LOGARITHMIC CURVE

 Other name: exponential curve.

 Logarithmic version:  Cartesian equation: . Transcendental curve. In the case where a = b: Curvilinear abscissa starting from the point with abscissa a: The Cartesian tangential angle is defined by . Radius of curvature: . Exponential version: Cartesian equation: . Transcendental curve. In the case a = b: Curvilinear abscissa starting from the point with zero abscissa: . The Cartesian tangential angle is defined by . Radius of curvature: .

The logarithmic curve is the plot of the logarithmic function (and also that of the exponential function) or its image by a dilatation.

NSC: curve with constant sub-tangent.

 The logarithmic curve is also characterized by the fact that translating it along its asymptote is equivalent to scaling it perpendicularly to this asymptote. Translation equivalent to a scaling in one direction In the case where a = b, its caustic by reflection for rays perpendicular to its asymptote is the catenary. For rays parallel to the asymptote, it is a pursuit curve.

 The logarithmic curve is the profile a tower (i.e. a solid of revolution) must have in order for the pressure applied on any horizontal section by the upper section to be constant. See tower of constant pressure.

See also the involute of an exponential, the ballistic curve, the catenary, and the tractrix.