - sl. 1 See: http://www.ibiblio.org/e-notes/Splines/Basis.htm

Powerful generalization of Bezier curves•local control

•opportunity to add control points without increasing the degree of the curve

•ability to interpolate or approximate data points

The B-spline curve defined by n+1 control points Pi consists of n – 2 curve

segments and is given by:

n

P(u ) Pi N i ,k (u ),

i 0

0 u umax

where Ni,k(u) are the B-spline (blending or basis) functions. The parameter k

controls the degree (k-1) of the B-spline curve.

Local control

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## 17. B-spline Curves - sl. 2 See: http://www.ibiblio.org/e-notes/Splines/Basis.htm

The B-spline curve defined by n+1 control points Pi consists of n – 2 curvesegments and is given by:

n

P(u ) Pi N i ,k (u ),

i 0

0 u umax

where Ni,k(u) are the B-spline (blending or basis) functions. The parameter k

controls the degree (k-1) of the B-spline curve.

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## 18. B-spline Curves - sl. 3 Basis Functions

• The function Ni,k determines how strongly control point Pi influencesthe curve at t. Its value is a real number – 0.25, 0.5…

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## 19. NURBS Curves - sl. 1

NURBS (Non-uniform Rational B-spline) curves are the generalization ofuniform B-spline curves.

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## 20. NURBS Curves - sl. 2

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