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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 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. • 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 ？
RATIONAL QUADRATIC CIRCLES ARE PARAMETRIZED BY CHORD LENGTH • Mathematica code:
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 • 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 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.
CURVES WITH CHORD LENGTH PARAMETERIZATION 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◦).
