1 / 59

Curves with chord length parameterization

Curves with chord length parameterization. Reporter: Hongguang Zhou Oct. 15th, 2008. What is chord length parameterization?. Given a curve p (t), t ∈[a, b] If t =chord (t) = So p (t) is chord-length parameterized . Chord length.

siobhan
Download Presentation

Curves with chord length parameterization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Curves with chord length parameterization Reporter: Hongguang Zhou Oct. 15th, 2008

  2. What is chord length parameterization? • Given a curve p(t), t ∈[a, b] If t =chord (t)= • So p(t) is chord-length parameterized.

  3. Chord length

  4. Chord-length vs arc-length parametrization

  5. Motivation: • Straight line segments is chord length parameterized. • What other types of parametric curve is chord length parameterized? Find

  6. Motivation: • Chord length parameterization has many useful properties: • Geometric parameter, unique; • No self-intersection; • Ease of point-curve testing; • Simplification of curve-curve/curve-surface intersecting.

  7. Outline • Interpreting chord-length parametrization • Defining bipolar coordinates • Circles and chord-length parametrization • Plane chord-length parameterization • Plane rational chord-length parameterization • Chord-length parameterization in higher-dimensional space

  8. References • Rational quadratic circles are parametrized by chord length Gerald Farin (CAGD 06) • Curves with rational chord-length parametrization J. Sánchez-Reyes, L. Fernández-Jambrina (CAGD 08) • Curves with chord length parameterization Wei Lü (CAGD In Press)

  9. Rational quadratic circles are parametrized by chord length Gerald Farin CAGD.(2006) 722–724

  10. About the author • Gerald Farin : Professor of Computer Science and Engineering at Arizona State University (ASU) since 1987. • His three main areas of research interest: • Curve and surface modelling; • NURBS; • Industrial curve and surface applications.

  11. About the author Editor in Chief in (CAGD) since 1994. Editor of the SIAM book series on Geometric Design for 10 years An editorial board member of the Springer Verlag series on Mathematics and Visualization. A member of a number of interdisciplinary project committees

  12. Rational quadratic circles the standard form of a rational quadratic curve An arc of a circle c0, c1, c2 form an isosceles triangle with base c0, c2 v1 = cos

  13. Chord length parametrization Rational quadratic circle segments in standard form: Its parametrization is not the arc length It is the chord length parametrization. It is the only one which is parametrized by chord length.

  14. Chord length parametrization chord(t) = t,

  15. Chord length parametrization • The complementary segment of the full circle is obtained by replacing v1 by −v1. • It is the chord length parametrization,too.

  16. Curves with chord length parameterization Wei Lü CAGD In press

  17. About the author Siemens PLM Software, 2000 Eastman Drive, Milford, OH 45150, USA

  18. Plane chord-length parameterization • Using complex analysis; • Parametric plane curve in P(t) = (x(t), y(t)) • Complex function Z(t) = x(t)+iy(t),

  19. Plane chord-length parameterization : The signed angle from Z1 −Z(t) to Z(t)−Z0 (1) -scheme of plane chord length parameterization.

  20. Plane chord-length parameterization (2) rational quadratic Bézier curve

  21. Plane chord-length parameterization The following properties of plane chord-length parameterization: • Z0, Z∗(t)and Z1 always form an isosceles triangle with the base angle , or , if • is constant other than 0 orπ, it becomes a circular arc. • = 0 (or π), it is a (unbounded) straight line segment through points Z0 and Z1. • , it is well defined and bounded. • End conditions

  22. Plane chord-length parameterization A curve Z = Z(t)admits a pre-specified chord length function Chord(t) =ξ(t)(0 ξ(t) 1)

  23. Rational chord-length parameterization • In complex field, a rational function is • The degree of a complex function, regarded asa curve, is at most twice of that in complex field. t in complex field Complexfunction regarded asa curve

  24. Rational chord-length parameterization • Lemma: A complex functionU = U(t) with |U(t)| = 1 is rational • There is a complex polynomialH = H(t) such that • Remark: • arg(U) = 2arg(H); • arg(U) = 2arg(H)+2π; • arg(U) = 2arg(H)−2π. • deg(U(t)) = 2deg(H(t))

