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Journal of Computing and Information Science in Engineering | Volume 8 | Issue 2 | Research Paper
Research Papers

Flattenable Mesh Surface Fitting on Boundary Curves

[+] Author and Article Information
Charlie C. Wang

Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, New Territory, Hong Kongcwang@mae.chuk.edu.hk

J. Comput. Inf. Sci. Eng 8(2), 021006 (Apr 30, 2008) (10 pages) doi:10.1115/1.2906695 History: Received February 08, 2007; Revised September 12, 2007; Published April 30, 2008
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This paper addresses the problem of fitting flattenable mesh surfaces in R3onto piecewise linear boundary curves, where a flattenable mesh surface inherits the isometric mapping to a planar region in R2. The developable surface in differential geometry shows the nice property. However, it is difficult to fit developable surfaces to a boundary with complex shape. The technique presented in this paper can model a piecewise linear flattenable surface that interpolates the given boundary curve and approximates the cross-tangent normal vectors on the boundary. At first, an optimal planar polygonal region is computed from the given boundary curve B∊R3, triangulated into a planar mesh surface, and warped into a mesh surface in R3, satisfying the continuities defined on B. Then, the fitted mesh surface is further optimized into a flattenable Laplacian (FL) mesh, which preserves the positional continuity and minimizes the variation of cross-tangential normals. Assembled set of such FL mesh patches can be employed to model complex products fabricated from sheets without stretching.
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Copyright © 2008 by American Society of Mechanical Engineers
Topics: Surface fitting
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References

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Figures

Grahic Jump Location
+Illustration for vertex flattenability—the inner angles before and after flattening the triangles around a vertex are identical
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Illustration for vertex flattenability—the inner angles before and after flattening the triangles around a vertex are identical
Figure 1

Illustration for vertex flattenability—the inner angles before and after flattening the triangles around a vertex are identical

Grahic Jump Location
+Method overview: (a) the given boundary curve (blue knots are endpoints of line segments) and the normal vectors (red arrows) defined on segments, (b) the computed optimal planar boundary and tessellated planar mesh patch, (c) the warped mesh surface from the planar mesh as the initial fitting patch (where the yellow arrows represent the normal vectors on the boundary of the warped patch), (d) the color map of flattenablity at vertices of the initial fitting patch, (e) the final flattenable mesh computed from (c), and (f) the corresponding flattenability map of (e)
View Large  |  Save Figure  |  Download Slide (.ppt)
Method overview: (a) the given boundary curve (blue knots are endpoints of line segments) and the normal vectors (red arrows) defined on segments, (b) the computed optimal planar boundary and tessellated planar mesh patch, (c) the warped mesh surface from the planar mesh as the initial fitting patch (where the yellow arrows represent the normal vectors on the boundary of the warped patch), (d) the color map of flattenablity at vertices of the initial fitting patch, (e) the final flattenable mesh computed from (c), and (f) the corresponding flattenability map of (e)
Figure 2

Method overview: (a) the given boundary curve (blue knots are endpoints of line segments) and the normal vectors (red arrows) defined on segments, (b) the computed optimal planar boundary and tessellated planar mesh patch, (c) the warped mesh surface from the planar mesh as the initial fitting patch (where the yellow arrows represent the normal vectors on the boundary of the warped patch), (d) the color map of flattenablity at vertices of the initial fitting patch, (e) the final flattenable mesh computed from (c), and (f) the corresponding flattenability map of (e)

Grahic Jump Location
+Angles on a boundary vertex vb: (a) the value of α(vb) can be predicted by the projected angle on the plane defined by vb and the normal vector 12(ne++ne−), and (b) the value of an optimal θ(vb)∊R2 has a lower bound as been addressed in Lemma 1
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Angles on a boundary vertex vb: (a) the value of α(vb) can be predicted by the projected angle on the plane defined by vb and the normal vector 12(ne++ne−), and (b) the value of an optimal θ(vb)∊R2 has a lower bound as been addressed in Lemma 1
Figure 3

Angles on a boundary vertex vb: (a) the value of α(vb) can be predicted by the projected angle on the plane defined by vb and the normal vector 12(ne++ne−), and (b) the value of an optimal θ(vb)∊R2 has a lower bound as been addressed in Lemma 1

