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Introduction to discrete differential geometry: the geometry of plane curves

The nascent field of discrete differential geometry deals with discrete geometric objects (such as polygons) which act as analogues to continuous geometric objects (such as curves). The discrete objects can be measured (length, area) and can interact with other discrete objects (collision/response). From a computational standpoint, the discrete objects are attractive, because they have been designed from the ground up with data-structures and algorithms in mind. From a mathematical standpoint, they present a great challenge: the discrete objects should have properties which are analogues of the properties of continuous objects. One important property of curves and surfaces is their curvature, which plays a significant role in many application areas (see, e.g., Chapters 4 and 5). In the continuous domain there are remarkable theorems dealing with curvature; a key requirement for a discrete curve with discrete curvature is that it satisfies analogous theorems. In this chapter we examine the curvature of continuous and discrete curves on the plane.