Abstract: A curve is said to osculate a second curve if the two touch only at a point. A straight line tangent to a curve is a familiar example. The straight line osculates the curve at the point where it touches the curve. A second example will be familiar to some calculus students: an osculating circle. In this case, the circle not only touches a given curve, but also matches the curvature of the curve at the point of touching.

Osculating curves approximate the curve they are touching in the neighbourhood of the contact point. For this reason, they are used a lot in Computer Aided Design (CAD) to speed up calculations and to ensure that curves and surfaces remain smooth at places where they join.

The thesis develops a new way of calculating osculating curves, without being restricted to straight lines or circles. This allows formulae of greater generality than before to be computed.