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Developable Surface Fitting to Point Clouds Martin Peternell Computer Aided Geometric Design 21(2004) 785-803 Reporter: Xingwang Zhang June 19, 2005

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About Martin Peternell Affiliation Institute of Discrete Mathematics and Geometry Vienna University of Technology Web http://www.geometrie.tuwien.ac.at/peternell People Helmut Pottmann Johannes Wallner etc.

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Research Interests Classical Geometry Computer Aided Geometric Design Reconstruction of geometric objects from dense 3D data Geometric Computing Industrial Geometry

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Overview Problem Developable surfaces Blaschke model Reconstruction of Developable Surfaces Q&A

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Problem Given: scattered data points from a developable surface Object: Construct a developable surface which fits best to the given data

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Ruled Surface directrix curve a generator A ruled surface Normal vector

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Developable Surface Each generator all points have the same tangent plane. Vectors and are linearly dependent Equivalent condition

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Developable Surface Three types of developable surfaces

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Geometric Properties of Developable Surface Gaussian curvature is zero Envelope of a one-parameter family of planes Dual approach: is a curve in dual projective 3-space.

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Singular Point A singular point doesn ’ t possess a tangent plane. Singular curve is determined by the parameter

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Three Different Classes Cylinder: singular curve degenerates to a single point at infinity Cone: singular curve degenerates to a single proper point, called vertex Tangent surface: tangent lines of a regular space curve, called singular curve

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Literature [Bodduluri, Ravani, 1993] duality between points and planes in 3-D space [Pottmann, Farin, 1995] projective algorithm, dual representation [Chalfant, Maekawa, 1998] optimization techniques [Pottmann, Wallner, 1999] a curve of dual projective 3-D space [Chu, Sequin, 2002] boundary curve, de Casteljau algorithm, equations [Aumann, 2003] affine transformation, de Casteljau algorithm

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General Fitting Technique Estimating parameter values Solving a linear problem in the unknown control points fitting unorganized data points Find an developable B-spline surface

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Two Difficult Problems Sorting scattered data Estimation of data parameters Estimation of approximated direction of the generating lines Guaranteeing resulting fitted surface is developable Leading a highly non-linear side condition in the control points

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Contributions of this Paper Avoid the above two problems Reconstruction of a 1-parameter family of planes close to the estimated tangent planes of the given data points Applicable Nearly developable surfaces Better slightly distorted developable surfaces

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Blaschke Model

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An oriented plane in Hesse normal form: Defining Blaschke mapping: Blaschke cylinder:

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Incidence of Point and Plane A fixed point, planes passing through this point Image points lie in the three space The intersection of is an ellipsoid.

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Blaschke Images of a Pencil of Lines and of Lines Tangent to a Circle Back

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Tangency of sphere and plane oriented sphere with center and signed radius Tangent planes: Blaschke image of tangent planes:

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Offset operation Maps a surface (as set of tangent planes) to its offset at distance is the offset surface of at distance Appearing in the Blaschke image as translation by the vector See Figure

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Laguerre Geometry satisfy : inverse Blaschke image tangent to a sphere form a constant angle with the direction vector

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The Tangent Planes of a Developable Surface be a 1-parameter family of planes Generating lines: Singular curve: Blaschke image is a curve on the Blaschke cylinder

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Classification

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Cylinder: Cone: Developable of constant slope: normal form a constant angle with a fixed direction Tangent to a sphere:

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Recognition of Developable Surfaces from Point Clouds

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Estimation of Tangent Planes, triangles, adjacent points Estimating tangent plane at Best fitting data points, MIN Original surface with measurement point developable, form a curve-like region on

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A Euclidean Metric in the Set of Planes Distance between and Geometric meaning: : intersection of with sphere

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Boundary Curves of Tolerance Regions of Center Lines

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A Cell Decomposition of the Blaschke Cylinder Tesselation of by subdividing an icosahedral net

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A Cell Decomposition of the Blaschke Cylinder (continued) Cell structure on the Blaschke cylinder 20 triangles, 12 vertices, 2 intervals 80 triangles, 42 vertices, 4 intervals 320 triangles, 162 vertices, 8 intervals 1280 triangles, 642 vertices, 16 intervals

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Analysis of the Blaschke Image

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Analysis of the Blaschke Image (continued) Check point cloud on fitted well by hyperplane Principal component analysis

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Principal Component Analysis (continued) Minimization Eigenvalue problem

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Principal Component Analysis (continued) Four small eigenvalues: The Blaschke image is a point-like cluster. The original surface is planar. Two small eigenvalues: The Blaschke image is a planar curve (conic). The original surface is a cone or cylinder of rotation. a cone of rotation. a cylinder of rotation.

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Principal Component Analysis (continued) One small eigenvalue and curve-like Blaschke image. The original surface is developable. a general cone a general cylinder a developable of constant slope. One small eigenvalue and surface-like Blaschke-image: The original surface is a sphere.

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Example Analysis of the Blaschke image – Sphere

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Example Cylinder of rotation

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Example Approximation of a developable of constant slope

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Example General cylinderTriangulated data points and approximation Original Blaschke image

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Example Developable of constant slope Triangulated data points and approximation Spherical image of the approximation with control points.

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Reconstruction of Developable Surfaces from Measurements

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Reconstruction Find a curve fitting best the tubular region defined by Determine 1-parameter family of tangent planes determined by Compute a point-representation of the corresponding developable approximation of the data points

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Parametrizing a Tubular Region Determine relevant cells of carrying points Thinning of the tubular region: Find cells carrying only few points and delete these cells and points Estimate parameter values for a reduced set of points (by moving least squares: marching through the tube) Compute an approximating curve on w.r.t. points

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Parametrizing a Tubular Region (continued)

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Curve Fitting Blaschke imageapproximating curve to thinned point cloud

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Curve Fitting (continued) support function (fourth coordinate)

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A Parameterization of the Developable Surface Approximating curve on determines the planes Compute planar boundary curves in planes (bounding box): Point representation of :

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Boundary Curves

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Example Developable surface approximating the data points Projection of the Blaschke image Approximating curve with control polygon

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Deviation Distance between estimated planes and the approximation Distance between measurements and the approximation

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Nearly Developable Nearly developable surfaceProjection of the original Blaschke image

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Nearly Developable Approximation developable approximationThinned Blaschke image with approximating curve

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Singular Points Singular points Data Points satisfy Singular points have to satisfy Singular curve is in the outer region of the bounding box.

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Conclusions Advantages Avoiding estimation of parameter values Avoiding estimation of direction of generators Guaranteeing approximation is developable Improving avoidance of singular points etc.

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Q&A Questions?

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Thanks all! Especial thanks to Dr Liu ’ s help

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