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.

( )

( ) ( )

p t A

p t A p t B

p(t) -A

p(t) -A + P(t) -B

Chord length

Chord-length vs arc-length parametrization

Motivation: Straight line segments is chord length

parameterized.

What other types of parametric curve is chord length parameterized?

Find

Motivation: Chord length parameterization has many usefu

l properties: Geometric parameter, unique;

No self-intersection;

Ease of point-curve testing;

Simplification of curve-curve/curve-surface intersecting.

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

References Rational quadratic circles are parametrized by chord len

gth

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)

Rational quadratic circles are parametrized by chord lengt

h

Gerald Farin CAGD.(2006) 722–724

About the author Gerald Farin : Professor of Com

puter 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.

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

Rational quadratic circles

An arc of a circle the standard form of a rationalquadratic curve

c0, c1, c2 form an isosceles triangle with base c0, c2v1 = cos

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.

Chord length parametrization

chord(t) = t, 0,1t

Chord length parametrization

The complementary segment of the full circle is obtained by replacing v1 by −v1.

It is the chord length parametrization,too.

Curves with chord length parameterization

Wei Lü CAGD In press

About the author Siemens PLM Software, 2000 Eastman D

rive, Milford, OH 45150, USA

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), t R

Plane chord-length parameterization

( )t : The signed angle from Z1 −Z(t) to Z(t)−Z0

(1) -scheme of plane chord length parameterization.

Plane chord-length parameterization

(2) rational quadratic Bézier curve

Plane chord-length parameterizationThe following properties of plane chord-length parameteri

zation: 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 Z

0 and Z1.

, it is well defined and bounded.

End conditions

( )t ( )t ( )2

t

( )t

( )t

12( )

Plane chord-length parameterization

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

Rational chord-length parameterization In complex field, a rational function is

The degree of a complex function, regarded as a curve, is at most twice of that in complex

field.

t in complex field

Complex function regarded as a curve

Rational chord-length parameterizationLemma:

A complex function U = U(t) with |U(t)| = 1 is rational

There is a complex polynomial H = H(t) such thatRemark:

arg(U) = 2arg(H); arg(U) = 2arg(H)+2π; arg(U) = 2arg(H)−2π. deg(U(t)) = 2deg(H(t))

Rational curves

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

Rational curves Each chord length parameterization is entirely determined b

y 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.

( )t

( )t

( )t

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

Rational cubics and G1 Hermite interpolationGiven: the end points Z0,Z1 and tangent directions T0,T1

Find: a rational curve Z(t) to interpolate Z0,Z1, T0,T1.

Rational cubics and G1 Hermite interpolation

0 00 170 , 60

0 00 1130 , 30

G1 Hermite interpolation with higher degree curves To Hermite interpolant the S-shaped curve data Use the higher-degree chord-length parameterizati

on.

α0: the signed angle from Z1 − Z0 to T0, α1: the signed angle from T1 to Z1 − Z0.

s: shape parameter

Examples:

Examples:

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)

Chord-length parameterization in higher-dimensional space

Construct: a chord-length parameterized curve in higher-

dimensional space.

(1) 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.

(α,β)-scheme for three-dimensional curves

L,M and N constitute 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.

Curves with rational chord-length parametrization

J. Sánchez-Reyes, L. Fernández-Jambrina CAGD 08

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 Vi

ctoria s/n, 28040-Madrid, Spain

Interpreting chord-length parametrization

Interpreted as a geometric coordinate Consider two fixed points A,B, and define the coordinate u of a gene

ric point p with 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.

Interpreting chord-length parametrization

chord-length parametrizations are preserving the modulus of the ratios.

Define:

Defining bipolar coordinatesThe bipolar coordinates (u, ):ϕ

of a point ϕ p is the angle between the segments pB and Ap. ∈ (−π,π]ϕ

Isoparametric curves with constant , uϕ

A fixed value u

Isoparametric curves with constant ϕ are again circular arcs, the locus of points : see a segment AB with constant angle (π − ϕ).

A circle, has a centre lying on the line AB

Interpreting chord-length parametrization

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

Complex inversion

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

Circles and chord-length parametrization

Setting a constant ϕ, p(u): a chord-length parametrization , u ∈ [0, 1]

Standard Bézier arcs

Planar rational curves with chord-length parametrization Planar curves p(u) admit rational chord-length par

ametrization on u ∈ [0, 1].

Control the quartic using the following shape handlesEndpoints 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 ABsetting the height:

Equilateral hyperbola and Lemniscate of Bernoulli

Limaçon of Pascal

Space curves with chord-length parametrizationq(u) = {x(u), y(u), z(u)} is chord-length only if its associated planar curve p(u) is chord-length, too.

3D rational curve q(u) with chord-length parametrization

c(u): a rational unit circle

Conclusions: Connection between bipolar coordinates and curves

with chord-length parametrization.

Identify the curves with chord-length parameterization in two or higher-dimension.

Identify the curves with Rational chord-length para

meterization

Thank you

Questions ?