Controlling the interpolation of NURBS curves and surfaces

The primary focus of this thesis is to determine the best methods for controlling the interpolation of NURBS curves and surfaces. The various factors that affect the quality of the interpolant are described, and existing methods for controlling them are reviewed. Improved methods are presented for calculating the parameter values, derivative magnitudes, data point spacing and twist vectors, with the aim of producing high quality interpolants with minimal data requirements. A new technique for obtaining the parameter values and derivative magnitudes is evaluated, which constructs a C$^1$ cubic spline with orthogonal first and second derivatives at specified parametric locations. When this data is used to create a C$^2$ spline, the resulting interpolant is superior to those constructed using existing parameterisation and derivative magnitude estimation methods. Consideration is given to the spacing of data points, which has a significant impact on the quality of the interpolant. Existing methods are shown to produce poor results with curves that are not circles. Three new methods are proposed that significantly reduce the positional error between the interpolant and original geometry. For constrained surface interpolation, twist vectors must be estimated. A method is proposed that builds on the Adini method, and is shown to have improved error characteristics. In numerical tests, the new method consistently outperforms Adini. Interpolated surfaces are often required to join together smoothly along their boundaries. The constraints for joining surfaces with parametric and geometric continuity are discussed, and the problem of joining $N$ patches to form an $N$-sided region is considered. It is shown that regions with odd $N$ can be joined with G$^1$ continuity, but those with even $N$ or requiring G$^2$ continuity can only be obtained for specific geometries

Color Computer Graphics as Applied to Introductory Calculus Instruction

The use of computer graphics to support and enhance the presentation of introductory calculus concepts is described. Computer graphics provides more accurate graph sketching, consistent presentations and the ability to develop mathematical models incrementally. The addition of extensive use of color aids even more, adding contrast, color keying, dimensionality, and interest to an illustration. Ten lessons have been designed, developed, and evaluated. They employ a set of subroutines which interface to the NBC APC microcomputer graphics software. These lessons as developed and evaluated may be used interactively in the classroom or by individuals, or noninteractively in the classroom by the use of photographic slides

A Reorganization in the Continuity of Subject Matter in Mathematics

This thesis considers a reorganization in the order of arrangement of certain topics in elementary and undergraduate mathematics; i.e. , arithmetic, algebra, plane geometry, solid geometry, trigonometry, analytic geometry, and calculus. Two terms important in the discussion are reorganization, the process of changing the relative position of topics or proofs in mathematics to an earlier or later place in the development of subject matter, and continuity, the logical order of topics arranged according to the need of one to explain the other. The purpose of the thesis is two-fold; First, to show what arrangement of topics may be desirable; and, Second, to justify the proposed changes by showing that such a reorganization will make it possible to give a simpler and more complete presentation of mathematics without affecting the logical sequence of topics. The discussion reviews the recent changes in elementary mathematics during the past forty years. These changes, in general, may be thought of as either of a general character indicating a trend or of a special character indicating a rearrangement in the order of particular topics. The general arrangement of the thesis is somewhat as follows. It is observed that propositions in elementary mathematics have been proved by methods of analytic geometry and calculus. Proofs of certain propositions in plane geometry are possible by coordinate methods. When they are presented in algebra, these proofs are not only simple but provide further understanding of topics in algebra, such as graphs, ratio and proportion, and the operations of algebra. Proofs of certain propositions, or formulas, from elementary mathematics are possible by means of integration. Such proofs by calculus are too difficult to be presented in algebra. These proofs should be postponed to calculus where the simple method of integration justifies the omission of any earlier type of proof of these propositions in elementary mathematics. In the conclusion of this discussion a rearrangement of topics in elementary mathematics (seventh year mathematics, eighth year mathematics, first year algebra, second course in algebra, and plane geometry) with special attention to the continuity of subject matter is given. Such a rearrangement, of necessity, implies changes in the order of some of the topics in later mathematics

A retelling of Newton’s work on Kepler’s Laws

AbstractThis paper details an investigation into Kepler’s Laws. Newton’s technique for deducing an inverse-square law from Kepler’s Laws is given a modern presentation, with necessary background material included. Kepler’s Laws are then deduced from the assumption of an inverse-square law. This is done in a geometric style, inspired by Newton’s work. Finally, a problem involving planetary orbits is stated and solved using the earlier results of the paper