Transcript Document

Smooth Spline Surfaces over
Irregular Topology
Hui-xia Xu
Wednesday, Apr. 4, 2007
 an inability of coping with surfaces of irregular topology,
i.e., requiring the control meshes to form a regular
quadrilateral structure
Improved Methods
 To overcome this limitation, a number of
methods have been proposed. Roughly speaking,
these methods are categorized into two groups:
Subdivision surfaces
Spline surfaces
Subdivision Surfaces
Subdivision Surfaces
---main idea
polygon mesh
resultant mesh
converging to
smooth surface
Subdivision Surfaces
---magnum opus
Catmull-Clark surfaces
 E Catmull and J Clark. Recursively generated B-spline surfaces
on arbitrary topological meshes, Computer Aided Design
10(1978) 350-355.
Doo-Sabin surfaces
 D Doo and M Sabin. Behaviour of recursive division surfaces
near extraordinary points, Computer Aided Design 10 (1978)
About Subdivision Surfaces
simplicity and
The subdivision surfaces do not admit a
closed analytical expression
Spline Surfaces
Method 1
the technology of manifolds
 C Grimm and J Huges. Modeling surfaces of arbitrary topology
using manifolds, Proceedings of SIGGRAPH (1995) 359-368
 J Cotrina Navau and N Pla Garcia. Modeling surfaces from
meshes of arbitrary topology, Computer Aided Geometric Design
17(2000) 643-671
Method 2
isolate irregular points
 C Loop and T DeRose. Generalised B-spline surfaces of arbitrary
topology, Proceedings of SIGGRAPH (1990) 347-356
 J Peters. Biquartic C1-surface splines over irregular meshes, ComputerAided Design 12(1995) 895-903
 J J Zheng et al. Smooth spline surface generation over meshes of
irregular topology, Visual Computer(2005) 858-864
 J J Zheng et al. C2 continuous spline surfaces over Catmull-Clark
meshes, Lecture Notes in Computer Science 3482(2005) 1003-1012
 J J Zheng and J J Zhang. Interactive deformation of irregular surface
models, Lecture Notes in Computer Science 2330(2002) 239-248
 etc.
Smooth Spline Surface Generation
over Meshes of Irregular Topology
J J Zheng , J J Zhang, H J Zhou and L G Shen
Visual Computer 21(2005), 858-864
What to Do
 In this paper, an efficient method generates a
generalized bi-quadratic B-spline surface and
achieves C1 smoothness.
Zheng-Ball Patch
 A Zheng-Ball patch is a generation of a Sabin
patch that is valid for 3- or 5-sided areas. For
more details, the following can be referred:
 J J Zheng and A A Ball. Control point surface over non-four sided
areas, Computer Aided Geometric Design 14(1997)807-820.
 M A Sabin. Non-rectangular surfaces suitable for inclusion in a Bspline surface, Hagen, T. (ed.) Eurographics (1983) 57-69.
Zheng-Ball Patch
 An n-sided Zheng-Ball patch of degree m is
defined by the following :
This patch model is able to smoothly
blend the surrounding regular patches
Zheng-Ball Patch
: the n-ple subscripts,
:n parameters of which only two are
: denotes the control points in
: the associated basis functions
,as shown in Fig 1.
Fig 1. Control points for a six-sided quadratic Zheng-Ball patch
Spline Surface Generation
---irregular closed mesh
Generate a new refined mesh
carry out a single Catmull-Clark subdivision over the
user-defined irregular mesh
Construct a C1 smooth spline surface
regular vertex---a bi-quadratic Bézier patch
Otherwise---a quadratic Zheng-Ball patch
Related Terms
The valence of a point is the number of its incident
Regular vertex
If its valence is 4, the vertex is said to be regular.
Regular face
A face is said to be regular if none of its vertices are
irregular vertices.
Catmull-Clark Surfaces
---subdivision rules
 Generation of geometric points
 Construction of topology
Geometric Points
 new face points
 averaging of the surrounding vertices of the corresponding
 new edge points
 averaging of the two vertices on the corresponding edge and
the new face points on the two faces adjacent to the edge
 new vertex points
 averaging of the corresponding vertices and surrounding vertices
 connect each new face point to the new edge
points surrounding it
 Connect each new vertex point to the new edge
points surrounding it
Mesh Subdivision
Fig 2. Applying Catmull-Clark subdivision once to vertex V with valence n
Mesh Subdivision
 new faces: four-sided
 The valence of a new edge point is 4
 The valence of the new vertex point v remains n
 The valence of a new face point is the number
of edges of the corresponding face of the initial
Patch Generation
 For a regular vertex, a bi-quadratic Bézier patch
is used
 For an extraordinary vertex, an n-sided
quadratic Zheng-Ball patch will be generated
Fig 3. Two adjacent patches joined with C1 continuity
Geometric Model
Fig 4. Closed irregular mesh and the resulting geometric model
Spline Surface Generation
---irregular open mesh
 Step 1: subdividing the mesh to make all faces
 Step 2: constructing a
corresponding to each vertex
 The main task is to deal with the mesh
Subdivision Rules for
Mesh Boundaries
Boundary mesh subdivision for 2- and
3-valent vertices
 face point: Centroid of the i-th face incident to V
 edge point: averaging of the two endpoints in
the associated edge
 vertex point: equivalent to n-valent vertex V of
the initial mesh
Fig 5. Subdivision around a boundary vertex v (n=3)
Boundary mesh subdivision for
 For each vertex V of valence>3, n new vertices
Wi (i=1,2, …,n) are created by
Convex Boundary Vertex
Fig 6. Left: Convex boundary vertex V0 of valence 4.
