siggraph 2003, san diego

SIGGRAPH 2003, San DiegoSIGGRAPH 2003, San Diego

Fair and Robust Circle Splines

Carlo Séquin, EECS, UCB

Kiha Lee, ME, UCB

Motivation: Smooth Curves on a SphereMotivation: Smooth Curves on a Sphere “Grand Tour” camera movements

to look inward towards an object. Sculpture Design:

“Viae Globi” (Pathways on a Sphere)Sweep along a smooth curve on a sphere.

Concepts and GoalsConcepts and Goals

Sparse set of control points.

These get interpolated exactly.

All symmetries are maintained.

Produces round, pleasing looking curves.

If circles are possible, they will result.

Like MVC*, but only local support.

* MVC = Minimum (Curvature) Variation Curve

Smooth Interpolating Curves on Sphere:

Circle-Splines (“C-Splines”)Circle-Splines (“C-Splines”)

on the sphere,in 3D space,

in the plane,

One simple scheme: …

Circle Splines: in the Plane (0)Circle Splines: in the Plane (0)

Original data points and control polygon

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

Circle Spline Construction (2)Circle Spline Construction (2)

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

Circle Spline Construction (3)Circle Spline Construction (3)

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

Circle Spline Construction (4)Circle Spline Construction (4)

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

Local SupportLocal Support

STRAIGHT LINE STRAIGHT LINE

Pi

Pi–1Pi–2 Pi+2Pi+1

A

D

C

B

PURE CIRCLE

PURE CIRCLE

BLEND

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.

?

Circle Blending: Previous Art (1)Circle Blending: Previous Art (1)

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.

SLinearly weighted:[ Wenz, 1996 ]

Circle Blending: Previous Art (2)Circle Blending: Previous Art (2)

A

B

D

CS

0 STEP u 1

WEIGHT

Trigonometrically weighted: Wu = Lu cos2(u /2) + Ru sin2(u /2)

[ Szilvasi-Nagy & Vendel, 2000 ]

Curve clings to base circleG2-continuity @ B

STEP u

Circle Blending: Previous Art (3)Circle Blending: Previous Art (3)

Angle-based blending [ Séquin & Yen, 2001 ]

Subdivision technique (find mid-point only, and recurse)

Curves looked good

Could not formally prove continuity

2003: Sampled (Morph) Construction

Curves look even better!

Can formally prove G2-continuity

Can also obtain C2 with re-parameterization

Blending With Intermediate Blending With Intermediate CirclesCircles (1) (1)

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.

Blending With Intermediate Blending With Intermediate CirclesCircles (2) (2)

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.

Blending With Intermediate Blending With Intermediate CirclesCircles (3) (3)

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.

Blending With Intermediate Blending With Intermediate CirclesCircles (4) (4)

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 uth point on uth circle.

Draw a bundle of regularly spaced Tangent Vectors.

G1-continuity @ B, C

Trigonometric Angle Blending (1)Trigonometric Angle Blending (1)

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 u

ANGLE

Trigonometric Angle Blending (2)Trigonometric Angle Blending (2)

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.

Trigonometric Angle Blending (3)Trigonometric Angle Blending (3)

A

B 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 D

Various Interpolation SchemesVarious Interpolation Schemes

TrigonometricallyAngle-Blended

C-SplineNO CUSPS !

CubicPolynomialSubdivision

Too “loopy”

TrigonometricallyPosition-Blended

Circular Arcs ( Szilvasi-Nagi )

Too sharp “Just right”

Comparing Interpolation SchemesComparing Interpolation Schemes A simple control polygon

( Test case 2 )

Comparing Interpolation SchemesComparing Interpolation Schemes Cubic polynomial subdivision scheme

[ Butterfly, Loop, Zorin ]

Too sharp

EXTREMECURVATURE

Comparing Interpolation SchemesComparing Interpolation Schemes (Non-uniform) Lagrange Interpolation

Too sharp

EXTREMECURVATURE

Comparing Interpolation SchemesComparing Interpolation Schemes Linearly position-blended circular arcs

[ Wenz, 1996 ]

Too “loopy”

Comparing Interpolation SchemesComparing Interpolation Schemes Trigonometrically position-blended circular arcs

[ Szilvasi-Nagi and Vendel, 2000 ]

Too “loopy”

Comparing Interpolation SchemesComparing Interpolation Schemes Angle-blended C-splines: C0 / G2

[ Séquin - Lee, 2003 ]

“Just right”

Comparing Interpolation SchemesComparing Interpolation Schemes Angle-blended C-splines, reparameterized: C2 / G2

[ Séquin - Lee, 2003 ]

“Just right”

Comparing Circle-Blending SchemesComparing Circle-Blending Schemes

Positional interpolationSzilvasi-Nagy & Vendel, 2000

Angle-based parametrization (new)

u=0

u=1

D

A

B CB C

PARAMETERLINESINTERSECT

SA

DC

SL

SR

AB

Averaged circle,based on angles !

