A simple matrix form for degree reduction of Bézier curves using Chebyshev–Bernstein basis transformations

Application of Chebyshev II–Bernstein basis transformations to degree reduction of Bézier curves

“…So, for a given error tolerance, the number of subdivisions cannot be known before degree reduction is conducted. Furthermore, our approach shares the common limitation of known methods using the basis transformation matrices [1,3,4,7,8,14,15,18,20,23]: although the transformation matrices are well-conditioned, degree reduction by them may be ill-conditioned (see Remark 2). It is an inherent limitation, since the Bernstein coefficients must be transferred to the specific forms before degree reduction and back after degree reduction.…”

“…In general, degree reduction cannot be done exactly so that it invokes approximation problems. Thus in recent twenty years, many works [1][2][3][4][6][7][8][13][14][15][16]18,20,21,23] relevant to the degree reduction of Bézier curves have been published.…”

“…The Chebyshev polynomials of the first kind are used to obtain the best approximation in L ∞ -norm [7,14,23] and in the 1/ √ 4t − 4t 2 -weighted square norm [18], whereas the Chebyshev polynomials of the second kind are used in L 1 -norm [13]. The Legendre polynomials are used in L 2 -norm [8,14,15].…”

Multiple Degree Reduction and Elevation of Bézier Curves Using Jacobi–Bernstein Basis Transformations

“…Also, by computing one more coefficient, the approximating polynomial of degree n can be easily computed from the approximating polynomial of matrix T n , the basis transformation matrices and the degree reduction matrix are derived. The Legendre-Bernstein basis transformation [15] and the Chebyshev-Bernstein basis transformation [17] become special cases.…”

Change of Basis Transformation from the Bernstein Polynomials to the Chebyshev Polynomials of the Fourth Kind

“…The Chebyshev polynomials have many applications in science and engineering; the best uniform approximation is characterized by the Chebyshev polynomials of first kind, the weighted least squares approximations are characterized by the relevant weight of Chebyshev polynomials of first, second, third, and fourth kinds. In Computer Aided Design (CAD), the weighted degree reduction of the Bézier curves is handled by the Chebyshev polynomials.For more details about the Chebyshev polynomials of the fourth kind and their applications, see [4,5,12]. Other related applications and properties are found in [13][14][15][16][17][18][19][20][21].…”