Reconstruction of Developable Surfaces

Martin Peternell,

Geometric Modeling and Industrial Geometry Group,

TU Wien,

<martin@geometrie.tuwien.ac.at>

GMP 2004, Beijing, China 1

Overview

1. The Problem

2. Geometric Properties of Developable Surfaces

3. The Basic Concept

4. The Algorithm

5. Examples

GMP 2004, Beijing, China 2

ProblemGiven: Triangulated datapoints from a developablesurface

Object: Construct a devel-opable surface which fits bestto the given data

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Literature

Hoschek, Schneider, Pottmann, Bodduluri, Ravani, Leopoldseder,Wallner, Redont, Farin, Aumann, Clements, Leon, ...

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Geometric Properties of Developable Surfaces

There are three types of developables:

Cylinder Cone Surface formed by

tangent lines of space curves

• The Gauss curvature of a developable surfaces is zero.

• A developable surface S is envelope of its one-parameter familyof tangent planes

T (t) : d(t) + a(t)x + b(t)y + c(t)z = 0.

• Dual approach: T (t) is a curve in dual projective 3-space.

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The Basic Concept

• The surface S is given by a data pointspi (measurements).

• Estimate tangent planes Ti at points pi.

• The estimated tangent planes Ti will lieclose to the true tangent planes T (t) ofthe surface S.

Idea: We compute a 1-parameter family of planes T (t) best fittingthe estimated tangent planes Ti.

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Tangent planes of a Developable

• Tangent planes T (t) : d(t) + a(t)x + b(t)y + c(t)z = 0 of adevelopable surface S.

• Normalizationa2 + b2 + c2 = 1

of the normal vectors n(t) = (a(t), b(t), c(t)) of S and T (t).• The image curve

C(t) = (a, b, c, d)(t) ∈ R4

is a curve on the

• Blaschke-Cylinder

B : x21 + x2

2 + x23 = 1 ⊂ R4.

• The normal vectors n(t) of S form acurve on the unit sphere S2.

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The Algorithm

• Compute estimates of the tangent planes

Ti : di + aix + biy + ciz = 0

at the data points pi. The points Ci = (ai, bi, ci, di) ∈ B ⊂ R4

should be arranged in a tubular region.

• Find a curve C(t) ⊂ B fitting best the tubular region definedby Ci.

• Determine the one-parameter family of tangent planes T (t)determined by C(t).

• Compute a point-representation of the correspondingdevelopable approximation S∗ of the data points.

• Avoid occurring singularities in the point representation.

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Estimating tangent planes

There are different methods:

• Compute a plane of regression w.r.t. adja-cent points of each data point.

• Compute a linear function z = ax + by + c

w.r.t. adjacent points in the l2-sense (or inthe l1-sense).

• Compute a quadratic function w.r.t. adja-cent points and find the tangent plane atthe considered data point.

• . . .

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Curve Fitting on the Blaschke-Cylinder

• Given: Estimated tangent planes

Ti : di + aix + biy + ciz = 0,

with a2i + b2

i + c2i = 1

and corresponding points Ci = (ai, bi, ci, di)on B (Blaschke cylinder).

• Object: Curve C(t) approximating the setof points Ci.

C(t) =∑

i

Bi(t)bi

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Curve Fitting on the Blaschke-Cylinder 2

• Parametrization problem: How to estimate parameter values ofthe given points Ci?

• Data access on the Blasche cylinder?

• How to handle outliers?

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Cell Structure on the Blaschke-Cylinder

• Tesselation of S2 by subdividing an ikosahedral net.

• Cell structure on the Blaschke-Cylinder B.

1. 20 triangles, 12 vertices, 2 intervals

2. 80 triangles, 42 vertices, 4 intervals

3. 320 triangles, 162 vertices, 8 intervals

4. 1280 triangles, 642 vertices, 16 intervals

5. 5120 triangles, 2562 vert., 32 intervals

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Parametrizing a tubular region

• Determine relevant cells of B carryingpoints Ci = (ai, bi, ci, di).

• Thinning of the tubular region: Find cellscarrying only few points and delete thesecells and points.

• Estimate parameter values for a reducedset of points Ck (by moving least squares:marching through the tube).

• Compute an approximating curve C(t) onB w.r.t. points Ck.

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Parametrizing a tubular region 2

Spherical image n = (a, b, c)(t) Support function (fourth coord. d(t))

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Point Representation of the Approximation

• Approximating curve C(t) = (a, b, c, d)(t) on B determines theplanes

E(t) : d(t) + a(t)x + b(t)y + c(t)z = 0,

which envelope the approximation S∗.

