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Precise Voronoi Cell Extraction Precise Voronoi Cell Extraction of Free-form Rational Planar of Free-form Rational Planar

Closed CurvesClosed CurvesIddo Hanniel, Ramanathan Muthuganapathy, Iddo Hanniel, Ramanathan Muthuganapathy,

Gershon ElberGershon ElberDepartment of Computer Science, Technion, IsraelDepartment of Computer Science, Technion, Israel..

&&Myung-Soo KimMyung-Soo Kim

School of Computer Science and Engineering,School of Computer Science and Engineering,Seoul National University, KoreaSeoul National University, Korea..

Center for Graphics and Geometric Computing, TechnionCenter for Graphics and Geometric Computing, Technion 2

Definition (Voronoi cell) Given - C0(t), C1(r1), ... , Cn(rn)

- disjoint rational planar closed regular C1 free-form curves.

The Voronoi cell of a curve C0(t) is the set of all points closer to C0(t) than to Cj(rj), for all j > 0.

Currently, our implementation assumes C0(t) to be convex.

C1(r1)

C2(r2)

C3(r3)

C4(r4)

C0(t)

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Definition (Voronoi cell (Contd.))

Boundary of the Voronoi cell.

Voronoi cell consists of points that are equidistant and minimal from two different curves.

C0(t)

C1(r1)

C2(r2)

C3(r3)

C4(r4)

C3(r3)C0(t),

C0(t), C4(r4)

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Definition (Voronoi cell (Contd.))

The above definition excludes non-minimal-distance bisector points.

This definition excludes self-Voronoi edges.

rr22rr33

rr11

tt

rr44

C0(t)

C1(r)rr

pp

qq

“The Voronoi cell consists of points that are equidistant and minimal from two different curves.”

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Definition (Voronoi diagram)

The Voronoi diagram is the union of the Voronoi cells of all the free-form curves. C0(t)

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Existing approaches for VD of free-formsFarouki and Ramamurthy 1999 – V.D. and medial axis.

Numerically traces the bisectors and trim them to get V.D. or MAT.Ramanathan and Gurumoorthy 2003 – Medial Axis (MA). Numerically traces the MAT segments directly. Elber and Kim 1999 – Bisector for planar rational curves.

An implicit representation of bisector.

Linear/circular arc approximation [Gursoy and Patrikalakis 1992, Held 1998, Ju-Hsein Kao 1999].

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Our approach

Using the original representation of the input curves (with no linear / circular approximation).

Generate an accurate implicit representation of the Voronoi cell.

Exact Approximated

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Outline of the algorithm

Implicit bisector function

Application of constraints

Lower envelope algorithm

Splitting into monotone pieces

trtr-space-spaceEuclidean Euclidean spacespace

C0(t)

C1(r)

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Implicit bisector functionGiven two regular C1 parametric curves C0(t) and

C1(r), we get an expression for a normal-intersection point: P(t,r) = (x(t,r), y(t,r)).

The implicit function F3 is defined by: 0)()(,

2

)()(),(),( 10

103

rCtC

rCtCrtPrtF

)(0 tC

)(1 rC ),( rtP

qq

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Computing points on the bisector

Points on the bisector are calculated using the (t,r) pairs of the zero-set of F3(t,r).

For every pair (t,r) the corresponding Euclidean point P is computed using the mapping: P(t,r) = (x(t,r), y(t,r)).

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Implicit bisector function – untrimmed bisector

tr

F3(t,r)

C1(r)

C0(t)

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Splitting the zero-set of the function into monotone pieces

t t

rr

Keyser et al., Keyser et al., Efficient and exact manipulation of algebraic Efficient and exact manipulation of algebraic points and curvespoints and curves, CAD, 32 (11), 2000, pp 649--662., CAD, 32 (11), 2000, pp 649--662.

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Constraints - orientation

Orientation (LL) Constraint – purge away points of the untrimmed bisector that do not lie on the desired side (we assume left side of both the curves as the desire side).

)( 22 tC

)( 11 tC

)( 11 tN L

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LL-constraint

Untrimmed bisector

C0(t)C0(t)

C1(r)C1(r)

Trimmed bisector

LLLL

LLRR

RRLL

RRRR

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Constraints - curvature

Curvature Constraint (CC) – purge away points of the untrimmed bisector whose distance to its footpoints (i.e., the radius of the Voronoi disk) is larger than the radius of curvature (i.e., 1/κ) at the footpoint.

1)(ˆ)(),(),( 21 iiiiii tNttCttP

)( 22 tC

)( 11 tC),( 21 ttP

)(ˆ111 tN

NN11//κκ11

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Illustration of curvature constraint1)(ˆ)(),(),( 21 iiiiii tNttCttP

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Application of curvature constraint

Before After

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Illustration of lower envelope

tt

DD

tt

tt

DD

DD(a)(a) (b)(b)

(c)(c)

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Lower envelope algorithm

Standard Divide and Conquer algorithm.

The main needed functions are:

Identifying intersections of curves.

Comparing two curves at a given parameter (above/below).

Splitting a curve at a given parameter.

||Di (t, ri)||2 = ||Dj (t, rj)||2 ,

F3(t, ri) = 0 ,

F3(t, rj) = 0.

Compare ||Di (t, ri)||2 and ||Dj(t,rj)||2 at the parametric values.

Split F3(t, ri) = 0 at the tri-parameter.

General Lower Envelope Voronoi Lower Envelope

Di (t, ri) = || P(t,ri) – C0(t)||A distance function D defined by

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Result1

C0(t)

C0(t)

C1(r1)C1(r1)

C2(r2)

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Result1 (Contd.)

C0(t)

C0(t)C1(r1)

C1(r1)

C2(r2)

C2(r2)

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Results

C0(t)

C0(t)

C1(r1)

C1(r1)C2(r2)

C2(r2)

C3(r3)

C3(r3)

C4(r4)

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Results

C0(t)

C0(t)

C1(r1)C1(r1)

C2(r2)C2(r2)

C3(r3)C4(r4)

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Discussion

Input rational curves are represented as Bezier/B-spline curves. The input curves are approximated neither by linear segments nor by circular arcs.

Implementation

1. Using IRIT software in C, developed at the Technion.

1. The bivariate function F3(t,r) is obtained using the symbolic library.

2. Constraints are solved using the multivariate library.

2. Uses floating-point arithmetic. Computation took from several seconds to two minutes.

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Future workExtend to open or non-C1 curves – extending the lower

envelope algorithm to point-curve bisectors.Generating Voronoi diagram and medial axis transform

(MAT) of free-form curves efficiently using the implicit representation and solver, and possibly using the lower envelope algorithm.

Implementation using exact arithmetic.

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The EndThe End