1 Introduction
Filling an sided hole with a multisided surface is an important problem in CAGD. Usually the patch should connect to the adjacent surfaces with at least continuity, but in some applications only positional (
) continuity is needed, and normal vectors or cross derivatives at the boundary curves are not available.
For , the Coons patch [1] solves this problem; in this paper we show how to generalize it to any number of sides.
2 Previous work
Most transfinite surface representation in the literature assume constraints, and the patch equations make use of the fixed crossderivatives at the boundary. This can be circumvented by generating a normal fence automatically, e.g. with a rotation minimizing frame [3]; however, in a setting this is an overkill, simpler methods exist.
One wellknown solution is the harmonic surface, which creates a “soap film” filling the boundary loop by solving the harmonic equation on a mesh with fixed boundaries. This, however, minimizes the total area of the surface, which often has unintuitive results, see an example in Section 4.
The basic idea of the proposed method, i.e., to define the surface as the weighted sum of Coons patches, each interpolating three consecutive sides, is the same as in the CR patch [2].
3 The multisided Coons patch
Let denote the th boundary curve. Let us also assume for all (with circular indexing). Then the ribbon is defined as a Coons patch interpolating , , , and – a cubic curve fitted onto the initial and (negated) end derivatives of sides and , respectively (see Figure 1).
Formally,
(1) 
where is defined as the Bézier curve^{1}^{1}1Except for , where degenerates to the point . determined by the control points
(2)  
(3) 
The surface is defined over a regular sided polygon. The Wachspress coordinates of a domain point are defined as
(4) 
where is the perpendicular distance of from the th edge of the domain polygon. Ribbon parameterization is based on these generalized barycentric coordinates:
(5) 
It is easy to see that , and that has the following properties:

on the th side.

on the “far” sides (all sides except , and ).

on the th side.
Finally, we define the patch as
(6) 
where is the blending function
(7) 
The interpolation property is satisfied due to the properties of mentioned above.
(Note: in Eq. (5) cannot be evaluated when , but at these locations the weight also vanishes.)
4 Examples
Figure 2 shows a comparison with the harmonic surface, which – due to its area minimizing property – results in an unnaturally flat patch.
Figure 3 shows a model with 5 patches: two 3sided, one 4sided, one 5sided, and one 6sided. The mean curvature map and contouring both show good surface quality.
Conclusion
We have defined a natural generalization of the Coons patch – a lightweight and efficient multisided surface representation, applicable when only positional data is available.
Acknowledgements
This work was supported by the Hungarian Scientific Research Fund (OTKA, No. 124727).
References
 [1] Steven Anson Coons. Surfaces for computeraided design of space forms. Technical Report MIT/LCS/TR41, Massachusetts Institute of Technology, 1967.
 [2] Péter Salvi, Tamás Várady, and Alyn Rockwood. Ribbonbased transfinite surfaces. Computer Aided Geometric Design, 31(9):613–630, 2014.
 [3] Wenping Wang, Bert Jüttler, Dayue Zheng, and Yang Liu. Computation of rotation minimizing frames. ACM Transactions on Graphics, 27(1):2, 2008.
Comments
There are no comments yet.