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CHORD LENGTH PARAMETERIZATIONPowerPoint Presentation

CHORD LENGTH PARAMETERIZATION

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### RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH(CAGD 2006)

### CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)(CAGD2008)

### CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)(CAGD2008)

### THANK YOU! LENGTH(CAGD 2006)

CHORD LENGTH PARAMETERIZATION

- Chord length:

CHORD LENGTH PARAMETERIZATION

- A curve is said to be chord-length parameterized if chord(t) = t.
- Geometric parameter
- No self-intersection
- Ease of point-curve testing
- Simplification of curve-curve intersecting

Gerald Farin

Computer Science

Arizona State University, USA

RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH LENGTH(CAGD 2006)

- An arc of a circle:
‖ ‖=‖ ‖

RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH LENGTH(CAGD 2006)

？

RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH LENGTH(CAGD 2006)

- Mathematica code:

RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH LENGTH(CAGD 2006)

Curves with rational chord-length parametrization LENGTH(CAGD 2006)

Curves with chord length parameterization

- Wei Lü

- J. Sánchez-Reyes,
- L. Fernández-Jambrina

J. Sánchez-Reyes

Instituto de Matemática Aplicada a la Ciencia e Ingeniería, ETS Ingenieros Industriales, Universidad de Castilla-La Mancha, Campus Universitario, 13071-Ciudad Real, Spain

L. Fernández-Jambrina

ETSI Navales, Universidad Politécnica de Madrid, Arco de la Victoria s/n, 28040-Madrid, Spain

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

- Chord length & bipolar coordinates

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

- Chord length & bipolar coordinates

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

- To construct chord-length parametrized curves p(u), simply choose an arbitrary function ϕ(u). Such curves can be thus regarded as the analogue, in bipolar coordinates (u,ϕ), of nonparametric curves (u, f (u)) in Cartesian coordinates (x, y), where one coordinate is explicitly expressed as a function of the other one.

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

- Quadratic circles:
= constant

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

- Quadratic circles:
- {0,1,1/2}-->{A, B, S}

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

- Rational representations of higher degree:
Any Bézier circle other than quadratic is degenerate.

(Berry and Patterson, 1997; Sánchez-Reyes, 1997)

There exist two types of degenerate circles:

- 1- Improperly parameterized:
A nonlinear rational parameter substitution.

No longer satisfy the chord-length condition.

- 2-Generalized degree elevation:
Preserve chord-length.

- The standard quadratic parametrization is the only rational chord-length parametrization of the circle.

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

=c(u)

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

We thus control the quartic using the following shape handles:

- Endpoints A,B, and angles α,β between the endpoint tangents and the segment AB.
- Angle σ between chords AS and SB at S = p(1/2).

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

Wei Lü

Siemens PLM Software, 2000 Eastman Drive, Milford, OH 45150, USA

CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

- always form an isosceles triangle.
- If α(t) is constant other than 0 or π, it is a circular arc.
- If α(t) = 0 (or π), the curve (5) is a (unbounded) straight line segment.
- For α( 1/2 ) ≠ π, the curve is well defined and bounded.
- End conditions.

CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

- is a complex function with | | = 1

CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

- A complex function U = U(t) with |U(t)| = 1 is rational if and only if there is a complex polynomial H = H(t) such that
- H(t) is not unique.
- Analyze and manipulate rational functions with just half degrees of the corresponding rational curves in Euclidean space.

CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

- Rational cubics and G1 Hermite interpolation

CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

- The cubic G1 Hermite interpolant is not able to reproduce a desired S-shape curve, as shown in dotted points (α0 = 70◦, α1 =−20◦).

CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

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