0
TECHNICAL PAPERS

Planar Parameterization for Closed Manifold Genus-g Meshes Using Any Type of Positive Weights

[+] Author and Article Information
D. Steiner

 Laboratory for CAD and Lifestyle Engineering, Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, Israel 32000ovir@tx.technion.ac.il

A. Fischer

 Laboratory for CAD and Lifestyle Engineering, Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, Israel 32000meranath@tx.technion.ac.il

J. Comput. Inf. Sci. Eng 5(2), 118-125 (Feb 04, 2005) (8 pages) doi:10.1115/1.1884132 History: Received August 17, 2004; Revised February 04, 2005

Parameterization of 3D meshes is important for many graphic and CAD applications, in particular for texture mapping, remeshing, and morphing. Current parameterization methods for closed manifold genus-g meshes usually involve cutting the mesh according to the object generators, adjusting the resulting boundary and then determining the 2D parameterization coordinates of the mesh vertices, such that the flattened triangles are not too distorted and do not overlap. Unfortunately, adjusting the boundary distorts the resulting parameterization, especially near the boundary. To overcome this problem for genus-g meshes we first address the special case of closed manifold genus-1 meshes by presenting cyclic boundary constraints. Then, we expand the idea of cyclic boundary constraints by presenting a new generalized method developed for planar parameterization of closed manifold genus-g meshes. A planar parameterization is constructed by exploiting the topological structure of the mesh. This planar parameterization can be represented by a surface which is defined over parallel g-planes that represents g-holes. The proposed parameterization method satisfies the nonoverlapping requirement for any type of positive barycentric weights, including asymmetric weights. Moreover, convergence is guaranteed according to the Gauss-Seidel method.

FIGURES IN THIS ARTICLE
<>
Copyright © 2005 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Constructing the second generator when merge occurs: (a) visited region (in eggplant purple) grows until merge occurs, (b) Dijkstra color map over the visited region

Grahic Jump Location
Figure 2

Illustration of Eq. 3. Vertices on the longitude are marked by stars and longitude-right-neighbors are marked by circles.

Grahic Jump Location
Figure 3

Edge states: (a) The boundary of the fan around vi, (b) penetrating a convex-hull edge, (c) penetrating an edge with concave vertex, (d) a boundary of a fan with convex region which is not on the convex-hull, (e) penetrating an edge of a convex region which is not on the convex-hull

Grahic Jump Location
Figure 4

Fixed vertex, overload state: (a) vfix on a concave region of the fan boundary, (b) penetrating an edge belongs to vfix, (c) subdivision of vfix fan

Grahic Jump Location
Figure 5

Torus parameterization: (a) torus and its two generators, (b) texture mapping using harmonic weights with fixed boundary, (c) parameterization space when using fixed boundary, (d) texture mapping using harmonic weights with cyclic boundary, (e) parameterization space when using cyclic boundary

Grahic Jump Location
Figure 6

Loop: (a) loop with its two generators, (b) loop, zooming on a problematic area with obtuse angles, (c) texture mapping using harmonic weights, (d) texture mapping using mean-value weights, (e) parameterization space using harmonic weights, (f) parameterization space using mean-value weights

Grahic Jump Location
Figure 7

Genus-2 object: (a) 8 generators left side (yellow), meridian right side (red), longitude right side (green) and singular points area (purple), (b) oriented generators above one another, crossing vertex (red) (c) zoomed crossing area

Grahic Jump Location
Figure 8

Flipping between meridian and longitude: (a) resulting twisted parameterization, (b) and (c) zoomed twisted area

Grahic Jump Location
Figure 9

Crossing point states: (a) figure-eight shape, its generators and connecting path, (b) outer circle: mapping to a plane and boundary directed edges, inner circle: stitching directions, (c) folding one-boundary surface into a cube without upper and lower face, (d) meridian to longitude crossing states, (e) flipping around meridian, (f) flipping around longitude, (g) longitude to meridian states, (h) flipping around diagonal, (i) flipping around diagonal

Grahic Jump Location
Figure 10

Four circular torus: (a) object with its generators, left (yellow) and right (green and red) sides marked, (b) the resulting texture when using one cut (cut-graph method), (c) resulting texture when using our method

Grahic Jump Location
Figure 11

Bagel with three holes: (a) the object with generators and singular points (purple), (b) the parameterization surface lifted (stretched in the Z direction for viewing purpose), (c) the resulting texture mapping

Grahic Jump Location
Figure 12

Four toruslike vases attached to a torus: (a) the object with generators and singular points (purple), (b) the parameterization surface lifted (stretched in the Z direction for viewing purpose), (c) the resulting texture mapping

Grahic Jump Location
Figure 13

Genus-7 object parameterization: the resulting texture mapping

Grahic Jump Location
Figure 14

Genus-3 object parameterization: (a) the original object, (b) the resulting texture mapping

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In