A survey of curve and surface methods in CAGD
Computer Aided Geometric Design
Markov chains and computer aided geometric design: Part II—examples and subdivision matrices
ACM Transactions on Graphics (TOG)
Markov chains and computer-aided geometric design: part I - problems and constraints
ACM Transactions on Graphics (TOG)
Computational geometry: an introduction
Computational geometry: an introduction
Subdivision algorithms converge quadratically
Journal of Computational and Applied Mathematics
Free-form deformation of solid geometric models
SIGGRAPH '86 Proceedings of the 13th annual conference on Computer graphics and interactive techniques
Triangular Berstein-Be´zier patches
Computer Aided Geometric Design
American Mathematical Monthly
Computing binomial coefficients
American Mathematical Monthly
Computer Aided Geometric Design - Special issue: Topics in CAGD
On the numerical condition of polynomials in Berstein form
Computer Aided Geometric Design
ACM Transactions on Graphics (TOG)
Algorithms for polynomials in Bernstein form
Computer Aided Geometric Design
On de Boor-like algorithms and blossoming
Computer Aided Geometric Design
Degree reduction of Be´zier curves
Computer-Aided Design
Chebyshev economization for parametric surfaces
Computer Aided Geometric Design
Curvature continuity and offsets for piecewise conics
ACM Transactions on Graphics (TOG)
A new multiaffine approach to B-splines
Computer Aided Geometric Design
The numerical problem of using Be´zier curves and surfaces in the power basis
Computer Aided Geometric Design
A multisided generalization of Bézier surfaces
ACM Transactions on Graphics (TOG)
Generating functionology
Computer Aided Geometric Design
Surfaces over Dirichlet Tessellations
Computer Aided Geometric Design
Shape preserving representations
Mathematical methods in computer aided geometric design
On the stability of transformations between power and Bernstein polynomial forms
Computer Aided Geometric Design
Creating multisided rational Bézier surfaces using base points
ACM Transactions on Graphics (TOG)
POLoynomials, POLar Forms, and InterPOLation
Mathematical methods in computer aided geometric design II
Curves and surfaces in geometrical optics
Mathematical methods in computer aided geometric design II
Degree reduction of Be´zier curves
Selected papers of the international symposium on Free-form curves and free-form surfaces
The conformal map z→z2 of the hodograph plane
Computer Aided Geometric Design
Polynomial real root finding in Bernstein form
Polynomial real root finding in Bernstein form
Multiresolution analysis of arbitrary meshes
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
Bernstein-Be´zier polynomials on spheres and sphere-like surfaces
Computer Aided Geometric Design
On the optimal stability of the Bernstein basis
Mathematics of Computation
The geometry of optimal degree reduction of Be´zier curves
Computer Aided Geometric Design
Robust analysis and design of control systems using interval arithmetic
Automatica (Journal of IFAC)
B-splines and optimal stability
Mathematics of Computation
Basis conversion among Bézier, Tchebyshev and Legendre
Computer Aided Geometric Design
Robust Schur Stability of Polynomials with Polynomial ParameterDependency
Multidimensional Systems and Signal Processing
Bernstein-Bézier Methods for the Computer-Aided Design of Free-Form Curves and Surfaces
Journal of the ACM (JACM)
De Casteljau's autobiography: my time at Citroën
Computer Aided Geometric Design - Special issue dedicated to Paul de Faget de Casteljau
Computer Aided Geometric Design - Special issue dedicated to Paul de Faget de Casteljau
Computer Aided Geometric Design - Special issue dedicated to Paul de Faget de Casteljau
Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon
Computer Aided Geometric Design - Special issue dedicated to Paul de Faget de Casteljau
Physical constraints on feedrates and feed accelerations along curved tool paths
Computer Aided Geometric Design
Legendre-Bernstein basis transformations
Journal of Computational and Applied Mathematics - Special issue/Dedicated to Prof. Larry L. Schumaker on the occasion of his 60th birthday
ACM Transactions on Mathematical Software (TOMS)
Curves and surfaces for CAGD: a practical guide
Curves and surfaces for CAGD: a practical guide
Pierre Bézier: an engineer and a mathematician
Computer Aided Geometric Design - Pierre Bézier
The Mathematical Basis of the UNISURF CAD System
The Mathematical Basis of the UNISURF CAD System
Robust Control: Systems with Uncertain Physical Parameters
Robust Control: Systems with Uncertain Physical Parameters
Journal of Computational and Applied Mathematics