• Generating lines L(t) of S∗ are

L(t) = E(t) ∩ E(t), with E(t) =d

dtE(t).

• Compute planar boundary curveski(t) in planes Hi (bounding box):

ki(t) = E(t) ∩ E(t) ∩Hi.

• Point representation of S∗:

x(t, u) = (1− u)k1(t) + uk2(t).

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Example

Left: Triangulated data points of a developable. Middle & Right:Spherical image.

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Example, cont.

Spherical image n = (a, b, c)(t)and support function d(t) of theapproximation S∗.

Approximating developable S∗

and data points.

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Singular Points of a Developable

• Singular points

s(t) = E(t) ∩ E(t) ∩ E(t).

are corresponding to osculating planes

σ : c(t) + λc(t) + µc(t)

of the Blaschke image C = c(t).

• Let N = c ∨ c ∨ c = (n1, . . . , n4). ⇒ s(t) = 1n4

(n1, n2, n3).

• Data points pi satisfy ‖pi‖ < 1 (or |pi1|, |pi2|, |pi3| < 1)

• Singular curve s(t) has to satisfy

‖s(t)‖ > 1, ( or |s1|, |s2|, |s3| > 1).

Singular curve s is in the outer region of the bounding box.

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Singular Points of a Developable

• The condition ‖s‖ > 1 is (highly) non-linear in thecontrol-points.

• C is a biarc, interpolating Hermiteelements Pj , Vj and Pj+1Vj+1. ⇒‖s‖ > 1 is a quadratic inequality inthe free parameter of the 1-par. so-lution.

• Pottmann and Wallner: Considerspecial parametrizations of devel-opable surfaces which make a con-trol of the singular curve easier.

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Blaschke Images of Special Shapes

Commonly used geometric objects, like cones, cylinders and spherescan be characterized by their images of the estimated tangentplanes on the Blaschke cylinder B:

Objects in R3 Images in the Blaschke-Cylinder

as set of tangent planes as point sets

cones and cylinders of revolution planar intersections (conics on B)

general cones and cylinders curves on B lying in a 3-space

developables of constant slope

spheres surfaces on B lying in a 3-space

(ellipsoids)

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Analysis of the Blaschke image

• Let Ti be reliable estimated tangent planes of a region R on theobject S.

• Let Rb be the Blaschke image ⊂ B of the considered region.

• Compute the ellipsoid of inertia (PCA) of Rb.(axes-hyperplanes and corresponding eigenvalues)

• Analyze the eigenvalues and eigenvectors

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Analysis of the Blaschke image 2

Let Hi be the 3-spaces with respect to eigenvalues λi,

Hi : hi0 + hi1x1 + hi2x2 + hi3x3 + hi4x4 = 0.

• Four small eigenvalues: The Blaschke image is a point-likecluster. All estimated tangent planes are nearly identical.

⇒ The original surface R is planar.

• Two small eigenvalues: The Blaschke image is a planar curve(conic).

⇒ The original surface R is a cone or cylinder of rotation.

– |h10 − h20| > ε: R is a cone of rotation.

– |h10 − h20| ≤ ε: R is a cylinder of rotation.

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Analysis of the Blaschke image 3

• One small eigenvalue and curve-like Blaschke-image:

⇒ The original surface R is developable.

– |h10| < ε: R is a general cone.

– |h10| < ε and |h14| < ε : R is a general cylinder. All planesare orthogonal to a fixed direction v.

– |h14| < ε : R is a developable of constant slope. All planesform a constant angle with a fixed direction v.

• One small eigenvalue and surface-like Blaschke-image:

⇒ The original surface R is a sphere.

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Analysis of the Blaschke image–Sphere

Let Hi be the 3-spaces with respect to eigenvalues λi,

Hi : hi0 + hi1x1 + hi2x2 + hi3x3 + hi4x4 = 0, h14 � ε.

• S is a sphere (λ1 < ε): Its center and radiusare determined by

m =1

h14(h11, h12, h13) , r =

−h10

h14

Region 1 contains 4166 data points.

Eigenvalues 0.00004 0.18598 0.36039 0.36837

Eigenvector -0.0024 0.0116 -0.3500 0.9367

-0.2972 0.9547 -0.0041 -0.0141

0.0015 -0.0087 -0.9368 -0.3499

0.9548 0.2972 -0.0006 -0.0015

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Cylinder and Cone of Rotation

Let Hi be the 3-spaces with respect to eigenvalues λi,

Hi : hi0 + hi1x1 + hi2x2 + hi3x3 + hi4x4 = 0, h14, h24 � ε.

