Transforming Trigonometric Knot Parameterizations into Rational Knot Parameterizations
نویسنده
چکیده
This paper develops a method for constructing rational parameterizations of knots, based on a trigonometric parameterization. It also introduces the class of torus knots and describes a method for constructing trigonometric and rational parameterizations of these knots. This research was conducted at the Mt. Holyoke REU, and was funded by the NSF through grant number DMS-9732228.
منابع مشابه
Parameterizations of 1-bridge Torus Knots
A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert’s normal form and the Conway’s normal form for 2-bridge knots. For a given Schubert’s normal form we give algorithms to determine the number of components and to compute the f...
متن کاملConstructing Polynomial Knots
Shastri proved[3] that every knot can be expressed as the image of a parametric function t 7→ (x(t), y(t), z(t)), where x, y, and z are polynomials in t. However, it is difficult based on his proof to actually find a polynomial knot of a given knot type. We present an algorithm for converting a piecewise linear parameterization of a knot into a polynomial parameterization of a knot of the same ...
متن کاملControl curves and knot insertion for trigonometric splines
We introduce control curves for trigonometric splines and show that they have properties similar to those for classical polynomial splines. In particular, we discuss knot insertion algorithms, and show that as more and more knots are inserted into a trigonometric spline, the associated control curves converge to the spline. In addition, we establish a convex-hull property and a variation-dimini...
متن کاملParameterizing surfaces with certain special support functions, including offsets of quadrics and rationally supported surfaces
We discuss rational parameterizations of surfaces whose support functions are rational functions of the coordinates specifying the normal vector and of a given non-degenerate quadratic form. The class of these surfaces is closed under offsetting. It comprises surfaces with rational support functions and non-developable quadric surfaces, and it is a subset of the class of rational surfaces with ...
متن کاملOn Parameterizations of Rational Orbits of Some Forms of Prehomogeneous Vector Spaces
In this paper, we consider the structure of rational orbit of some prehomogeneous vector spaces originally motivated by Wright and Yukie. After we give a refinement of WrightYukie’s construction, we apply it to the inner forms of the D5 and E7 types. Parameterizations of reducible algebras for the cubic cases are included.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002