A Differential Approach to Geometry: Geometric Trilogy III
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This book presents the classical theory of curves in the plane and three-dimensional space, and the classical theory of surfaces in three-dimensional space. It pays particular attention to the historical development of the theory and the preliminary approaches that support contemporary geometrical notions. It includes a chapter that lists a very wide scope of plane curves and their properties. The book approaches the threshold of algebraic topology, providing an integrated presentation fully accessible to undergraduate-level students.
At the end of the 17th century, Newton and Leibniz developed differential calculus, thus making available the very wide range of differentiable functions, not just those constructed from polynomials. During the 18th century, Euler applied these ideas to establish what is still today the classical theory of most general curves and surfaces, largely used in engineering. Enter this fascinating world through amazing theorems and a wide supply of surprising examples. Reach the doors of algebraic topology by discovering just how an integer (= the Euler-Poincaré characteristics) associated with a surface gives you a lot of interesting information on the shape of the surface. And penetrate the intriguing world of Riemannian geometry, the geometry that underlies the theory of relativity.
The book is of interest to all those who teach classical differential geometry up to quite an advanced level. The chapter on Riemannian geometry is of great interest to those who have to “intuitively” introduce students to the highly technical nature of this branch of mathematics, in particular when preparing students for courses on relativity.
corresponding integrals on each (t≠t′) are zero, by definition of a polygonal decomposition (see Definition 7.12.1). Next, we take care of the integral of the geodesic curvature. Since we have a covering of the whole surface, every edge (which is contained in a local map, by Definition 7.11.5) necessarily appears as an edge of two different faces. Moreover, since the surface is orientable, if we travel positively along the borders of these two faces, we travel along the mutual edge in opposite
strophoid is the pedal curve of a parabola with respect to the intersection of the directrix and the axis of symmetry of that parabola. It is also the inverse of a rectangular hyperbola with respect to one of its vertices. The strophoid is its own inverse for an inversion with center the origin and power a 2 (see Sect. 5.7 in , Trilogy I). A secant through the point A=(−a,0) makes equal angles with the tangents to the strophoid, at the two intersection points. 3.16 The Tractrix Imagine a
are equal to zero. Otherwise we have where θ(s) is the angle between and η(s). By Proposition 6.5.1, η(s) is of length 1. But by Proposition 6.7.5, since c is given in normal representation, is proportional to η(s), thus cosθ(s)=±1. Therefore which forces the conclusion. □ Definition 6.9.7 Consider a Riemann patch of class and a regular curve of class given in normal representation. The relative geodesic curvature is the quantity where η is the normal vector field to the curve (see
attention to the surfaces with constant Gaussian curvature and in particular, to the sphere. Of course, arriving at the end of this trilogy, we also “open some doors” to further fascinating developments of geometry. We achieve this by drawing the reader’s attention to some striking results whose proofs often rely on some deep topological results which are beyond the scope of this book (such as the Jordan curve theorem). We switch to the study of curve polygons drawn on a surface. Making clear
example. Take a sheet of paper: everybody knows how to roll it up to get a cylinder, or to get a cone. As a consequence, putting these two operations together, you know how to transform a piece of a cylinder into a piece of a cone. This is of course done “without any stretching”, since a piece of paper cannot be stretched! Mathematically “without any stretching” means that if you consider the various curves that you can draw on the piece of paper, the lengths of all these curves are preserved