Grahic Jump Location
+The wrinkle patch example: (a) the given boundary curve and the cross-tangent normals defined on it, (b) the initial warped patch M0 and its color map for flattenability, and (c) the result FL mesh surface patch its the color map
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The wrinkle patch example: (a) the given boundary curve and the cross-tangent normals defined on it, (b) the initial warped patch M0 and its color map for flattenability, and (c) the result FL mesh surface patch its the color map
Figure 8

The wrinkle patch example: (a) the given boundary curve and the cross-tangent normals defined on it, (b) the initial warped patch M0 and its color map for flattenability, and (c) the result FL mesh surface patch its the color map

Grahic Jump Location
+The hole filling example: (a) the given moai model with a large hole at the back of the model, (b) the 2D mesh patch computed from the hole boundary, (c) the result FL mesh surface patch for filling, and (d) the color map for flattenability on the result FL mesh
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The hole filling example: (a) the given moai model with a large hole at the back of the model, (b) the 2D mesh patch computed from the hole boundary, (c) the result FL mesh surface patch for filling, and (d) the color map for flattenability on the result FL mesh
Figure 9

The hole filling example: (a) the given moai model with a large hole at the back of the model, (b) the 2D mesh patch computed from the hole boundary, (c) the result FL mesh surface patch for filling, and (d) the color map for flattenability on the result FL mesh

Grahic Jump Location
+An example for the flattenable surface modeling in apparel industry: (a) the curve network for fitting flattenable mesh surfaces—the normals of cross-tangent surfaces are defined on the curves (see the yellow arrows), (b) the mesh surface after initial fitting and its flattenability map, and (c) the final FL mesh surface patches fitting the given curve network
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An example for the flattenable surface modeling in apparel industry: (a) the curve network for fitting flattenable mesh surfaces—the normals of cross-tangent surfaces are defined on the curves (see the yellow arrows), (b) the mesh surface after initial fitting and its flattenability map, and (c) the final FL mesh surface patches fitting the given curve network
Figure 10

An example for the flattenable surface modeling in apparel industry: (a) the curve network for fitting flattenable mesh surfaces—the normals of cross-tangent surfaces are defined on the curves (see the yellow arrows), (b) the mesh surface after initial fitting and its flattenability map, and (c) the final FL mesh surface patches fitting the given curve network

Grahic Jump Location
+An example for modeling a shoe: (a) the input curve network with normal vectors, (b) the surface after initial fitting and the color map for illustrating vertex flattenability, and (c) the final FL mesh surface patches (unexpected wrinkles are generated at the region near arch)
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An example for modeling a shoe: (a) the input curve network with normal vectors, (b) the surface after initial fitting and the color map for illustrating vertex flattenability, and (c) the final FL mesh surface patches (unexpected wrinkles are generated at the region near arch)
Figure 11

An example for modeling a shoe: (a) the input curve network with normal vectors, (b) the surface after initial fitting and the color map for illustrating vertex flattenability, and (c) the final FL mesh surface patches (unexpected wrinkles are generated at the region near arch)

Grahic Jump Location
+A refined design for the shoe cover: (a) the input curve networks with normal vectors, (b) the surface after RBF warping (i.e., the initial fitting result), and (c) the final FL mesh patches that are automatically constructed by our approach
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A refined design for the shoe cover: (a) the input curve networks with normal vectors, (b) the surface after RBF warping (i.e., the initial fitting result), and (c) the final FL mesh patches that are automatically constructed by our approach
Figure 12

A refined design for the shoe cover: (a) the input curve networks with normal vectors, (b) the surface after RBF warping (i.e., the initial fitting result), and (c) the final FL mesh patches that are automatically constructed by our approach

Grahic Jump Location
+The 2D layout of leather patterns for making the shoe shown in the left, where the patterns are generated by Ref. 16 from the patches shown in Fig. 1
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The 2D layout of leather patterns for making the shoe shown in the left, where the patterns are generated by Ref. 16 from the patches shown in Fig. 1
Figure 13

The 2D layout of leather patterns for making the shoe shown in the left, where the patterns are generated by Ref. 16 from the patches shown in Fig. 1

