A construction of rational manifold surfaces of arbitrary topology and smoothness from triangular meshes Presented by: LiuGang

Authors Giovanni Della Vecchia research assistant at Bert Jüttler Research interest: CAGD, Applied Geometry, kinematics Associate Editor of CAGD Myung-Soo Kim Research interest: Computer Graphics, Computer Animation, Geometric Modeling and processing Associate Editor of CAGD, CAD, computer graphics forum Johannes Kepler University Linz, Institute of Applied Geometry, Austria Seoul National University, School of Computer Science and Engineering, South Korea

Outline Introduction Past works Overview of the construction The notion of blending manifolds Construction for rational blending manifolds Examples

Introduction Given triangular mesh consisting of list of vertices, normal vectors (optional), oriented triangles Construct a smooth free-form surface described as a collection of rational patches

Past works Two groups methods Patch–based methods By joining polynomial or rational surface patches with various degrees of geometric continuity Manifold-type constructions Based on a traditional concepts: Manifold

Patch–based methods J. Peters (2002b). C 2 free–form surfaces of degree (3, 5). Computer Aided Geometric Design U. Reif (1998). TURBS - topologically unrestricted rational B-splines. Constructive Approximation Using singularly parameterized surfaces deal with where three or more than four quadrilateral surface patches meet in a common point H. Prautzsch (1997). Freeform splines. Computer Aided Geometric Design Avoids singular points by composing the parameterization of the geometry at extraordinary points with piecewise polynomial uses tensor-product patches of degree (3, 5) to construct curvature continuous free- form surfaces of degree 2 Contribution: The patch–based constructions are able to generate smooth free- form surfaces of relatively low degree. Typically they require a special treatment for “extraordinary” points.

What is a manifold? Define the overlap U ij to be the part of chart i that overlaps with chart j. May be empty. Transition function y ij maps from U ij to U ji. Given: Surface S of dimension m embedded in Construct a set of charts, each of which maps a region of S to a disk in Mapping must be 1-1, onto, continuous (hold for the inverse) Every point in S must be in the domain of at least one chart Collection of charts is called an atlas Note: A surface is manifold if such an atlas can be constructed

Manifold cont. Given a set of charts and transition function, define manifold to be quotient –Transition functions Reflexive  ii (x) = x Transitive (  ik (  kj (x)) =  ij (x) Symmetric  ij (  ji (x)) = x –Quotient: if two points are associated via a transition function, then they ’ re the same point

Manifold-type constructions C. M. Grimm and J. F. Hughes (1995). Modeling surfaces of arbitrary topology using manifolds. Siggraph’95 First who presented a constructive manifold surface construction. The desired surface is specified using a sketch mesh where all vertices have valence four. L. Ying and D. Zorin. A simple manifold-based construction of surfaces of arbitrary smoothness. Siggraph’04 Transition functions are chosen from a particular class of holomorphic functions

Notations: Given triangular mesh M in R 3 m V is the number of vertices; m F is the number of faces; m E is the number of edges The mesh are oriented by outward pointing normals. Blending manifolds associated with triangular meshes

Definition of indices Set of vertex indices: Ordered list of neighboring vertices of i-th vertex: Set of edge indices: Set of face indices:

Charts For each vertex of the triangular mesh i, we define a chart C i ⊂ R 2 as a circular disk. From the topology of the triangular mesh we have three charts overlapping

Subcharts face subcharts Charts C i edge subcharts Innermost part

i j k l Edge subchart Face subchart

Parameterization of subcharts edge subchart parameterizations Requirement: smooth, surjective, orientation preserving (regular) Remarks: E2 and E4 are mapped to the lower and upper boundaries of

Parameterization of subcharts face subchart parameterizations Requirement: smooth, surjective, orientation preserving (regular)

Transition functions and atlas

Transition functions and atlas cont. The transition function between C i and C j The triplet will be called the smooth parameterized atlas of the manifold, provided that all subchart parameterizations are valid.

