CHORD LENGTH PARAMETERIZATION

支德佳 2008.10.30

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

CHORD LENGTH PARAMETERIZATION

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

Gerald FarinComputer 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

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Mathematica code:

RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH

RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH

Wei Lü

J. Sánchez-Reyes, L. Fernández-Jambrina

Curves with rational chord-length parametrization

Curves with chord length parameterization

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION(CAGD2008)

J. Sánchez-ReyesInstituto 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

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

Quadratic circles:

= constant

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

Quadratic circles: {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.

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

=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 RATIONAL CHORD LENGTH PARAMETRIZATION

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

CURVES WITH CHORD LENGTH PARAMETERIZATION(CAGD2008)

Wei LüSiemens PLM Software, 2000 Eastman Drive, Milford, OH 45150, USA

CURVES WITH CHORD LENGTH PARAMETERIZATION

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.

CURVES WITH CHORD LENGTH PARAMETERIZATION

CURVES WITH CHORD LENGTH PARAMETERIZATION is rational

is rational

Rational cubics and G1 Hermite interpolation

CURVES WITH CHORD LENGTH PARAMETERIZATION

CURVES WITH CHORD LENGTH PARAMETERIZATION

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◦).

CURVES WITH CHORD LENGTH PARAMETERIZATION

THANK YOU!