On approximating contours of the piecewise trilinear interpolant using triangular rationalquadratic bézier patches

Bernd Hamann; Issac J. Trotts; Gerald E. Farin

doi:10.1109/2945.620489

On approximating contours of the piecewise trilinear interpolant using triangular rationalquadratic bézier patches

Bernd Hamann, Issac J. Trotts, Gerald E. Farin

Research output: Contribution to journal › Article › peer-review

19 Scopus citations

Abstract

Given a three-dimensional (3D) array of function values F_i,j,k on a rectilinear grid, the marching cubes (MC) method is the most common technique used for computing a surface triangulation T approximating a contour (isosurface) F(x, y, z) = T. We describe the construction of a C⁰ -continuous surface consisting of rational-quadratic surface patches interpolating the triangles in T. We determine the Bézier control points of a single rational-quadratic surface patch based on the coordinates of the vertices of the underlying triangle and the gradients and Hessians associated with the vertices.

Original language	English (US)
Pages (from-to)	215-227
Number of pages	13
Journal	IEEE Transactions on Visualization and Computer Graphics
Volume	3
Issue number	3
DOIs	https://doi.org/10.1109/2945.620489
State	Published - 1997
Externally published	Yes

Keywords

Approximation
Contour
Isosurface
Marching cubes
Rational bézier curve
Rational bézier surface
Triangular patch
Triangulation
Trilinear interpolation
Visualization

ASJC Scopus subject areas

Software
Signal Processing
Computer Vision and Pattern Recognition
Computer Graphics and Computer-Aided Design

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10.1109/2945.620489

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title = "On approximating contours of the piecewise trilinear interpolant using triangular rationalquadratic b{\'e}zier patches",

abstract = "Given a three-dimensional (3D) array of function values Fi,j,k on a rectilinear grid, the marching cubes (MC) method is the most common technique used for computing a surface triangulation T approximating a contour (isosurface) F(x, y, z) = T. We describe the construction of a C0 -continuous surface consisting of rational-quadratic surface patches interpolating the triangles in T. We determine the B{\'e}zier control points of a single rational-quadratic surface patch based on the coordinates of the vertices of the underlying triangle and the gradients and Hessians associated with the vertices.",

keywords = "Approximation, Contour, Isosurface, Marching cubes, Rational b{\'e}zier curve, Rational b{\'e}zier surface, Triangular patch, Triangulation, Trilinear interpolation, Visualization",

author = "Bernd Hamann and Trotts, {Issac J.} and Farin, {Gerald E.}",

note = "Funding Information: This work was supported by the U.S. National Science Foundation under contracts ASC-9210439 (Research Initiation Award) and ASC 9624034 (CAREER Award) to Mississippi State University and the University of California, Davis. In addition, this project was funded by the U.S. Office of Naval Research under contract N00014-97-1-0222 and the U.S. Army Research Office under contract ARO 36598-MA-RIP to the University of California, Davis.",

year = "1997",

doi = "10.1109/2945.620489",

language = "English (US)",

volume = "3",

pages = "215--227",

journal = "IEEE Transactions on Visualization and Computer Graphics",

issn = "1077-2626",

publisher = "IEEE Computer Society",

number = "3",

}

TY - JOUR

T1 - On approximating contours of the piecewise trilinear interpolant using triangular rationalquadratic bézier patches

AU - Hamann, Bernd

AU - Trotts, Issac J.

AU - Farin, Gerald E.

N1 - Funding Information: This work was supported by the U.S. National Science Foundation under contracts ASC-9210439 (Research Initiation Award) and ASC 9624034 (CAREER Award) to Mississippi State University and the University of California, Davis. In addition, this project was funded by the U.S. Office of Naval Research under contract N00014-97-1-0222 and the U.S. Army Research Office under contract ARO 36598-MA-RIP to the University of California, Davis.

PY - 1997

Y1 - 1997

N2 - Given a three-dimensional (3D) array of function values Fi,j,k on a rectilinear grid, the marching cubes (MC) method is the most common technique used for computing a surface triangulation T approximating a contour (isosurface) F(x, y, z) = T. We describe the construction of a C0 -continuous surface consisting of rational-quadratic surface patches interpolating the triangles in T. We determine the Bézier control points of a single rational-quadratic surface patch based on the coordinates of the vertices of the underlying triangle and the gradients and Hessians associated with the vertices.

AB - Given a three-dimensional (3D) array of function values Fi,j,k on a rectilinear grid, the marching cubes (MC) method is the most common technique used for computing a surface triangulation T approximating a contour (isosurface) F(x, y, z) = T. We describe the construction of a C0 -continuous surface consisting of rational-quadratic surface patches interpolating the triangles in T. We determine the Bézier control points of a single rational-quadratic surface patch based on the coordinates of the vertices of the underlying triangle and the gradients and Hessians associated with the vertices.

KW - Approximation

KW - Contour

KW - Isosurface

KW - Marching cubes

KW - Rational bézier curve

KW - Rational bézier surface

KW - Triangular patch

KW - Triangulation

KW - Trilinear interpolation

KW - Visualization

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JO - IEEE Transactions on Visualization and Computer Graphics

JF - IEEE Transactions on Visualization and Computer Graphics

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ER -

On approximating contours of the piecewise trilinear interpolant using triangular rationalquadratic bézier patches

Abstract

Keywords

ASJC Scopus subject areas

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