ABSTRACT
In general there is a unique circle having the same curvature and same tangent direction as the curve at a common point which is regular for the curve. The centre of this circle is called the centre of curvature at the point and the radius of the circle is called the radius of curvature at the point. The centre of curvature can also be described as the limiting position of the point of intersection of 'consecutive' normals. We now define the centre of curvature and the circle of curvature. Definition 9.1. The standard parametrisation of a circle having curvature K (K, ^ 0), and centre ZQ, is
z(t) = z0 H-elKt or r(t) = ro H-(cos Ktì sin KÌ) J K KJ
Notice that the standard parametrisation of a circle is a unit speed (arclength) parametrisation (i.e., \z'\ = 1), and that t is not the argument (polar angle) of z — zo. This circle is positively oriented (counter-clockwise) if K > 0, and is negatively oriented (clockwise) if K < 0. Definition 9.2. At a regular point r(£o), respectively z(to), of the curve t i-> r(t), respectively t *-» z(£), the centre of curvature is
z+ = z*(to) = ^C^o)H-— t /At ^,, or, equivalently,
r* = r*(*0) =r(* 0 ) +
K\z'(to)\
1 /c|r'(*0)l
an 'oriented distance' - to the 'left' of the curve. This means that the centre K
of curvature lies on the 'left' if K > 0, and lies on the 'right' if K < 0. 'Left' and 'right' here are determined by the direction of the parametrisation. We shall use the notations t »->• r(£) and t »-> z(t) interchangeably for a curve. Many statements, proofs, and solutions are simpler using the curve t »-> z(t) in the Argand diagram. Definition 9.3. At a regular point r(£o) of the curve t ■-» r(t) the radius of curvature is 7-- for K ^ 0, and is 00 for K = 0.