Influence and geometry function. Definition: For any i ∈ V, consider a scalar–valued function : R 2 → R which satisfies the following three conditions: The geometry function is an embedding function for each chart

Spline manifold surface The i–th vertex patch face patch The collection of vertex, edge and face patches is said to be the blending manifold surface which is associated with the C s smooth parameterized atlas A and the geometry and influence functions. edge patch

Construction for rational blending manifolds

Given triangular mesh M m V vertices, m F oriented triangles. Normal vector for each vertex and tangent plane

is the bisectors of the arcs from is chosen as the point which divides the arc from by the ratio 1 : 5 is chosen as the point which divides the arc fromby the ratio 1 : 5.

Face subchart parameterization : planar rational Bézier triangle of degree two is equilateral triangle Choose as the intersection of the circle tangent at and Control points: The associated weights: We choose the face subchart parameterization as a planar rational Bézier triangle of degree two

Edge subcharts Parameterization Once the edge subchart parameterizations are known, we have two parameterizations of the overlapping regions. We choose the edge subchart parameterizations such that these two parameterizations of the overlapping regions are smooth. has to satisfy the following two conditions: It has a smooth joint with the tensor-product patches along its edges E 4 and E 2, respectively. Its boundary is contained in the boundary

Möbius transformation 1. A Möbius transformation is a special mapping of the plane into itself 2. Möbius transformations can map circles onto circles. 3. The inverse of a Möbius transformation is again a Möbius transformation. 4. A Möbius transformation is uniquely determined by prescribing three different images for three different points.

Construction Edge subchart parameterization is a a rational tensor product patch of degree (4, 4s + 2)

Example

Innermost part Geometry functions can be chosen as: t i is a linear parameterization of the tangent plane at the vertex i of the mesh n i is the normal vector and q i (u, v) is a quadratic polynomial. γis a shrinking factor which controls the size of the embedding of the chart. λis a flatness factor the flatness of the chart embedding. The parameters γ and λ control the distance between the manifold surface and the control mesh. In this case v(i) = 5 Bézier triangles are needed.

Using λ = 0.5, γ varying between 0.7 for valence 3 and 0.3 for valence 10 in the following examples. Influence function: s is the desired order of smoothness

Examples Double torus: the mesh has 284 faces and 140 vertices. The yellow, blue and red regions correspond to the vertex, edge and face patches, respectively. The surface was rendered using 17,890 triangles.

Comparison between adaptive (top row) and non-adaptive method (bottom row) for the double torus model

Various C 2 smooth surfaces which demonstrate the possibilities of the presented construction. The surfaces were rendered using 31,626, 6,048, and 9,072 triangles, respectively.

C 2 manifold surface obtained from a star shaped polyhedron, for different values of the shape parameters and controlling the geometry functions. The surface was rendered using 14,336 triangles.

Conclusion Construct a rational manifold surface with arbitrary order s of smoothness from a given triangular mesh. The given triangular mesh is used both to guide the geometry functions and to define the connectivity of the charts. The transition functions are obtained via subchart parameterizations. The manifold surface can be described as a collection of quadrangular and triangular (untrimmed) rational surface patches.

Contribution The use of manifold surfaces provides an explicit parameterization of the surface. It is possible to get any order of smoothness and there are no difficulties associated with “extraordinary” vertices. Constructions can generate surfaces from triangular meshes. The point-wise evaluation requires only rational operations. The subcharts and transition functions can be adapted to the geometry of the given triangular mesh.

Future works The complete algorithm (charts and geometry functions generation, embedding, blending and triangulation of the surfaces) is evaluated in approximately 14 sec per vertex. The optimal choice of charts (which do not need to be circular) and geometry functions. Address boundary conditions and sharp features (edges) that may be present in a given object. Investigate other manifold constructions which are based on subchart parameterization.

Thanks you