Apollonius tenth problem via radius adjustment and Mobius transformations

1. Apollonius tenth problem via radius adjustment and Mobius transformations Donguk Kim,Deok-Soo Kim,Kokichi Sugihara Computer-Aided Design 2006

2. Author • Donguk Kim:a research assistant • professor at Hanyang University,Korea • Deok-Soo Kim:a professor in • Department of Industrial Engineering • Hanyang University,Korea • Kokichi Sugihara:a professor in • Mathematical Engineering at Tokyo • University,Japan • Interest:Computational Geometry,Computer Graphics,Computer Vision

3. Introduction • 1.three geometric objects-circles • 2.tangent circles

4. Organization • Configurations of Apollonius circles • Radius adjustment transformations • Mobius transformation • Algorithm • Examples • Applications • Conclusions

5. Configurations • Generator circles Gi=(ci,ri),i=0,1,2 • Center:ci=(xi,yi);radius:ri&ri≤rj where i<j

6. Definition 1.Suppose that there are two circles,a target circle and a neighboring circle.Then,a status value is assigned to the target circle as follows: • (i)the target is assigned with O,if both the target and the neighbor are outside each other • (ii)the target is assigned with X,if the target is contained inside the neighbor • (iii)the target is assigned with ∆,if the target contains the neighbor

7. Theorem 1.In a configuration,X cannot follow ∆. • Corollary 2.Among 27 possibilities,the following 7 configurations are infeasible:∆XO,O∆X,∆OX,X∆X,∆XX,∆∆X,and ∆X∆.

9. Radius adjustment transformations • Pre-adjustment

11. Fig.(a),(b),(c):the status value of the smallest generator is O; • Fig.(d),(e),(f):X; • Fig.(g),(h):∆. • Target:three circles→two circles and one dot

12. Post-adjustment: • (a,b,c):shrink the tangent circle by the radius of the smallest generator • (d,e,f):enlarge the radius of the tangent circle • (g,h):shrink the tangent circle

13. Adjustment rules: • If the value of the small generator is O,we shrink all O generators while we enlarge all X-and ∆-generators.Once the tangent circles is obtained,we shrink the tangent circle to get the desired Apollonius circle. • If X,we shrink all X and ∆-generators while we enlarge all O generators.Once the tangent circle is obtained,we enlarge the tangent circle to get. • If ∆,we shrink all ∆-generators while we enlarge all O-generators.Once obtained,we shrink the tangent circle and shift the sign of radius.

15. Mobius transformation • O-plane:the original Euclidean plane where we want to find Apollonius circles • Z-plane:the radius adjusted complex plane • W-plane:the complex plane mapped by W(z)

16. Definition 2. • Let Tij be a circle tangent to both radius adjusted circles Z1 and Z2 with status values i andj,respectively,where i,j∈{O,X,∆}

18. Lemma 3.Mobius transformation W(z)=1/(z-z0) has the following properties. • It transforms lines and circles passing through z0 in the Z-plane to straight lines in the W-plane • It transforms lines and circles not passing through z0 in the Z-plane to circles in the W-plane • It transforms the point at infinity in the Z-plane to the origin in the W-plane • It transforms the point z0 in the Z-plane to the infinity in the W-plane

20. Definition 3.Suppose that a circle,called a target circle,is tangent to a line,called the tangent line. • (i)the target is assigned with O,if the target and the origin are located in the same half space with respect to the tangent line,while the origin is not contained in the target • (ii)the target is assigned with X,if the target and the origin are located in the opposite half space with respect to the tangent line • (iii)the target is assigned with ∆,if the origin is located in the target

21. Theorem 4.By the mapping W(z),the tangent circle Tij in the Z-plane maps to a line Lij tangent to W1 and W2 in the W-plane,where i,j∈{O,X,∆}.

22. Algorithm • Input:G0,G1,G2,and the configuration of desired Apollonius circle • Output:a computed Apollonius circle • Procedure: • 1.Adjust radii of generators according to Column C in Table 2,and obtain Z1,Z2 and z0. • 2.Transform Z1 and Z2 into the W-plane by W(z). • 3.Compute the corresponding tangent line of W1 and W2 in the W-plane according to Column D in Table 2. • 4.Transform the tangent line obtained in Step 3 into tangent circle in the Z-plane by Z(w). • 5.Adjust the radius of the tangent circle according to Column Ein Table 2.

23. Example • Configuration is OX∆; • Configuration is ∆O∆; • Configuration is XO∆.

24. Application • The Euclidean Voronoi diagram for circles in a plane:

25. Conclusion • The observations which are summarized as a compact rule indeed provide an easy-to-code algorithm to compute any desired Apollonius circle.It turns out that the algorithm is computationally efficient and robust.

26. Thanks