Chord length parameterization
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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

CHORD LENGTH PARAMETERIZATION

支德佳

2008.10.30


Chord length parameterization1

CHORD LENGTH PARAMETERIZATION

  • Chord length:


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

  • An arc of a circle:

    ‖ ‖=‖ ‖


Rational quadratic circles are parametrized by chord length1

RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH


Rational quadratic circles are parametrized by chord length2

RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH

  • Mathematica code:


Rational quadratic circles are parametrized by chord length3

RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH


Chord length parameterization

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

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

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

  • Chord length & bipolar coordinates


Chord length parameterization

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

  • Chord length & bipolar coordinates


Chord length parameterization

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.


Chord length parameterization

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

  • Quadratic circles:

    = constant


Curves with rational chord length parametrization1

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

  • Quadratic circles:

  • {0,1,1/2}-->{A, B, S}


Curves with rational chord length parametrization2

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 parametrization3

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION

=c(u)


Curves with rational chord length parametrization4

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 parametrization5

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION


Curves with rational chord length parametrization6

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION


Curves with rational chord length parametrization7

CURVES WITH RATIONAL CHORD LENGTH PARAMETRIZATION


Curves with chord length parameterization cagd2008

CURVES WITH CHORD LENGTH PARAMETERIZATION(CAGD2008)

Wei Lü

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


Curves with chord length parameterization

CURVES WITH CHORD LENGTH PARAMETERIZATION


Curves with chord length parameterization1

CURVES WITH CHORD LENGTH PARAMETERIZATION


Curves with chord length parameterization2

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 parameterization3

CURVES WITH CHORD LENGTH PARAMETERIZATION

  • is a complex function with | | = 1


Curves with chord length parameterization4

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 parameterization5

CURVES WITH CHORD LENGTH PARAMETERIZATION

is rational

is rational


Curves with chord length parameterization6

CURVES WITH CHORD LENGTH PARAMETERIZATION

  • Rational cubics and G1 Hermite interpolation


Curves with chord length parameterization7

CURVES WITH CHORD LENGTH PARAMETERIZATION


Curves with chord length parameterization8

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 parameterization9

CURVES WITH CHORD LENGTH PARAMETERIZATION


Thank you

THANK YOU!


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