6 - Differential geometry

Abstract

Differential geometry is the application of differential calculus to the study of curves and surfaces. A continuous curve corresponds to each continuous function. A smooth curve, that is, a curve without discontinuities and breaks, corresponds to each differentiable function. This chapter focuses on plane curves. It reviews the main elements of plane curves. Cycloid, epicycloids, hypocycloid, Cassini oval, catenary, tractrix, cartesian oval, cissoid, strophoid, and conchoid are a few important plane curves. If a circle is rolled along a straight line, the curve traced out by a point on its circumference is called a (common) cycloid. The path described by a point on the circumference of a circle rolled on the exterior of another circle is called an epicycloid. The path described by a point on the circumference of a circle that is rolled on the inner side of the circumference of another circle is called a hypocycloid. A Cassini oval is the set of all points of which distances from two fixed points have a constant product a2. A cable, perfectly flexible and heavy, suspended at two points in equilibrium takes the form of a catenary. The chapter discusses these plane curves. It also explains space curves and their arc elements and curved surfaces.