 The curvature of a "logarithmic spiral" formula_32 is formula_33

The area of a sector of a curve (see diagram) with polar equation formula_13 is

For a spiral with equation formula_36 one gets


The formula for a "logarithmic spiral" formula_32 is
formula_40

The length of an arc of a curve with polar equation formula_13 is

For the spiral formula_36 the length is

Not all these integrals can be solved by a suitable table.In case of a Fermat's spiral the intgral can be expressed by elliptic integrals only.The arc length of a "logarithmic spiral" formula_32 is formula_46

The inversion at the unit circle has in polar coordinates the simple description