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An intuitive way for constructing parametric quadric triangles

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Abstract

We present an intuitive algorithm for providing quadric surface design elements with shape parameters. To this end, we construct rational parametric triangular quadratic patches which lie on quadrics. The input of the algorithm is three vertex data points in 3D and normals at these points. It emanates from a thorough analysis of two existing methods for the construction of rational parametric Bézier triangles on quadrics, that allows to establish an interesting geometric relation between them. The sufficient condition for a configuration of vertex and normal data to allow for the existence of a rational triangular quadratic patch lying on a quadric whose tangent planes at the vertices are those prescribed by the given normals is the concurrence of certain cevians. When these conditions are not met we offer an optimization procedure to tweak the normals, without varying the vertex data, so that for the new normals there is a rational triangular quadratic patch that lies on a quadric. The resulting quadric design element offers three free shape parameters.

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Notes

  1. In the classical geometry literature, see for example Coxeter’s book Geometry Revisited (Coxeter and Greitzer 1967), the lines \(\mathbf{b}_{A}\mathbf{c}_{BC}\), \(\mathbf{b}_B\mathbf{c}_{CA}\) and \(\mathbf{b}_C\mathbf{c}_{AB}\) are called cevians, after Giovanni Ceva (1642–1734), Italian geometer and engineer.

  2. Note that the scaling of \(\mathbf{b}_i\) to \(\mathbf{b}_i/\lambda _i\) assumes that \(\mathbf{b}_i\) is expressed in terms of a coordinate system with origin at \(\mathbf s\).

  3. In general for any six points \((x_i,y_i)\) in the plane, no four collinear, and six numbers \(f_i\) there exists a unique paraboloid (possibly degenerate, i.e. a plane) \(f(x,y)\) such that \(f(x_i,y_i)=f_i\).

  4. These weights result from (9) and from the application of a rational linear parameter transformation producing a patch in standard form, i.e., with unit corner weights, see, e.g., Albrecht (1995).

  5. By edge we mean the line containing the edge.

  6. If \(\mathbf{s}\) lies at infinity, according to Remark 3.1 we have \(\lambda _{AB} = cr(\mathbf{d}_{AB}, \mathbf{b}_{AB}, \mathbf{s}, \mathbf{s}) = 1\) and analogously \(\lambda _{CA} = \lambda _{BC} = 1\).

  7. Some of the triangles (usually less than 10 %) might be left out because of very high aspect ratios which causes the optimization algorithm not to converge.

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Acknowledgments

The authors thank Université de Valenciennes, DIME, Universidad Nacional de Colombia, SedeMedellín as well as Ecos Nord (grant no. C13M01) and The Abdus Salam International Centre for Theoretical Physics (grant no. VS-481) for financial support of this work. They also wish to thank Samir Posada for carrying out the numerical tests yielding Tables 1 and 2. The authors are especially thankful to the anonymous referees for their very thorough reviewing which led us to improve the readability of the paper.

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Authors and Affiliations

  1. Univ Lille Nord de France, UVHC, LAMAV, FR CNRS 2956, 59313, Valenciennes, France

    Gudrun Albrecht

  2. Escuela de Matemáticas, Universidad Nacional de Colombia, Sede Medellín, Calle 59 A No 63-20, Medellín, Colombia

    Marco Paluszny & Marianela Lentini

Authors
  1. Gudrun Albrecht
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  2. Marco Paluszny
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  3. Marianela Lentini
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Correspondence to Gudrun Albrecht.

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Communicated by Jinyun Yuan.

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Albrecht, G., Paluszny, M. & Lentini, M. An intuitive way for constructing parametric quadric triangles. Comp. Appl. Math. 35, 595–617 (2016). https://doi.org/10.1007/s40314-014-0207-y

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  • DOI: https://doi.org/10.1007/s40314-014-0207-y

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