Generalized barycentric coordinates on irregular polygons
Journal of Graphics Tools
Solving Algebraic Systems in Bernstein-Bézier Representation
CG '91 Proceedings of the International Workshop on Computational Geometry - Methods, Algorithms and Applications
Computer Aided Geometric Design
Lower bound functions for polynomials
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
Theoretical Computer Science - Algebraic and numerical algorithm
Recursive de Casteljau bisection and rounding errors
Computer Aided Geometric Design
Mean value coordinates for arbitrary planar polygons
ACM Transactions on Graphics (TOG)
A unified, integral construction for coordinates over closed curves
Computer Aided Geometric Design
A two-steps algorithm for approximating real roots of a polynomial in Bernstein basis
Mathematics and Computers in Simulation
A unified approach to resultant matrices for Bernstein basis polynomials
Computer Aided Geometric Design
Gauss-Lobatto to Bernstein polynomials transformation
Journal of Computational and Applied Mathematics
Division algorithms for Bernstein polynomials
Computer Aided Geometric Design
Transfinite mean value interpolation
Computer Aided Geometric Design
Fast construction of constant bound functions for sparse polynomials
Journal of Global Optimization
Subdivision methods for solving polynomial equations
Journal of Symbolic Computation
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Complex rational Bézier curves
Computer Aided Geometric Design
An efficient algorithm for range computation of polynomials using the Bernstein form
Journal of Global Optimization
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
On the convergence of derivatives of Bernstein approximation
Journal of Approximation Theory
Optimal Conditioning of Bernstein Collocation Matrices
SIAM Journal on Matrix Analysis and Applications
Maximum entropy coordinates for arbitrary polytopes
SGP '08 Proceedings of the Symposium on Geometry Processing
Complex Bézier curves and the geometry of polygons
Computer Aided Geometric Design
Journal of Global Optimization
Fast simplicial finite element algorithms using Bernstein polynomials
Numerische Mathematik
Barycentric coordinates for polytopes
Computers & Mathematics with Applications
h-Blossoming: A new approach to algorithms and identities for h-Bernstein bases and h-Bézier curves
Computer Aided Geometric Design
Journal of Approximation Theory
Polytope-based computation of polynomial ranges
Computer Aided Geometric Design
Explicit formula for improved filter sharpening polynomial
IEEE Transactions on Signal Processing
Fast simplicial quadrature-based finite element operators using Bernstein polynomials
Numerische Mathematik
Technical Communique: Robust D-stability via positivity
Automatica (Journal of IFAC)
Planar piecewise algebraic curves
Computer Aided Geometric Design
Short communication: The octant of a sphere as a non-degenerate triangular Bézier patch
Computer Aided Geometric Design
Piecewise algebraic surface patches
Computer Aided Geometric Design
Rates of convergence of control polygons
Computer Aided Geometric Design
A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
IEEE Transactions on Neural Networks
Bernstein-Bézier Finite Elements of Arbitrary Order and Optimal Assembly Procedures
SIAM Journal on Scientific Computing
Visualizing scientific graphics for astronomy and astrophysics
ACM SIGGRAPH 2013 Posters
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One hundred years after the introduction of the Bernstein polynomial basis, we survey the historical development and current state of theory, algorithms, and applications associated with this remarkable method of representing polynomials over finite domains. Originally introduced by Sergei Natanovich Bernstein to facilitate a constructive proof of the Weierstrass approximation theorem, the leisurely convergence rate of Bernstein polynomial approximations to continuous functions caused them to languish in obscurity, pending the advent of digital computers. With the desire to exploit the power of computers for geometric design applications, however, the Bernstein form began to enjoy widespread use as a versatile means of intuitively constructing and manipulating geometric shapes, spurring further development of basic theory, simple and efficient recursive algorithms, recognition of its excellent numerical stability properties, and an increasing diversification of its repertoire of applications. This survey provides a brief historical perspective on the evolution of the Bernstein polynomial basis, and a synopsis of the current state of associated algorithms and applications.