CHSCHSUCBUCB

Inspiration: Inspiration:

Brent Collins’ Brent Collins’

“Pax Mundi”“Pax Mundi”

a sweep path a sweep path on a sphereon a sphere

CHSCHSUCBUCB Circle-Splines (C-Splines)Circle-Splines (C-Splines)

on the sphere.

in 3D space.

in the plane.

CHSCHSUCBUCB Circle Splines: in the Plane (1)Circle Splines: in the Plane (1)

Original data points and control polygon

CHSCHSUCBUCB Circle Spline Construction (1)Circle Spline Construction (1)

Original data points and control polygon

A

D

C

B

Focus on 4 consecutive points: A, B, C, D

CHSCHSUCBUCB Circle Spline Construction (1)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

CHSCHSUCBUCB Circle Spline Construction (1)Circle Spline Construction (1)

Original data points and control polygon

LEFT CIRCLE thru A, B, CRIGHT CIRCLE thru B, C, D

A

D

C

B

Focus on 4 consecutive points: A, B, C, D

CHSCHSUCBUCB Circle Spline Construction (1)Circle Spline Construction (1)

Original data points and control polygon

LEFT CIRCLE thru A, B, CRIGHT CIRCLE thru B, C, D

BLEND CURVE between B and C

A

D

C

B

Focus on 4 consecutive points: A, B, C, D

CHSCHSUCBUCB How to do the Blending ?How to do the Blending ?

A

B

D

C

Left Circle thru: A, B, C; Right Circle thru: B, C, D.

CHSCHSUCBUCB Blending With Intermediate CirclesBlending 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.

CHSCHSUCBUCB Blending With Intermediate CirclesBlending 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.

CHSCHSUCBUCB Blending With Intermediate CirclesBlending 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.

CHSCHSUCBUCB Blending With Intermediate CirclesBlending 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 andchoose ith point on ith circle.

Draw a bundle of regularly spaced Tangent Vectors.

CHSCHSUCBUCB Trigonometric Angle BlendingTrigonometric 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

CHSCHSUCBUCB Trigonometric Angle BlendingTrigonometric 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.

CHSCHSUCBUCB Trigonometric Angle BlendingTrigonometric 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

CHSCHSUCBUCB Previous Work with CirclesPrevious Work with Circles

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.

M. Szilvasi-Nagi & T.P. Vendel (CAGD 2000)“Generating curves and swept surfaces by blended circles.”Trigonometrical blend: L(i) *cos2(i) + R(i) *sin2(i) G-2 Continuity at B, C. But Cusps are still possible !!

0

i

n

CHSCHSUCBUCB Circle Blending: Previous ArtCircle 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

CHSCHSUCBUCB The Generated Curve SegmentsThe Generated Curve Segments

CHSCHSUCBUCB Previous Methods (comparison)Previous Methods (comparison)

CHSCHSUCBUCB CurvatureCurvature

Symmetrical S-CurvesBetween Points (±1, 0)

Angle = 3.125 rad

Angle = 3.100 radAngle = 3.050 rad

Max Curvature = 4

Angle : range 3.125 -- 1.125

CHSCHSUCBUCB Concept: Swivel Planes thru B,CConcept: 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 <lin./trig.> angle-steps.

CHSCHSUCBUCB Implementation HintsImplementation Hints

Avoid calculations that explicitly involve the centers of the circular arcs,since these will go off to infinity, when the arcs become straight.

Calculate points along arc as an offset from end point B or C.

CB

Pi

Linear steps, ti

CHSCHSUCBUCB ConclusionsConclusions

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