Grahic Jump Location
+Discussion: (a) the computed optimal planar boundary may globally self-intersected (top) although the local self-intersection (bottom) has been prevented and (b) tangential continuity cannot always be preserved on the resultant patches (comparing normal vectors on the boundary curves at heel and other regions)
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Discussion: (a) the computed optimal planar boundary may globally self-intersected (top) although the local self-intersection (bottom) has been prevented and (b) tangential continuity cannot always be preserved on the resultant patches (comparing normal vectors on the boundary curves at heel and other regions)
Figure 14

Discussion: (a) the computed optimal planar boundary may globally self-intersected (top) although the local self-intersection (bottom) has been prevented and (b) tangential continuity cannot always be preserved on the resultant patches (comparing normal vectors on the boundary curves at heel and other regions)

Grahic Jump Location
+The closed-path constraint and the position coincident constraint on the planar boundary
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The closed-path constraint and the position coincident constraint on the planar boundary
Figure 4

The closed-path constraint and the position coincident constraint on the planar boundary

Grahic Jump Location
+The steps for computing an initial patch fitting to the given boundary: (a) the given boundary and normal vectors for cross-tangent planes, (b) the computed optimal planar boundary, (c) the inner region is filled with regular triangles, (d) the boundary region is tessellated by the CDT (29) and the inner vertices are repositioned through an area-based smoothing, (e) the planar mesh is warped to interpolate the given boundary and cross-tangent planes, and (f) the color map of flattenability at vertices of the warped patch
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The steps for computing an initial patch fitting to the given boundary: (a) the given boundary and normal vectors for cross-tangent planes, (b) the computed optimal planar boundary, (c) the inner region is filled with regular triangles, (d) the boundary region is tessellated by the CDT (29) and the inner vertices are repositioned through an area-based smoothing, (e) the planar mesh is warped to interpolate the given boundary and cross-tangent planes, and (f) the color map of flattenability at vertices of the warped patch
Figure 5

The steps for computing an initial patch fitting to the given boundary: (a) the given boundary and normal vectors for cross-tangent planes, (b) the computed optimal planar boundary, (c) the inner region is filled with regular triangles, (d) the boundary region is tessellated by the CDT (29) and the inner vertices are repositioned through an area-based smoothing, (e) the planar mesh is warped to interpolate the given boundary and cross-tangent planes, and (f) the color map of flattenability at vertices of the warped patch

Grahic Jump Location
+Different constraints on the boundary yield different surfaces by the RBF-based warping: (a) only the positions and the normal vectors are constrained, and (b) all the positions, the normal vectors, and the cross-tangents are constrained
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Different constraints on the boundary yield different surfaces by the RBF-based warping: (a) only the positions and the normal vectors are constrained, and (b) all the positions, the normal vectors, and the cross-tangents are constrained
Figure 6

Different constraints on the boundary yield different surfaces by the RBF-based warping: (a) only the positions and the normal vectors are constrained, and (b) all the positions, the normal vectors, and the cross-tangents are constrained

Grahic Jump Location
+The comparison of (a) direct Newton’s routine versus (b) our multiple-loop algorithm for computing a FL mesh surface from an input of Fig. 5, where the results after 200, 400, and 800 iterations of direct Newton’s routine are shown in (a) and the results after 5, 10, and 20 outer loop iterations (i.e., accumulated 25, 50, and 100 steps) are shown in (b)
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The comparison of (a) direct Newton’s routine versus (b) our multiple-loop algorithm for computing a FL mesh surface from an input of Fig. 5, where the results after 200, 400, and 800 iterations of direct Newton’s routine are shown in (a) and the results after 5, 10, and 20 outer loop iterations (i.e., accumulated 25, 50, and 100 steps) are shown in (b)
Figure 7

The comparison of (a) direct Newton’s routine versus (b) our multiple-loop algorithm for computing a FL mesh surface from an input of Fig. 5, where the results after 200, 400, and 800 iterations of direct Newton’s routine are shown in (a) and the results after 5, 10, and 20 outer loop iterations (i.e., accumulated 25, 50, and 100 steps) are shown in (b)

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