Right: New boundary vertices V0 , W1 and W4 of valence 2 or 3
Concave Boundary Vertex
Fig 7. Left: Concave inner boundary vertex V of valence 4.
Right: New boundary vertices W1 and W4 of valence 3
Boundary Patches
Some Definitions
 Boundary vertex: vertex on the boundary of the new
 Boundary face: at lease one of its vertices is a boundary
 Intermediate vertex: not a boundary vertex, but at least
one of its surrounding faces is a boundary face
 Inner vertex: none of the faces surrounding is a
boundary face
Generation Rules
--- intermediate vertex
d is a central control point
d2i is a corner point if its valence is 2
d2i-1 is a mid-edge control point if its valence is 3
½*(di + di+1 ) is a corner control point if the valences of
di and di+1 are 3.
 ½*(d2i-1 + d) and ½*(d2i+1 + d) are the two mid-edge
control points if fi is not a boundary face.
 The centroid of face fi is a corner point if fi is not a
boundary face.
Generation Rules
--- intermediate vertex
Fig 8. Intermediate vertex d (valence 5). Control points (○) for
the patch corresponding to it
Geometric Model
Fig 9. Two models generated from open meshes by proposed method
Fig 10. Sphere produced with Loop’s method (left ) and with the
proposed method (right )
Interactive Deformation of
Irregular Surface Models
J J Zheng and J J Zhang
LNCS 2330(2002), 239-248
 Interactive deformation of surface models is an
important research topic in surface modeling.
 However, the presence of irregular surface
patches has posed a difficulty in surface
 Interactive deformation involves possibly the
following user-controlled deformation operations
moving control points of a patch
specifying geometric constraints for a patch
deforming a patch by exerting virtual forces
 By far the most difficult task is to all these
operations without violating their connection
Outline of the Proposed Research
This paper will concentrate on two issues
modeling of irregular surface patches
Zheng-Ball model
the connection between different patches
formulate an explicit formula to degree elevation
and to insert a necessary number of extra control
Zheng-Ball Patch
 This patch model can have any number of sides
and is able to smoothly blend the surrounding
regular patches
 This surface model is control-point based and to
a large extent similar to Bézier surfaces
Zheng-Ball Patch
Fig 11. 3-sided cubic Zheng-Ball Patch with its control points
Explicit Formula of Degree Elevation
Explicit Formula of Degree Elevation
 The functions
are defined by
 The functions
are defined by
After Degree Elevation
Fig 12. Quartic patches with control points after degree elevation. The
circles represent the control points contributing to the C0 condition, the
black dots represent the control points contributing G1 condition, and the
square in the middle represents the free central control point
Central Control Point
 The central control point has provided an extra
degree of freedom.
 Moving this control point will deform the shape
of the blending patch intuitively, without
violating the continuity conditions
Energy function
 For an arbitrary patch
defined by :
, an energy function is
where Vi, Ki and Fi are the control point vector,
stiffness matrix and force vector, respectively.
Global Energy Function
 The new global energy functional is given by
Deformation Function
 The continuity constraints are defined by the
following linear matrix equation:
Minimising the global energy function subject to the
continuity constraints leads to the production of a
deformed model consisting of both regular and
irregular patches !
 Typical G1 continuity constraints for the two patches
can be expressed by the following:
Fig 13. Two cubic patches share a common boundary
Fig 14. Model with 3- and 5-sided patches (green patches). (Middle and
Right) Deformed models. There are eight triangular patches on the outer
corners of the model, and eight pentagonal patches on the inner corners
of the model.
Algorithm for Interactive Deforming
 If physical forces are applied to the surface, the
following linear system is generated by minimising the
quadratic form
 Subjuect to linear constraints
Algorithm for Interactive Deforming
Fig 16. Algorithm if interactive deformation
Algorithm for Interactive Deforming
 l>k. There are free variable left in linear
constraints. So linear system can be solved.
 l<=k. There is no free variable left in linear
constraints. So linear system is not solvable.
 In the latter case, extra degrees of freedom are
needed to solve linear system.
Smooth Models
Fig 17. A smooth model with 3- and 5-sided cubic surface
patches (left). Deformed model after twice degree elevation
(right). Arrows indicate the forces applied on the surface points.
 Proposed a surface deformation technique
 no assumption is made for the degrees of freedom
 all surface patches can be deformed in the unified form
 during deformation process, the smoothness conditions between
patches will be maintained
 Derived an explicit formula for degree elevation
of irregular patches
Thank you!