Linearly averaged point position.SP

Comparison of Location of Midpoint SComparison of Location of Midpoint S

SA

DC

SL

SR

AB

SP

Problem of Position AveragingProblem of Position Averaging

It would be difficult to reach tangent direction at C from mid-point SP

Key Points about Circle SplinesKey Points about Circle Splines

Going from Polynomial Splines to Blended Circles:

Give up affine invariance

Give up linear precision

Avoid cusps and hair-pin turns

Get nicely rounded, loopy curves

Approximate the behavior of the MVC

Bounded CurvatureBounded Curvature For non-extreme control polygons,

the resulting curvature is bounded.

D CB

A179° SWITCHBACK @ B MAX. CURVATURE = 10

Circle-Splines Circle-Splines on the Sphereon the Sphere

Examples from a prototype editor program.

We only need one small conceptual change …

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 Plane.

B, C, D Right Plane.

Intermediate planes / arcs at <lin./trig.> angle-steps.

A

P(u)

(u)

Discrete Sample Points on Swivel PlaneDiscrete Sample Points on Swivel Plane

C

DA

B

END VIEW SIDE VIEW

C

DA

B

“Trigonometric Swivel” for G2-continuity

Circle-Splines in 3D Free SpaceCircle-Splines in 3D Free Space

We want to have a seamless, unnoticeable transition from sphere to plane:

No “if” statements in the code associated with a check for planarity !

In 2001, we had a very cumbersome wayto bring spheres and planes under one hat.

Today, the 3D case and the spherical casenaturally turn out to be the same !

Circle-Splines in 3DCircle-Splines in 3D

A

B

D

C

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

Mirror Tangent Vectors for both circles at B and C.

Draw a bundle of suitably spaced Tangent Vectors.

Corresponding tangent pairs define intermediate arcs.

Two different planes

MIRROR

Circle-Splines on SphereCircle-Splines on Sphere

A

B

D

C

There is always at least one sphere through A, B, C, D.

Left Circle and Right Circle lie on that sphere;

The two tangent bundles are tangent to this sphere;

All intermediate arcs lie on sphere through A,B,C,D;

All generated points lie on sphere !

Tangent planes

MIRROR

A 3D Circle-Spline (Figure-8 Knot)A 3D Circle-Spline (Figure-8 Knot)

(Cross-eye stereo view)

Important Implementation IssuesImportant Implementation Issues

P(u)

Linear steps, t(u)

B C

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 (B,C)as an offset from end point B.

Parametric Representation of Blend CurveParametric Representation of Blend Curve

For: u = 0 1 t(u) = ti cos2(u p/2) + ti+1 sin2(u p/2)

(Trigonometric angle parametrization)

f(u) = d sin(u t(u)) / sin(t(u)) (u) = (1-u) t(u)

(Blend curve P(u) in polar coordinates)

Bf(u)

(u)C

P(u)t(u)

How to Find Arc Through 3 Data Points ?How to Find Arc Through 3 Data Points ?(We don’t really need arc, just need tangents!)(We don’t really need arc, just need tangents!)

B

a

C

Ac

b

Do not try to find center, radius !

Key InsightsKey Insights The typical polynomial spline

blends between the straight-line segments of the control polygon.

Circle splines start with an overlapping sequence of circular arc segmentsand then blend between those.

The relevant parameter in this contextare ( turning / tangent ) angles.

Why This WorksWhy This Works

We want to emulate a (loopy) MVC.

The ideal MVC result is a circle.

If you want to see circles, start with circles !

Affine invariance dimensional collapse.

Circles always consider problem in a 2D way, giving equal weight to both coordinate axes.

CaveatCaveat These curves are not MVC’s !

While they are G2 (C2) continuous, theymay possess unnecessary undulations.

A local scheme cannot compete with a global scheme !

BAD TANGENT

UNNECESSARY INFLECTION POINTS

ExtensionsExtensions

Extend range of local support to >3 points.

Better approximation to MVC: Cornu spiral or clothoid (curvature is linear with arc length).

In 3D also consider a helix as a primitive.

Issues are : How to find robustly and efficiently

the (unique?) matching primitives ?

How to blend them ?

An “Evolutionary” Process…An “Evolutionary” Process…

From rulers…

Polynomial splines with their linear interpolation

are reminiscent of straight edge rulers…

An Evolutionary Process…An Evolutionary Process…

From rulers…

to compasses…

Circle splines capture the power of compasses…

An Evolutionary Process…An Evolutionary Process…

From rulers…

to compasses…

to French Curves !

QUESTIONSQUESTIONS ??