If h14 < ε, then (hi1, hi2, hi3) = a is the axis of R.

• λ1, λ2 < ε ⇒ centers and radii of two spheres

m1 = 1h14

(h11, h12, h13) , r1 = −h10h14

m2 = 1h24

(h21, h22, h23) , r2 = −h20h24

, with

• |r1 − r2| < ε: R is a cylinder of rot. with axis and radius

a = m2 −m1, r = mean(r1, r2).

• |r1 − r2| � ε: R is a cone of rot. with vertex, axis and angle

v =1

r2 − r1(r2m1− r1m2), a = m2−m1, sinφ =

r2 − r1

‖m2 −m1‖.

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Example: Cylinder of rotation

Region 1 contains 3553 data points.

Eigenvalues 0.00013 0.00024 0.48129 0.51860

Eigenvector 0.0245 0.0017 -0.9958 0.0881

0.1074 -0.9942 0.0014 0.0050

-0.0239 0.0026 0.0875 0.9959

0.9936 0.1075 0.0265 0.0212

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Developable of Constant Slope

Let Hi be the 3-spaces with respect to eigenvalues λi,

Hi : hi0 + hi1x1 + hi2x2 + hi3x3 + hi4x4 = 0.

• λ1 < ε and curve-like Blaschke-image:

⇒ The original surface R is developable.

– |h14| < ε : R is a developable of constant slope. All planesform a constant angle with a fixed direction v.

Eigenvalues 0.0000 0.0028 0.1473 0.3175

Eigenvector 0.0013 0.9815 0.1849 0.0498

0.0006 0.1407 -0.8727 0.4675

1.0000 -0.0014 0.0004 -0.0002

-0.0002 -0.1299 0.4518 0.8826

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Segmentation of Geometric Objects

• Estimate tangent planes Ti with respect to given triangulateddata points pi of the object S.

• Edge detection to perform a pre-segmentation of the object.

• Compute connected components of the object (w.r.ttriangulation).

• Compute Blaschke images T bi of the estimated tangent planes

Ti.

• Segmentation and analysis of the Blaschke image Sb.

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Example: Simple composed object

Region 1 contains 1933 data points.

Eigenvalues: 0.00000 0.28204 0.37189 0.47503

Eigenvector 0.5705 -0.0035 0.0927 -0.8161

-0.2269 0.6355 0.7339 -0.0779

0.1618 -0.6957 0.6729 0.1926

0.7726 0.3349 0.0061 0.5393

Region 2 contains 1933 data points.

Eigenvalues: 0.00000 0.27870 0.36828 0.42823

Eigenvector: -0.4899 0.1305 0.1641 0.8462

-0.2408 0.5691 -0.7826 -0.0754

0.1718 -0.7100 -0.6005 0.3254

0.8201 0.3938 -0.0060 0.4152

Region 3 contains 811 data points.

Eigenvalues: 0.00000 0.00000 0.00001 0.00001

Eigenvector: .0052 -0.2686 0.9624 0.0410

0.0003 0.2839 0.1198 -0.9513

0.9992 -0.0361 -0.0149 -0.0124

0.0406 0.9198 0.2435 0.3051

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Example: Simple composed object 2

Region 4 contains 822 data points.

Eigenvalues: 0.00000 0.00000 0.06475 0.46085

Eigenvector 0.1916 0.1297 -0.1339 0.9636

-0.5603 0.8283 0.0001 -0.0000

0.3355 0.2269 0.9138 0.0297

-0.7326 -0.4956 0.3834 0.2657

Region 5 contains 333 data points.

Eigenvalues: 0.00000 0.00000 0.00002 0.00006

Eigenvector: 0.0394 0.4624 0.5194 0.7175

0.9980 -0.0234 -0.0592 0.0032

0.0497 0.2202 0.7146 -0.6621

0.0067 -0.8586 0.4647 0.2166

Region 6 contains 153 data points.

Eigenvalues: 0.00000 0.00000 0.00001 0.00036

Eigenvector: -0.0473 0.5308 0.8455 -0.0327

0.0367 0.2322 -0.1807 -0.9550

0.9807 0.1684 -0.0475 0.0876

-0.1863 0.7975 -0.5002 0.2813

. . .

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Future research

• Continuous transformations of developable surfaces.

• Treatment of nearly developable surfaces.

• Segmentation using both points pi and tangent planes Ti.

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Thank You for Your attention!

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