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

About Martin Peternell Affiliation Institute of Discrete Mathematics and Geometry Vienna University of Technology Web People Helmut Pottmann Johannes Wallner etc.

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

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

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

Ruled Surface directrix curve a generator A ruled surface Normal vector

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

Developable Surface Three types of developable surfaces

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.

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

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

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

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

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

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

Blaschke Model

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

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.

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

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

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

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

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

Classification

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

Recognition of Developable Surfaces from Point Clouds

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

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

Boundary Curves of Tolerance Regions of Center Lines

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

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

Analysis of the Blaschke Image

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

Principal Component Analysis (continued) Minimization Eigenvalue problem

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.

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.

Example Analysis of the Blaschke image – Sphere

Example Cylinder of rotation

Example Approximation of a developable of constant slope

Example General cylinderTriangulated data points and approximation Original Blaschke image

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

Reconstruction of Developable Surfaces from Measurements

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

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

Parametrizing a Tubular Region (continued)

Curve Fitting Blaschke imageapproximating curve to thinned point cloud

Curve Fitting (continued) support function (fourth coordinate)

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

Boundary Curves

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

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

Nearly Developable Nearly developable surfaceProjection of the original Blaschke image

Nearly Developable Approximation developable approximationThinned Blaschke image with approximating curve

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

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

Q&A Questions?

Thanks all! Especial thanks to Dr Liu ’ s help