Polynomial surfaces interpolating arbitrary triangulations

Stefanie Hahmann; Georges-Pierre Bonneau

doi:10.1109/TVCG.2003.1175100

Journal Articles IEEE Transactions on Visualization and Computer Graphics Year : 2003

Polynomial surfaces interpolating arbitrary triangulations

(1) , (2)

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Stefanie Hahmann

Function : Correspondent author
PersonId : 177946
IdHAL : stefanie-hahmann
ORCID : 0000-0001-8623-7225
IdRef : 056534647

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Laboratoire de Modélisation et Calcul

Georges-Pierre Bonneau

Function : Author
PersonId : 17497
IdHAL : georges-pierre-bonneau
ORCID : 0000-0002-2872-3580
IdRef : 061406945

Models, Algorithms and Geometry for Computer Generated Image Graphics

Abstract

Triangular Bezier patches are an important tool for defining smooth surfaces over arbitrary triangular meshes. The previously introduced 4-split method interpolates the vertices of a 2-manifold triangle mesh by a set of tangent plane continuous triangular Bezier patches of degree five. The resulting surface has an explicit closed form representation and is defined locally. In this paper, we introduce a new method for visually smooth interpolation of arbitrary triangle meshes based on a regular 4-split of the domain triangles. Ensuring tangent plane continuity of the surface is not enough for producing an overall fair shape. Interpolation of irregular control-polygons, be that in 1D or in 2D, often yields unwanted undulations. Note that this undulation problem is not particular to parametric interpolation, but also occur with interpolatory subdivision surfaces. Our new method avoids unwanted undulations by relaxing the constraint of the first derivatives at the input mesh vertices: the tangent directions of the boundary curves at the mesh vertices are now completely free. Irregular triangulations can be handled much better in the sense that unwanted undulations due to flat triangles in the mesh are now avoided.

Keywords

Triangulation irregular 3D meshes arbitrary topology modeling surfaces triangular patches piecewise polynomial patches interpolation arbitrary tangent vectors reconstruction

Domains

Computational Geometry [cs.CG] Graphics [cs.GR]

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HB03-tvcg.pdf (3.31 Mo)

zoom_2b_opt.jpg (25.72 Ko)

sph1.jpg (25.2 Ko)

sph2.jpg (15.8 Ko)

sph_gfd_C.jpg (18.37 Ko)

sph_opt_C.jpg (19.38 Ko)

tore_b.jpg (15.82 Ko)

tore_c.jpg (24.04 Ko)

val_creux_C.jpg (23.89 Ko)

val_round_C.jpg (22.58 Ko)

zoom_2b_gfd.jpg (22.64 Ko)

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https://hal.science/hal-00319652

Submitted on : Tuesday, September 9, 2008-2:50:58 PM

Last modification on : Thursday, April 4, 2024-9:28:35 PM

Long-term archiving on: Monday, October 8, 2012-1:01:51 PM

Dates and versions

hal-00319652 , version 1 (09-09-2008)

Identifiers

HAL Id : hal-00319652 , version 1
DOI : 10.1109/TVCG.2003.1175100

Cite

Stefanie Hahmann, Georges-Pierre Bonneau. Polynomial surfaces interpolating arbitrary triangulations. IEEE Transactions on Visualization and Computer Graphics, 2003, 9 (1), pp.99-109. ⟨10.1109/TVCG.2003.1175100⟩. ⟨hal-00319652⟩

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Polynomial surfaces interpolating arbitrary triangulations

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