(i.) If ψ denotes the angle which the radius vector drawn from the origin makes with the tangent to a curve at a point whose polar coordinates are r, θ and if p denotes the perpendicular from the origin to the tangent, then
cos ψ = dr/ds, sin ψ = rdθ/ds = p/r,
where ds denotes the element of arc. The curve may be determined by an equation connecting p with r.
(ii.) The locus of the foot of the perpendicular let fall from the origin upon the tangent to a curve at a point is called the pedal of the curve with respect to the origin. The angle ψ for the pedal is the same as the angle ψ for the curve. Hence the (p, r) equation of the pedal can be deduced. If the pedal is regarded as the primary curve, the curve of which it is the pedal is the “negative pedal” of the primary. We may have pedals of pedals and so on, also negative pedals of negative pedals and so on. Negative pedals are usually determined as envelopes.
(iii.) If φ denotes the angle which the tangent at any point makes with a fixed line, we have
r2 = p2 + (dp/dφ)2.
(iv.) The “average curvature” of the arc Δs of a curve between two points is measured by the quotient
| Δφ Δs |
| Δφ |
| Δs |
where the upright lines denote, as usual, that the absolute value of the included expression is to be taken, and φ is the angle which the tangent makes with a fixed line, so that Δφ is the angle between the tangents (or normals) at the points. As one of the points moves up to coincidence with the other this average curvature tends to a limit which is the “curvature” of the curve at the point. It is denoted by
| dφ ds |
| dφ |
| ds |