In the paper, the Lagrange geometric interpolation by spatial rational cubic Bézier curves is studied. It is shown that under some natural conditions the solution of the interpolation problem exists ...and is unique. Furthermore, it is given in a simple closed form which makes it attractive for practical applications. Asymptotic analysis confirms the expected approximation order, i.e., order six. Numerical examples pave the way for a promising nonlinear geometric subdivision scheme.
► Lagrange geometric interpolation of spatial data by rational cubic curves. ► Analysis of the solvability conditions and the solution given in a closed form. ► Optimal approximation order established. ► Nonlinear subdivision based on the developed Lagrange scheme derived.
Lattices on simplicial partitions Jaklič, Gašper; Kozak, Jernej; Krajnc, Marjeta ...
Journal of computational and applied mathematics,
02/2010, Letnik:
233, Številka:
7
Journal Article, Conference Proceeding
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In this paper,
(
d
+
1
)
-pencil lattices on simplicial partitions in
R
d
are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a ...continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of this fact leads to an efficient computer algorithm for the design of a lattice.
In this paper, the geometric Lagrange interpolation of four points by planar cubic Pythagorean-hodograph (PH) curves is studied. It is shown that such an interpolatory curve exists provided that the ...data polygon, formed by the interpolation points, is convex, and satisfies an additional restriction on its angles. The approximation order is 4. This gives rise to a conjecture that a PH curve of degree
n can, under some natural restrictions on data points, interpolate up to
n
+
1
points.
In this paper, geometric interpolation of certain circle-like curves by parametric polynomial curves is studied. It is shown that such an interpolating curve of degree
n achieves the optimal ...approximation order 2
n, the fact already known for particular small values of
n. Furthermore, numerical experiments suggest that the error decreases exponentially with growing
n.
In this paper, the geometric interpolation of planar data points and boundary tangent directions by a cubic G² Pythagorean-hodograph (PH) spline curve is studied. It is shown that such an interpolant ...exists under some natural assumptions on the data. The construction of the spline is based upon the solution of a tridiagonal system of nonlinear equations. The asymptotic approximation order 4 is confirmed.
In this paper, geometric interpolation by parametric polynomial curves is considered. Discussion is focused on the case where the number of interpolated points is equal to r + 2, and n=r denotes the ...degree of the interpolating polynomial curve. The interpolation takes place in $\mathbb R^d$ with d=n. Even though the problem is nonlinear, simple necessary and sufficient conditions for existence of the solution are stated. These conditions are entirely geometric and do not depend on the asymptotic analysis. Furthermore, they provide an efficient and stable way to the numeric solution of the problem.
In this paper, geometric interpolation by parametric polynomial curves is considered. Discussion is focused on the case where the number of interpolated points is equal to r + 2, and n = r denotes ...the degree of the interpolating polynomial curve. The interpolation takes place in Rdwith d = n. Even though the problem is nonlinear, simple necessary and sufficient conditions for existence of the solution are stated. These conditions are entirely geometric and do not depend on the asymptotic analysis. Furthermore, they provide an efficient and stable way to the numeric solution of the problem.
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