a construction of rational manifold surfaces of arbitrary topology and smoothness from triangular...
TRANSCRIPT
A construction of rational manifold surfaces of arbitrary topology and
smoothness from triangular meshes
Presented by: LiuGang
2008-12-11
Authors• Giovanni Della Vecch
ia
• research assistant
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-formsurface 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). C2 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 Uij to be the part of chart i that overlaps with chart j. May be empty.
Transition function yij maps from Uij to Uji.
m
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 chartCollection of charts is called an atlas
Note: A surface is manifold if such an atlas can be constructed
n
ij
1i j
1j i
iU jUi jU U
Manifoldcont.
• 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 R3
mV is the number of vertices;
mF is the number of faces;
mE 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
Ci⊂R2 as a circular disk.
• From the topology of the triangular mesh we have three charts overlapping
Subcharts
• face subcharts
• Charts Ci
• edge subcharts
• Innermost part
i
j
k
lEdge 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
l
k
i j
l
k
i j
Transition functions and atlascont.
• The transition function between Ci and Cj
{ | 1,2,..., }iVC C i m
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 : R2 → R which satisfies the following three conditions:
i
2
( ) ,
( ) ( ) 0,
( ) ( ) 0, \
i s
i i
i i
i C
ii x x C
iii x if x R C
连续
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 blendingmanifold surface which is associated with the Cs smooth parameterized atlas A and the geometry and influence functions.
• edge patch
Construction for rational blending manifolds
• Given triangular mesh M
• mV vertices, mF 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 from by the ratio 1 : 5.
Face subchart parameterization
• : planar rational Bézier triangle of degree two
200 020 002 1
is equilateral triangle
Choose as the intersection of the circle tangent at and
Control points: The associated weights:
011 101 1
1110 020 2002cos ( ,0, )b b
We choose the face subchart parameterizationas 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 E4 and E2, 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 arational tensor product patch of degree (4, 4s + 2)
Example
Innermost part
Geometry functions can be chosen as:
ti is a linear parameterization of the tangent plane at the vertex i of the mesh
ni is the normal vector and qi(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 andred 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 C2 smooth surfaces which demonstrate the possibilities of the presented construction. The surfaces were rendered using 31,626, 6,048, and 9,072 triangles, respectively.
•C2 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