Invitation to Geometry

Z. A. Melzak

• Publisher: Dover Publications
• Year: 2014
• ISBN:
  9780486789484
• Topic: Mathematik
• Subtopic: Algebraische Geometrie

Intended for students of many different backgrounds with only a modest knowledge of mathematics, this text features self-contained chapters that can be adapted to several types of geometry courses. Only a slight acquaintance with mathematics beyond the high-school level is necessary, including some familiarity with calculus and linear algebra.
This text's introductions to several branches of geometry feature topics and treatments based on memorability and relevance. The author emphasizes connections with calculus and simple mechanics, focusing on developing students' grasp of spatial relationships. Subjects include classical Euclidean material, polygonal and circle isoperimetry, conics and Pascal's theorem, geometrical optimization, geometry and trigonometry on a sphere, graphs, convexity, and elements of differential geometry of curves. Additional material may be conveniently introduced in several places, and each chapter concludes with exercises of varying degrees of difficulty.

CHAPTER TEN

CURVES IN SPACE AND CURVES ON SURFACES

This section is devoted to some elements of classical differential geometry: a branch of geometry that uses calculus and linear algebra to investigate the behavior of curves and surfaces. Some knowledge of the rudiments of vector calculus and of linear algebra will therefore be assumed, and it will be seen how these two subjects interact to help us derive our principal result, the Frenet-Serret equations, which describe the curving and twisting of curves in space. Then we shall consider a few special cases of geodesies: curves of shortest length which join two points of a surface and lie entirely on that surface.

By way of introduction and motivation let us start with the following question: What property do the line L, the circle C, and the circular helix H have in common, that no other curve has? This question is rather loose and a variety of answers could be given. Or, perhaps, the one correct answer could be phrased in a variety of ways. Looking at the three curves locally, piece by piece, we observe that all three are somehow homogeneous or uniform, in the sense that a neighborhood of a point on any one curve looks exactly like the neighborhood of any other point on that curve. This self-congruence “in the small” suggests looking at the three curves “in the large” or globally, that is, looking at the whole curves. It might occur to us to move the whole curve; the motion is supposed to be rigid so that the curve may be imagined to be made of thin, stiff wire. We observe then that each curve acts as its own track, it can be moved “within” itself: L by being translated along itself, C by being rotated on itself, and H by a screw motion over itself. Or, to put the same thing differently, we can take two copies of any one of our three curves, one copy is stationary and we call it the track, the other copy is movable and we call it the train. It is now possible to move each train on its track without any distortion: the train and the track, as wholes, coincide at all times.

We propose to pursue the matter further, and for this purpose we introduce the concept of a local property of a curve. Let K...