Modeling of Curves and Surfaces with MATLAB®

verfasst von: Vladimir Rovenski

Verlag: Springer New York

Buchreihe : Springer Undergraduate Texts in Mathematics and Technology

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

Functions and Transformations

Frontmatter

Chapter 1. Functions and Graphs

Abstract

Section 1.1 discusses the geometry of (integer, real, complex, and quaternion) numbers. and the necessary notations from algebra. In Section 1.2 we plot graphs of some elementary, special, and piecewise functions, and investigate functions using derivatives. In Section 1.3 we study piecewise functions of one variable that are defined by several formulae for different values (intervals) of that variable. Section 1.4 is an excursion to into remarkable curves (graphs) in polar coordinates. In Sections 1.5, and 1.6 we use several MATLAB ^® functions that implement various interpolation and approximation algorithms. Chapter 1 can be considered as the introduction to MATLAB ^® symbolic/numeric calculations, programming, and basic graphing as needed in later chapters.

Vladimir Rovenski

Chapter 2. Rigid Motions (Isometries)

Abstract

Section 2.1 discusses the vectors and their use in analytic geometry. In Section 2.2 we study rigid motions of \({\mathbb{R}}^{n}\) (n≥2), including the complex and quaternionic approaches. Section 2.3 is devoted to the geometry on a sphere (induced by Euclidean geometry of the space), it and also introduces the stereographic projection. The study of polyhedra is organized in Section 2.4 in into the a sequence of themes: the notion of a polyhedron, Platonic solids, symmetries of geometrical figures, star-shaped polyhedra, and Archimedean solids. Section 2.5 (Appendix) surveys matrices and groups.

Vladimir Rovenski

Chapter 3. Affine and Projective Transformations

Abstract

In addition to isometries, there are two kinds of mappings that preserve lines: affine (Section 3.1) and projective (Section 3.2) transformations. Affine transformations f of \({\mathbb{R}}^{n}\) have the following property: If l is a line then f(l) is also a line, and if l ∥ k then f(l) ∥ f(k). A line in \({\mathbb{R}}^{n}\) means a set of the form {r ₀+r:r∈W}, where \({\mathbf{r}}_{0} \in {\mathbb{R}}^{n}\) and \(W \subset {\mathbb{R}}^{n}\) is a one-dimensional subspace. Projective transformations f of \({\mathbb{R}}^{n}\) map lines to lines, preserving the cross-ratio of four points. We also use homogeneous coordinates \(\mathbf{x} = ({x}_{1} : \ldots : {x}_{n+1})\) in \({\mathbb{R}}^{n+1}\). Section 3.3 describes transformation matrices in homogeneous coordinates.

Vladimir Rovenski

Chapter 4. Möbius Transformations

Abstract

In Section 4.1 we review reflections in a sphere and the inversive geometry of a plane. Sections 4.2 and 4.3 are devoted to Möbius transformations and their applications in spherical and hyperbolic geometries. In Section 4.4 we develop several MATLAB ^® procedures to solve problems and visualize solutions in a half-plane and half-space (the Poincaré model).

Vladimir Rovenski

Curves and Surfaces

Frontmatter

Chapter 5. Examples of Curves

Abstract

Section 5.1 starts from the basic notion of a regular curve. Then we investigate cycloidal curves and other remarkable parametric curves, as well as curves given implicitly as level curves of functions in two variables. The level sets are useful in solving problems with conditional extrema. We study the variational calculus for functions of one variable and employ it to derive Euler’s spiral. In Section 5.2 we study some fractal curves. Section 5.3 discusses the basic MATLAB^® capabilities for plotting space curves, surveys the parallel and perspective projections, and presents curves with shadows on planar, cylindrical, or spherical displays. Section 5.4 introduces helix-type curves on surfaces of revolution and studies curves obtained by the intersection of pairs of surfaces.

Vladimir Rovenski

Chapter 6. Geometry of Curves

Abstract

We will study some constructions and visualizations of the differential geometry of curves: the tangent line and Frenet frame (Section 6.1), the singular points (Section 6.2), the length and center of mass (Section 6.3), and the curvature and torsion (Section 6.4).

Vladimir Rovenski

Chapter 7. Geometry of Surfaces

In Section 7.1 we consider the basic notions of a parametric surface and a regular surface, and use a number of MATLAB® commands to produce surfaces by various methods. (Similar definitions for curves were studied in Section 5.1). In Section 7.2 we calculate and plot the tangent planes and normal vectors of a surface. As an application we solve conditional extremum problems in space. In Section 7.3 we consider parametric and implicitly defined surfaces with singularities. In Section 7.4 we use changes in coordinates and linear transformations in space to calculate and plot the osculating paraboloid at a point of a surface. This elementary approach is given only for methodical reasons. In Section 7.5 we calculate characteristics related to the first and second fundamental forms, the Gaussian and mean curvatures, write down the equations of geodesics using the M-file (program) of Section A.8, and plot geodesics on surfaces.

Vladimir Rovenski

Chapter 8. Examples of Surfaces

In this chapter we will study three very important and commonly occurring classes of surfaces (algebraic surfaces, surfaces of revolution, and ruled surfaces) and envelopes of surfaces. In Section 8.1 we will study surfaces of revolutions and plot the Möbius band and a map on a torus that cannot be colored with six colors. In Section 8.2 we will develop an M-file (the resultant) to deduce the polynomial equations of surfaces. In Section 8.3 we will study ruled surfaces of various types and calculate their striction curves and the distribution parameter. In Section 8.4, using the notion of a tangent plane (see Section 7.2), we will plot some envelopes of families of surfaces (see also Section 6.1 for curves).

Vladimir Rovenski

Chapter 9. Piecewise Curves and Surfaces

After an introductory section, in Section 9.2 we will consider Bézier curves. Then we will discuss the important notions of parametric C ^k- continuity and geometric C ^k-continuity of piecewise curves. Section 9.3 is devoted to (parametrically C ^k-continuous) Hermite interpolation and its applications to spline curves. In Section 9.4 we will study β-splines, which are geometrically C ^k-continuous. We will also briefly survey their particular case, B-splines, which are parametrically C ^k-continuous. In Sections 9.5 and 9.6 we will study similar problems for piecewise surfaces. Along the way we will create and apply several M-files that are given in Section A.4.

Vladimir Rovenski

Backmatter

Titel: Modeling of Curves and Surfaces with MATLAB®
verfasst von: Vladimir Rovenski
Verlag: Springer New York
Electronic ISBN: 978-0-387-71278-9
Print ISBN: 978-0-387-71277-2
DOI: https://doi.org/10.1007/978-0-387-71278-9