1. Inspiration: Brent Collins’ “Pax Mundi” a sweep path on a sphere

2. Circle-Splines (C-Splines) on the sphere. in 3D space. in the plane.

3. Circle Splines: in the Plane (1) Original data points and control polygon

4. Circle Spline Construction (1) Original data points and control polygon A D C B Focus on 4 consecutive points: A, B, C, D

5. Circle Spline Construction (1) Original data points and control polygon LEFT CIRCLE thru A, B, C A D C B Focus on 4 consecutive points: A, B, C, D

6. Circle Spline Construction (1) Original data points and control polygon LEFT CIRCLE thru A, B, C RIGHT CIRCLE thru B, C, D A D C B Focus on 4 consecutive points: A, B, C, D

7. Circle Spline Construction (1) Original data points and control polygon LEFT CIRCLE thru A, B, C RIGHT CIRCLE thru B, C, D BLEND CURVE between B and C A D C B Focus on 4 consecutive points: A, B, C, D

8. How to do the Blending ? A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D.

9. Blending With Intermediate Circles A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. Draw Tangent Vectors for both circles at B and C.

10. Blending With Intermediate Circles A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. Draw Tangent Vectors for both circles at B and C. Draw a bundle of regularly spaced Tangent Vectors.

11. Blending With Intermediate Circles A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. Draw n equal-angle-spaced Circles from B to C. Draw Tangent Vectors for both circles at B and C. Draw a bundle of regularly spaced Tangent Vectors.

12. Blending With Intermediate Circles A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. S Draw n equal-angle-spaced Circles from B to C. Draw Tangent Vectors for both circles at B and C. Make n equal segments on each arc and choose i th point on i th circle. Draw a bundle of regularly spaced Tangent Vectors.

13. Trigonometric Angle Blending A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. Draw Tangent Vectors for both circles at B and C. Draw a bundle of trigonometrically spaced tangents. STEP i ANGLE

14. Trigonometric Angle Blending A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. Draw n trigonometrically-spaced Circles from B to C. Draw Tangent Vectors for both circles at B and C. Draw a bundle of trigonometrically spaced Tangents.

15. Trigonometric Angle Blending A B D C Left Circle thru: A, B, C; Right Circle thru: B, C, D. Draw n trigonometrically-spaced Circles from B to C. Draw Tangent Vectors for both circles at B and C. Draw a bundle of trigonometrically spaced Tangents. S Blend curve “hugs” initial circles longer: --> G2

16. Previous Work with Circles u H.- J. Wenz (CAGD 1996) “Interpolation of curve data by blended generalized circles.” Linear interpolation: L(i) *(1-i) + R(i) *(i)  G-1 Continuity at B, C. u M. Szilvasi-Nagi & T.P. Vendel (CAGD 2000) “Generating curves and swept surfaces by blended circles.” Trigonometrical blend: L(i) *cos 2 (i) + R(i) *sin 2 (i)  G-2 Continuity at B, C. But Cusps are still possible !! 0 i n

17. Circle Blending: Previous Art A B D C Left Circle thru: A, B, C. Right Circle thru: B, C, D. n points on Left Circle. n points on Right Circle. Interpolate positions between corresponding points. S

18. The Generated Curve Segments

19. Previous Methods (comparison)

20. Curvature Symmetrical S-Curves Between Points (±1, 0) Angle = 3.125 rad Angle = 3.100 rad Angle = 3.050 rad Max Curvature = 4 Angle : range 3.125 -- 1.125

21. Concept: Swivel Planes thru B,C 3 consecutive points define a plane and a circle on it. A, B, C  Left Circle. B, C, D  Right Circle. Intermediate planes / arcs at angle-steps.

22. Implementation Hints u Avoid calculations that explicitly involve the centers of the circular arcs, since these will go off to infinity, when the arcs become straight. u Calculate points along arc as an offset from end point B or C. C B PiPi Linear steps, t i

23. Conclusions Angle-Averaged Circles (C-Splines) are useful for making smooth shapes on a sphere, in the plane, and in 3D.