- مبلغ: ۸۶,۰۰۰ تومان
- مبلغ: ۹۱,۰۰۰ تومان
Extended Chebyshev spaces that also comprise the constants represent large families of functions that can be used in real-life modeling or engineering applications that also involve important (e.g. transcendental) integral or rational curves and surfaces. Concerning CAGD, the unique normalized B-bases of such vector spaces ensure optimal shape preserving properties, important evaluation or subdivision algorithms and useful shape parameters. Therefore, we propose global explicit formulas for the entries of those transformation matrices that map these normalized B-bases to the traditional (or ordinary) bases of the underlying vector spaces. Then, we also describe general and ready to use control point configurations for the exact representation of those traditional integral parametric curves and (hybrid) surfaces that are specified by coordinate functions given as (products of separable) linear combinations of ordinary basis functions. The obtained results are also extended to the control point and weight based exact description of the rational counterpart of these integral parametric curves and surfaces. The universal applicability of our methods is presented through polynomial, trigonometric, hyperbolic or mixed extended Chebyshev vector spaces.
5. Final remarks
As listed in Section 1, concerning geometric modeling, the normalized B-bases (of EC spaces that also comprise the constant functions) ensure many optimal shape preserving properties and algorithms. Moreover, they may also provide useful design or shape parameters that can arbitrarily be specified by the user or the engineer. In Section 3, we have seen that polynomial, trigonometric, hyperbolic or mixed EC spaces allow us to obtain the control point based exact description of many (rational) curves and surfaces that are important in several areas of applied mathematics. The investigated large classes of vector spaces also ensure the description of famous geometrical objects (like ellipses; epi- and hypocycloids; Lissajous curves; torus knots; foliums; rose curves; the witch of Agnesi; the cissoid of Diocles; Bernoulli’s lemniscate; Zhukovsky airfoil profiles; cycloids; hyperbolas; helices; catenaries; Archimedean and logarithmic spirals; ellipsoids; tori; hyperboloids; catenoids; helicoids; ring, horn and spindle Dupin cyclides; non-orientable surfaces such as Boy’s and Steiner’s surfaces and the Klein Bottle of Gray).