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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

CHORD LENGTH PARAMETERIZATION

支德佳

2008.10.30



Chord length parameterization2
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

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

  • An arc of a circle:

    ‖ ‖=‖ ‖





Chord length parameterization

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

Curves with chord length parameterization

  • Wei Lü

  • J. Sánchez-Reyes,

  • L. Fernández-Jambrina


Curves with rational chord length parametrization cagd2008

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

  • Chord length & bipolar coordinates


Chord length parameterization

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

  • Chord length & bipolar coordinates


Chord length parameterization

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.


Chord length parameterization

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

  • Quadratic circles:

    = constant


Curves with rational chord length parametrization1
CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION LENGTH(CAGD 2006)

  • Quadratic circles:

  • {0,1,1/2}-->{A, B, S}


Curves with rational chord length parametrization2
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 parametrization4
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 chord length parameterization cagd2008

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

Wei Lü

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




Curves with chord length parameterization2
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 parameterization3
CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

  • is a complex function with | | = 1


Curves with chord length parameterization4
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 parameterization5
CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

is rational

is rational


Curves with chord length parameterization6
CURVES WITH CHORD LENGTH PARAMETERIZATION LENGTH(CAGD 2006)

  • Rational cubics and G1 Hermite interpolation



Curves with chord length parameterization8
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◦).



Thank you

THANK YOU! LENGTH(CAGD 2006)