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.
whose Cartesian equation is the product of the Cartesian equations of the two individual circles, that is, As we have seen, f and g are parametric representations of different curves having precisely this support. It is also useful to draw attention to the fact that Propositions 2.3.4 and 2.3.5 give the false impression that regular point and simple point are closely related notions. The following counterexamples throw more light on this question. Counterexample 2.3.6 The right strophoid
suggests that this should indeed be the envelope, perhaps with a problem of regularity at the point (0,1). But verifying the conditions in Definition 2.6.2 “from scratch” would be a serious challenge! Fortunately, the considerations of Sect. 2.11 will take care of that. □ The first study of the envelope of an arbitrary family of curves (not just straight lines) was probably due to Torricelli, around 1642: this student of Galileo has already been mentioned several times in Chap. 1. Galileo
short section is devoted to proving a celebrated result on principal directions: Theorem 5.13.1 (Rodrigues Formula) Given a regular parametric representation f(u,v) of class of a surface, consider the function : “the normal vector to the surface”. Consider further the differentials of f and at a non-umbilical point with parameters (u 0,v 0): The direction is principal if and only if there exists a scalar κ such that Under these conditions, κ is the corresponding principal curvature.
of E, F, G. So the Gaussian curvature is a Riemannian notion, while the normal curvature is not. To prove this, in view of the formula of Proposition 5.16.3, it suffices of course to prove that the quantity LN−M 2 can be expressed as a function of E, F, G. For this, let us switch back to the notation h ij and g ij of Definitions 6.6.2 and 6.1.1. Definition 6.11.1 Consider a regular parametric representation of class of a surface: The Riemann tensor of this surface consists of the family of
Fig. 1.13 Very trivially, such a definition does not work at all for arbitrary curves. Just have a look at Fig. 1.14: a tangent can cut the curve at a second point, and a line which cuts the curve at exactly one point has no reason to be a tangent. Fig. 1.14 Consider the trivial case of a straight line: the tangent to a straight line should be the line itself, which certainly takes us very far from a “unique” point of intersection, globally or locally. Also keep in mind that a tangent can