Surfaces

Reporter: Gang XuMay 30, 2006

Aug 1, 2006, 8:30-10:15 am

Session Chair: Leif.P Kobbelt.

Surfaces Session

Papers Real-Time GPU Rendering of Piecewise Algebraic Surfaces

Charles Loop, Jim Blinn (Microsoft Research) Point-Sampled Cell Complexes

Anders Adamson (Technische Universität Darmstadt), Marc Alexa (Technische Universität Berlin) Geometric Modeling with Conical Meshes and Developable Surfaces

Yang Liu (University of Hong Kong), Helmut Pottmann, Johannes Wallner (Technische Universität Wien ), Wenping Wang (University of Hong Kong), Yong-Liang Yang (Tsinghua University) Mesh Quilting For Geometric Texture Synthesis

Kun Zhou, Xin Huang, Xi Wang (Microsoft Research Asia), Yiying Tong, Mathieu Desbrun (California Institute of Technology), Baining Guo, Heung-Yeung Shum (Microsoft Research Asia)

Paper One Real-Time GPU Rendering of Piecewise Algebraic Surfaces Direct rendering of Bézier tetrahedra by specially encoding coefficient data on vertices and rasterizing faces. A pixel shader then robustly solves the (up to fourth-order) equations using analytic techniques.

Paper four Mesh Quilting for Geometric Texture Synthesis A novel geometry texture synthesis algorithm, called mesh quilting, that seamlessly synthesizes a 3D texture sample given in the form of a triangle mesh over arbitrary surfaces.

Geometric modeling with conical meshes and developable surface

Liu Yang, H.Pottman, J. Wallner, Wang Wen-Ping, Yang Yong-Liang

Application Background(1) Requirement of architectural design with glass structures

Application Background(2)

Conical Meshes

Previous Work (1) Discrete differential geometry Course in Siggraph 2005

Previous Work (2) PQ meshes: quad meshes with planar faces, discrete counterpart of conjugate curve net works on surfaces (Sauer,1970)

Previous work(3)Circular meshes: quad meshes that all quads possess a circumcircle, discrete counterpart of the network of principal curvature lines (Martin et al, 1986)

Quad mesh Computation of quad-dominant meshes from smoothed principal curvature lines (Alliez et al, 03) Variational shape approximation (Cohen, 04)

Previous work(4)

Previous work(5) Developable surface Arrangement of n planar quads in a single row.

Surface for architecture and

aesthetic design (Sequin, 2005)

Previous work(6)

Contribution Introduce conical meshes and demonstrate their superiority PQ perturbation algorithm How to produce conical meshes

and circular meshes?

PQ meshes(1)

1,i j

2,i j

4,i j

3,i j

1 2 3 4, , , , , , 2 0pq i j i j i j i j i jC

PQ perturbation: optimization problem with constraints

, , , , , , , ,,

, ( ), ( ) Tpq pq pq i j pq i j pq pq i j pq pq i j

i j

C C C C

PQ meshes(2) A thin planar quad which converges to a straight line segment, above constraints serve to maintain con

vexity but they can’t express the planarity.

PQ meshes(3) Two energy terms:

Fair shape:

Close to the original mesh:

Objective function

Small or medium case: Sequential Quadratic Programming (SQP)

Large case: penalty method

PQ mesh(4)

PQ mesh(5)

Subdividing PQ meshes

Generate a PQ mesh from a coarse mesh : combine the PQ perturbation

algorithm with a DS or CC algorithm

Subdivision developable surfaces

Applying a curve subdivision rule to its boundaries

Subsequent application of PQ perturbation

Conical Meshes(1) Conical vertex: all the four planes

meeting at the vertex are tangent to a common sphere (or cone of revolution)

Conical Meshes(2) Conical mesh: all of the its vertex of

valance four are conical. Conical mesh is the discrete

counterpart of the system of principal curvature

lines.

Conical Meshes(3) Types of conical mesh vertexes elliptic parabolic

hyperbolic

Conical Meshes(4) An angle criterion for conical meshes A vertex of a quad mesh is a conical

vertex if and only if the angle balance is satisfied.

1 3 2 4

Conical Meshes(5) Offsetting conical meshes Offsetting all faces by a fixed distance

leads again to a mesh with the same connectivity

Conical Meshes(6) The offsetting property is a characterizing property of conical meshes. General Laguerre transformations map conical meshes to conical meshes Mobius transformations map circular meshes to circular meshes

Conical Meshes(7)Neighboring axes(discrete surface normals)are co-planar, they are contained in a planeorthogonal to the mesh in a discrete sense.