  25. Rational curves Z(t)can be converted into a rational Bézier form with its Bézier control points in Euclidean space

  26. Rational curves • Each chord length parameterization is entirely determined by a unit-circular parameterization. • Bounded straight line segments are the only curves with the polynomial (linear) chord-length parameterization ( ≡ 0). • Unbounded straight line segments are the only curves having a rational linear chord-length parameterization ( ≡ π). • Circular arcs are the only curves admitting a rational quadratic chord-length parameterization with constant other than 0 and π.

  27. Rational cubics and G1 Hermite interpolation • H(t) =h0(1−t)+h1t +i, h0 , h1 R The form of rational cubic Bézier If h0=h1, Z(t) a circular arc

  28. Rational cubics and G1 Hermite interpolation Given: the end points Z0,Z1 and tangent directions T0,T1 Find: a rational curve Z(t) to interpolate Z0,Z1, T0,T1.

  29. Rational cubics and G1 Hermite interpolation

  30. G1 Hermite interpolation with higher degree curves • To Hermite interpolant the S-shaped curve data • Use the higher-degree chord-length parameterization. • α0: the signed angle from Z1 − Z0 to T0, α1: the signed angle from T1 to Z1 − Z0. • s: shape parameter

  31. Examples:

  32. Examples:

  33. Chord-length parameterization in higher-dimensional space to be the curve with chord-length parameterization P(0)=p0, P(1)=p1 , Q(t):a unit-vector valued function being always perpendicular to L; (x(t), y(t)): a plane chord-length parameterization curve interpolating two end points P0 = (− l/2,0) (at t = 0) and P1 = (l/2, 0) (at t = 1)

  34. Chord-length parameterization in higher-dimensional space • Construct: a chord-length parameterized curve in higher- • dimensional space. • Create a unit vector-valued function Q(t) perpendicular to L. • (2) Build up a planar chord-length parameterization (x(t), y(t)). • If Q(t): a fixed unit vector. • Represents a planar chord-length parameterized curve.

  35. (α,β)-scheme for three-dimensional curves L,M and Nconstitute an orthogonal system. 3-dimensional chord-length parameterized curve is essentially decided by two angular functions α(t), β(t). α(t): its chord-length parameterization , β(t): its rotation around the axis through two end points.

  36. Curves with rational chord-length parametrization J. Sánchez-Reyes, L. Fernández-Jambrina CAGD 08

  37. About the author • 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

  38. Interpreting chord-length parametrization • Interpreted as a geometric coordinate • Consider two fixed points A,B, and define the coordinate uof a generic point pwith respect to A,B as: • A curve p(u)over a unit domain u ∈ [0, 1], with A = p(0), B = p(1), • If its parameter • The curve is chord-length parameterized.

  39. Interpreting chord-length parametrization Define: chord-length parametrizations are preserving the modulus of the ratios.

  40. Defining bipolar coordinates The bipolar coordinates (u,ϕ): ϕ of a point p is the angle between the segments pB and Ap. ϕ ∈ (−π,π]

  41. Isoparametric curves with constant ϕ, u A fixed value u A circle, has a centre lying on the line AB Isoparametric curves with constant ϕare again circular arcs, the locus of points : see a segment AB with constant angle (π − ϕ).

  42. Interpreting chord-length parametrization To construct chord-length parametrized curves p(u), simply choose an arbitrary function ϕ(u).

  43. Complex inversion The inverse 1/p(u)of a chord-length parametrized curve p(u): chord-length parametrization, too.

  44. Circles and chord-length parametrization Setting a constant ϕ, p(u): a chord-length parametrization , u ∈ [0, 1] Standard Bézier arcs

  45. Planar rational curves with chord-length parametrization • Planar curves p(u)admit rational chord-length parametrization on u ∈ [0, 1].

  46. 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). Choose the position of S on the bisectorof AB setting the height:

  47. Equilateral hyperbola and Lemniscate of Bernoulli

More Related