Discrete Probability Models Of Generalized Bernstein Operators And Curve Design

Bernstein polynomial is a very important operator in the field of approximation theory and geometric design.With the development of quantum calculus,the h-Bernstein operator based on h-calculus and the Lupas q-Bernstein operator and Phillips q-Bernstein operator based on q-calculus appear.The thesis studies the properties of q-Bernstein operator and h-Bernstein operator by constructing their discrete probability models and applies h-Bernstein operator to curve design.The main research result is as follows:The basic properties and new algorithms of Lupas q-Bernstein operator are studied based on the generating function of Lupas q-Bernstein operator.By constructing the gen-erating function identity of Lupas q-Bernstein operator,we obtain the basic properties of Lupas q-Bernstein operator such as linear reproducibility and degree elevation algorithm,and we deduce the q-Marsden's identity,degree reduction algorithm and rational function equation of Lupas q-Bernstein operator.The discrete probability model of generalized Bernstein operator is constructed and the properties of generalized Bernstein operator are studied from the perspective of prob-ability.On one hand,we construct the discrete probability model of Lupas q-Bernstein operator by selecting balls from different urns without repetition.From the perspec-tive of probability,we analyze the basic properties of Lupas q-Bernstein operator such as non-negativity,partition of unity and degree elevation,discuss the degree reduction algorithm and rational function equation of Lupas q-Bernstein operator derived from generating function,define the discrete convolution of Lupas q-Bernstein operator.By extending the model of Lupas q-Bernstein operator,the q-inverse symmetry of Lupas q-Bernstein operator is analyzed and the re-parameterized Lupas q-Bernstein operator is generated.On the other hand,we construct the discrete probability model of Phillips q-Bernstein operator by selecting balls from different urns with repetition,obtain the basic properties such as non-negativity and partition of unity and the degree reduction algorithm for Phillips q-Bernstein operator from the perspective of probability.We con-struct the discrete probability model of(q,h)-Bernstein operator by extending the discrete probability model of Phillips q-Bernstein operator,analyze the basic properties and al-gorithms of(q,h)-Bernstein operator from the perspective of probability and discuss the basic properties and algorithms of h-Bernstein operator.By adding positive real numbers as weights to h-Bernstein operator,the rational h-Bezier curve is defined and the conic section is represented accurately.First,we analyze the basic properties of rational h-Bezier curves,derive the degree elevation algorithm and de Casteljau algorithm of rational h-Bezier curve by applying the basic properties and algorithms of h-Bernstein operator.Then,we discuss the classification of conic sections represented by quadratic rational h-Bezier curves from the perspective of algebra and geometry.Finally,we also give the modeling examples of fountain and arch.Numer-ical examples show that rational h-Bezier curves have more modeling superiority and flexibility than h-Bezier curves and classical rational Bezier curves.