For around three decades one of the distinguishing features of my department has been a second year course on the geometry of curves, in which (following an earlier set of precepts) plane curves are studied simultaneously from the algebraic and differentiable viewpoints. It has a proven history of success, providing students with a wonderful introduction to visually attractive geometry. My experience in teaching this course convinced me that the time was ripe to raise the profile of undergraduate geometry. The algebraic viewpoint developed into my text, ‘Elementary Geometry of Algebraic Curves’ (Cambridge University Press, 1998). The present text is intended as a companion volume, representing the differentiable viewpoint.

Differential Geometry

Differential geometry is a fascinating area of mathematics, of substantial and increasing importance in the physical sciences. Despite that, the subject has a low billing in most undergraduate curricula, either not appearing or relegated to a final year optional course. That is a shame since much can be achieved with minimal mathematical preparation in the second year by restricting attention to plane curves: those who then wish to develop their interest can proceed to final year courses studying more general objects. Plane curves live in an environment familiar from school mathematics (the Euclidean plane), and have features readily visible on a computer screen: moreover, their study uses foundational mathematics (calculus, linear algebra and complex numbers) in a useful way, with only a handful of results using basic facts in real analysis.