Conical Meshes(8) An edge of the mesh, the discrete normals at its endpoints, and the corresponding edge of any offset mesh lie on a common plane.

Conical Meshes(9) Successive discrete normals of a conical mesh along a row or column are co-planar and therefore form a discrete developable surface.

If a subdivision process, which preserves the conical property, refines a conical mesh and in the limit produces a curve network on a surface, then this limit curve network is the network of principal curvature lines .

Conical Meshes(9)

Conical Meshes(10) If there are meshes which are both circular and conical?

1 3 2 4, , , , 0i j i j i j i j

Computing Conical/Circular Meshes

Similar with the PQ perturbation algorithm.

1 3 2 4, , , ,0, 0 i j i j i j i j

Similar with the PQ perturbation algorithm.

Results(1)

Results(2)

Future work in the paper Computation of special discrete

surfacesin a principal mesh and especially

conical mesh representation.

Other future work(1) There are many surfaces that their parameter curves are their lines of curvatures, such as surface of revolution, Enneper’s minimal surfaces.

Other future work(2) Change the constraint condition with angles to the constraint condition with edges. 托勒密定理：設四邊形 ABCD 內接於圓，則有AB×CD + AD×BC=AB×BD 。 註：在凸四邊形 ABCD 中，有 AB×CD + AD×BC ≧AB×BD 。等號成立的充要條件是 ABCD為圓內接四邊形。

Other future work(3)

Can we use the PQ perturbation algorithm and subdivision algorithm to modeling other special surfaces?

Architectural design using minimal surface

Point-Sampled Cell Complexes

Anders Adamson Marc AlexaTU Darmstadt TU Berlin

Surface modeling method NURBS (T-Spline): parameter domain Subdivision surface: each face PDE surface FFD method Physics-based method

Contribution Provide a versatile shape

representation for modeling that allows

prescribing features exactly

Point-sampled surface Defining a point-sampled surface

is difficult.

It is usually defined as the level set of

a smooth function.

Definition of the Point surface(1)

Wendland’s radial functionsDefinition of the Point surface(2)

Projection procedure:

Definition of the Point surface(3) The point set surface is defined as the set of stationary points under projection

Cell complex( 胞腔复形 )(1) 流形结构 : N 维拓扑形体只能由 (N-1) 维拓扑形体组成 . 非流形结构 :N 维拓扑形体不仅由有限个(N-1) 维拓扑形体 , 而且还包含 (N-2),(N-3), …., 0 维形体的信息 .

Cell complex( 胞腔复形 )(2)

Cell complex( 胞腔复形 )(3) In a cell complex, each surface patch should be open and attached to curve segments forming a boundary; and each curve segment would have endpoints as boundary. (homotopy) Relaxed understanding: closed curves and surfaces

Cell complex( 胞腔复形 )(4)

Cell complex( 胞腔复形 )(5)

Cell complex( 胞腔复形 )(6) The surface defined by this complex

will be defined as the set of the stationary

points of a projection operator ,

How to construct the projection operator ?

( )P x

{ ( ) }x P x x

Construction of projection operator

Define the geometry of individual cells

Extend the corresponding projection operator to respect the connectivity structure of the complex.

Defining sub-manifolds (1)

Point set surface: a hyper-surface of dimension (n-1). ( a vector-valued function that defines approximate normals of this hyper-surface) Definition of the Point surface(3) In this paper, the author uses the tangent frames.

Defining sub-manifolds (2)

For each points in the support of d-cell M, define the tangent space . Let be a matrix representation of . ( 曲面 S 上所有过 P 点的曲线的切向量构成一个二维的线性空间 ) is the eigenvectors of C that correspond to large eigenvalues.

xxT M

0 1 1( , , , )dT t t t

xT

it

Defining sub-manifolds (3) The sub-manifolds is the set of points x whose locally weighted centroid is contained in the tangent space approximated at x.

Defining sub-manifolds (4)

The projection operator

Result

Projecting onto the complex (1)

The process.

Projecting onto the complex (2) Interpolating attached cells of

lower dimension

Constraint

Application

Comparison Definition domain: NURBS (T-Spline): parameter domain Subdivision surface: each face of control mesh Point-sample Cell complex: sampled points Gluing together surface patches: a cell is always glued to a bounding cell of lower dimension Interactive modeling: Only change the locations of sampled points or other attached information

Adaptive sampling

Application: trees, blood vessels,…….

Future work

Thanks!