A Simple Affine-Invariant Spline Interpolation over Triangular Meshes
Abstract
:1. Introduction
2. Main Results
3. Preliminaries and Straightforward Observations
- (i)
- Given and , we have and ;
- (ii)
- Given two adjacent mesh triangles , with common edge , for the points on the line connecting , we have and for any .
4. Version in the Pure Algebraic Setting
- (i)
- , whenever for some n;
- (ii)
- whenever and ;
- (iii)
- whenever .
Algorithm 1:. Triangular -spline with first order data |
Require: for the number of mesh points; |
List of mesh points; |
List for function data at mesh points; |
List , |
for the number of mesh triangles; |
of linear forms for prescribed derivatives at mesh points; |
List of indices with |
such that ; |
List of vectors |
such that ; |
Ensure: List of polynomials with coefficients in . |
Calculation: Consecutively, for each index , we compute the polynomial by applying Theorem 1 and (18) as follows: |
For , let: |
, , , |
, , ; |
For technical reasons, for , we set also |
, , , ; |
After setting the actual values for using the formulas in the theorem, for , |
establish the barycentric weights and their derivatives as affine, resp. linear, forms, |
in terms of the outer product (see [6] Berger(1987)):
|
For cyclic indexing, we set also: |
, . |
Then, for , we compute the correction coefficients by means of (18):
|
Finally, we let:
|
5. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
References
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Stachó, L.L. A Simple Affine-Invariant Spline Interpolation over Triangular Meshes. Mathematics 2022, 10, 776. https://doi.org/10.3390/math10050776
Stachó LL. A Simple Affine-Invariant Spline Interpolation over Triangular Meshes. Mathematics. 2022; 10(5):776. https://doi.org/10.3390/math10050776
Chicago/Turabian StyleStachó, László L. 2022. "A Simple Affine-Invariant Spline Interpolation over Triangular Meshes" Mathematics 10, no. 5: 776. https://doi.org/10.3390/math10050776
APA StyleStachó, L. L. (2022). A Simple Affine-Invariant Spline Interpolation over Triangular Meshes. Mathematics, 10(5), 776. https://doi.org/10.3390/math10050776