[Paper Review] Bezier curves and surfaces based on modified Bernstein polynomials

Khalid Khan, D. K. Lobiyal+1 more|arXiv (Cornell University)|Nov 20, 2015

Advanced Numerical Analysis Techniques13 references15 citations

TL;DR

This paper introduces a novel family of Bézier curves and surfaces using modified Bernstein basis functions with shifted knots, enabling shape-preserving approximation over extended parameter intervals. The approach generalizes classical Bézier curves by incorporating shape parameters α and β, and establishes a de Casteljau algorithm, degree elevation, and degree reduction techniques, with the classical Bézier curves recovered when α = β = 0.

ABSTRACT

In this paper, we use the blending functions of Bernstein polynomials with shifted knots for construction of Bezier curves and surfaces. We study the nature of degree elevation and degree reduction for Bezier Bernstein functions with shifted knots. Parametric curves are represented using these modified Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We get Bezier curve defined on [0, 1] when we set the parameter α=βto the value 0. We also present a de Casteljau algorithm to compute Bernstein Bezier curves and surfaces with shifted knots. The new curves have some properties similar to Bezier curves. Furthermore, some fundamental properties for Bernstein Bezier curves and surfaces are discussed.

Motivation & Objective

• To develop a generalized Bézier curve and surface model using Bernstein basis functions with shifted knots for improved shape control.
• To investigate degree elevation and degree reduction properties for these modified basis functions over non-standard intervals.
• To extend the de Casteljau algorithm to compute points on Bézier curves and surfaces with shifted knots.
• To analyze geometric and shape-preserving properties such as convex hull, corner interpolation, and isoparametric curve behavior.
• To demonstrate that classical Bézier curves and surfaces are recovered when α = β = 0.

Proposed method

• The modified Bernstein basis functions are defined as $ G_{n,α,β}^{k}(t) = \binom{n}{k} \left(\frac{n+\beta}{n}\right)^n \left(t - \frac{\alpha}{n+\beta}\right)^k \left(\frac{n+\alpha}{n+\beta} - t\right)^{n-k} $, valid for $ t \in \left[\frac{\alpha}{n+\beta}, \frac{n+\alpha}{n+\beta}\right] $.
• The de Casteljau algorithm is extended to compute points on Bézier curves and surfaces using recursive blending of control points with the modified basis functions.
• Degree elevation is achieved by deriving new control points $ \mathbf{P}_{i,j}^{\ast} $ through weighted combinations involving parameters $ \alpha_i = 1 - \frac{m+1-i}{m+1} $ and $ \beta_j = 1 - \frac{n+1-j}{n+1} $.
• Tensor product Bézier surfaces are constructed using the product of univariate modified basis functions, preserving properties like convex hull and corner interpolation.
• Total positivity and shape-preserving properties are analyzed using the structure of the modified basis functions and their non-negativity and partition of unity.
• The method generalizes classical Bézier curves by allowing the parameter domain to shift via $ \alpha $ and $ \beta $, with recovery of the classical case when $ \alpha = \beta = 0 $.

Experimental results

Research questions

• RQ1 How can Bernstein basis functions be modified with shifted knots to extend the domain of Bézier curves beyond [0,1]?
• RQ2 What are the degree elevation and degree reduction properties for Bézier curves based on modified Bernstein polynomials with shifted knots?
• RQ3 Can a de Casteljau algorithm be generalized to compute points on Bézier curves and surfaces using these modified basis functions?
• RQ4 What geometric and shape-preserving properties are retained or modified in the new Bézier model?
• RQ5 How do the parameters $ \alpha $ and $ \beta $ influence the shape and behavior of the resulting curves and surfaces?

Key findings

• The modified Bernstein basis functions $ G_{n,\alpha,\beta}^{k}(t) $ are non-negative and satisfy the partition of unity property over the interval $ \left[\frac{\alpha}{n+\beta}, \frac{n+\alpha}{n+\beta}\right] $.
• The basis functions exhibit end-point interpolation, ensuring that the curve passes through the first and last control points when $ t = \frac{\alpha}{n+\beta} $ and $ t = \frac{n+\alpha}{n+\beta} $.
• The tensor product Bézier surface defined by $ \mathbf{P}(u,v) = \sum_{i=0}^{m}\sum_{j=0}^{n} \mathbf{P}_{i,j} G_{m,\alpha,\beta}^{i}(u) G_{n,\alpha,\beta}^{j}(v) $ satisfies the convex hull property and corner point interpolation.
• The isoparametric curves of the surface are themselves Bézier curves with shifted knots, confirming the hierarchical structure of the model.
• When $ \alpha = \beta = 0 $, the modified basis functions reduce to classical Bernstein polynomials, and the model recovers standard Bézier curves and surfaces.
• A new de Casteljau algorithm is derived for evaluating points on Bézier surfaces with shifted knots, using recursive blending with the modified basis functions.