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CHORD LENGTH PARAMETERIZATION. 支德佳 2008.10.30. CHORD LENGTH PARAMETERIZATION. Chord length:. CHORD LENGTH PARAMETERIZATION. A curve is said to be chord-length parameterized if chord (t) = t.

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CHORD LENGTH PARAMETERIZATION

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## CHORD LENGTH PARAMETERIZATION

2008.10.30

### 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

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

Gerald Farin

Computer Science

Arizona State University, USA

### RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH

• An arc of a circle:

‖ ‖=‖ ‖

### RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH

• Mathematica code:

### RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH

Curves with rational chord-length parametrization

Curves with chord length parameterization

• Wei Lü

• J. Sánchez-Reyes,

• L. Fernández-Jambrina

## CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION(CAGD2008)

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

• Chord length & bipolar coordinates

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

• Chord length & bipolar coordinates

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

• 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

= constant

### CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

• {0,1,1/2}-->{A, B, S}

### CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

• 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.

=c(u)

### CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

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 CHORD LENGTH PARAMETERIZATION(CAGD2008)

Wei Lü

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

### CURVES WITH CHORD LENGTH PARAMETERIZATION

• 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

• is a complex function with | | = 1

### CURVES WITH CHORD LENGTH PARAMETERIZATION

• 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.

is rational

is rational

### CURVES WITH CHORD LENGTH PARAMETERIZATION

• Rational cubics and G1 Hermite interpolation

### CURVES WITH CHORD LENGTH PARAMETERIZATION

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