Inspiration: Brent Collins Pax Mundi a sweep path on a sphere

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Inspiration Brent Collins Pax Mundia
sweep path on a sphere
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Circle-Splines (C-Splines)
in the plane.
on the sphere.
in 3D space.
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Circle Splines in the Plane (1)
Original data points and control polygon
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Circle Spline Construction (1)
Original data points and control polygon
Focus on 4 consecutive points A, B, C, D
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Circle Spline Construction (1)
Original data points and control polygon
Focus on 4 consecutive points A, B, C, D
LEFT CIRCLE thru A, B, C
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Circle Spline Construction (1)
Original data points and control polygon
Focus on 4 consecutive points A, B, C, D
LEFT CIRCLE thru A, B, C
RIGHT CIRCLE thru B, C, D
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Circle Spline Construction (1)
Original data points and control polygon
Focus on 4 consecutive points A, B, C, D
LEFT CIRCLE thru A, B, C
RIGHT CIRCLE thru B, C, D
BLEND CURVE between B and C
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How to do the Blending ?

• Left Circle thru A, B, C Right Circle thru B,
  C, D.

D
B
C
A
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Blending With Intermediate Circles

• Left Circle thru A, B, C Right Circle thru B,
  C, D.

Draw Tangent Vectors for both circles at B and C.
D
B
C
A
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Blending With Intermediate Circles

• 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.
D
B
C
A
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Blending With Intermediate Circles

• 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.
Draw n equal-angle-spaced Circles from B to C.
D
B
C
A
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Blending With Intermediate Circles

• 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.
Draw n equal-angle-spaced Circles from B to C.
D
Make n equal segments on each arc andchoose ith
point on ith circle.
B
C
A
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Trigonometric Angle Blending

• 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.
D
ANGLE
B
C
A
STEP i
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Trigonometric Angle Blending

• 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.
Draw n trigonometrically-spaced Circles from B to
C.
D
B
C
A
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Trigonometric Angle Blending

• 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.
Draw n trigonometrically-spaced Circles from B to
C.
D
Blend curve hugs initial circles longer --gt G2
B
C
S
A
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Previous 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 !!

i
0
n
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Circle Blending Previous Art

• 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.

D
C
B
S
A
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The Generated Curve Segments
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Previous Methods (comparison)
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Curvature
Symmetrical S-Curves Between Points (1, 0)
Angle range 3.125 -- 1.125
Max Curvature 4
Angle 3.050 rad
Angle 3.100 rad
Angle 3.125 rad
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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
ltlin./trig.gt angle-steps.
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Implementation 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.

Pi

• B

C
Linear steps, ti
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Conclusions

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

QUESTIONS ?