Progressive iterative approximation and bases with the fastest convergence rates
Computer Aided Geometric Design
Interpolation by geometric algorithm
Computer-Aided Design
Volumetric parameterization and trivariate B-spline fitting using harmonic functions
Computer Aided Geometric Design
Loop subdivision surface based progressive interpolation
Journal of Computer Science and Technology
Totally positive bases and progressive iteration approximation
Computers & Mathematics with Applications
Local progressive-iterative approximation format for blending curves and patches
Computer Aided Geometric Design
Progressive interpolation using loop subdivision surfaces
GMP'08 Proceedings of the 5th international conference on Advances in geometric modeling and processing
Technical Section: An extended iterative format for the progressive-iteration approximation
Computers and Graphics
Technical note: Progressive iteration approximation and the geometric algorithm
Computer-Aided Design
B-spline surface fitting by iterative geometric interpolation/approximation algorithms
Computer-Aided Design
Geometric point interpolation method in R3 space with tangent directional constraint
Computer-Aided Design
Computer Graphics in China: Convergence analysis for B-spline geometric interpolation
Computers and Graphics
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The geometric interpolation algorithm is proposed by Maekawa et al. in [Maekawa T, Matsumoto Y, Namiki K. Interpolation by geometric algorithm. Computer-Aided Design 2007;39:313-23]. Without solving a system of equations, the algorithm generates a curve (surface) sequence, of which the limit curve (surface) interpolates the given data points. However, the convergence of the algorithm is a conjecture in the reference above, and demonstrated by lots of empirical examples. In this paper, we prove the conjecture given in the reference in theory, that is, the geometric interpolation algorithm is convergent for a blending curve (surface) with normalized totally positive basis, under the condition that the minimal eigenvalue @l"m"i"n(D"k) of the collocation matrix D"k of the totally positive basis in each iteration satisfies @l"m"i"n(D"k)=@a0. As a consequence, the geometric interpolation algorithm is convergent for Bezier, B-spline, rational Bezier, and NURBS curve (surface) if they satisfy the condition aforementioned, since Bernstein basis and B-spline basis are both normalized totally positive.