Area-Preserving Geometric Hermite Interpolation
For researchers in geometric modeling and computer-aided design, this provides a novel interpolation method that exactly preserves area while achieving higher accuracy than standard approaches.
This paper develops a framework for planar geometric interpolation using cubic Bézier polynomials that exactly preserves area, achieving 5th-order accuracy (one order higher than standard geometric cubic Hermite interpolation) when endpoint curvature is non-zero, and 4th-order accuracy otherwise. The method is computationally efficient with an explicit formula for endpoint parametrization speeds.
In this paper we establish a framework for planar geometric interpolation with exact area preservation using cubic Bézier polynomials. We show there exists a family of such curves which are $5^{th}$ order accurate, one order higher than standard geometric cubic Hermite interpolation. We prove this result is valid when the curvature at the endpoints does not vanish, and in the case of vanishing curvature, the interpolation is $4^{th}$ order accurate. The method is computationally efficient and prescribes the parametrization speed at endpoints through an explicit formula based on the given data. Additional accuracy (i.e. same order but lower error constant) may be obtained through an iterative process to find optimal parametrization speeds which further reduces the error while still preserving the prescribed area exactly.