References
The bibliography supports the mathematical background and the application formulations discussed in this documentation. Use the links from each chapter to identify the result relevant to its assumptions. A theorem about the reference optimization problem does not by itself give an error bound for a finite-data Bézier fit.
Fitting, approximation and statistical behavior
Kobayashi et al. [KHS+19] introduces Bézier simplex fitting for Pareto fronts and its approximation perspective. Tanaka et al. [TSKH20] studies statistical risk. Optimal skeleton sampling extends its optimal sample ratio to every dimension and degree, and Theory of the skeleton risk proves the result, using the polynomial small-ball inequality of Carbery and Wright [CW01]. Read these alongside What is Bézier simplex fitting? and Cross-validation and degree search: expressiveness at a sufficiently large degree and accuracy from a particular finite sample are distinct questions.
Hikima et al. [HKT+25] concerns objective-based stochastic optimization of a Bézier representation. Its setting is discussed in Objective minimization; it should be distinguished from linear least-squares regression on known parameter-response pairs.
Topology and domain construction
Mizota et al. [MHI21] provides the unconstrained strongly convex results stated in What is Bézier simplex fitting?. Check smoothness, feasibility and objective assumptions before transferring a statement to a constrained application. Weak simpliciality permits a many-to-one parameterization; it is stronger to claim that the image is homeomorphic to a simplex.
Hamada [Ham20], Hamada and Goto [HG18] discuss topology assessment from data under their stated assumptions. A sampled picture or failure to detect a hole is not independently a proof of simplex topology.
For product cells and their faces, see Peltier et al. [PFL09], Peltier and Lienhardt [PL18] and the geometric introduction in Single and assembled domains. The mathematical continuation in The Category of Simploidal Sets uses the category-theory background of Mac Lane [ML98], Mac Lane and Moerdijk [MLM92].
Shape-preserving interpolation on simplicial meshes (Shape-preserving interpolation on simplicial meshes) uses the square part of the Fritsch–Carlson monotonicity region [FC80] to limit one-dimensional cubic Hermite segments. The default interpolant is C0, not generally C1. Supplying PCHIP slopes, with the Fritsch–Butland weighted harmonic mean at interior vertices [FB84] and PCHIP’s one-sided endpoint rule, recovers PCHIP up to float64 roundoff. Its face controls use the degree elevation of Bézier triangles described by Farin [Far02].
Application sources and reproducible examples
Each application links its formulation to relevant literature and names the script used for its figures. These scripts are small numerical illustrations of specified targets. Consult the page’s parameter range, validation set and feasibility checks when interpreting its reported error; a reference to a broader application does not make the toy experiment an implementation of that complete system.
Bibliography
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