A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation
of a hyperbola. Because it can be generated by a circle inversion of an Archimedean spiral, it is called reciproke spiral, too.
Pierre Varignon first studied the curve in 1704.Later Johann Bernoulli and Roger Cotes worked on the curve as well.
the hyperbolic spiral with the polar equation
can be represented in cartesian coordinates by
The hyperbola has in the --plane the coordinate axes as asymptotes. The hyperbolic spiral (in the --plane) approaches for the origin as asymptotic point. For the curve has an asymptotic line (see next section).
From the polar equation and one gets a representation by an equation:
the curve has an
From vector calculus in polar coordinates one gets the formula for the polar slope and its angle between the tangent of a curve and the tangent of the corresponding polar circle.
For the hyperbolic spiral the polar slope is
The curvature of a curve with polar equation is
From the equation and the derivatives and one gets the curvature of a hyperbolic spiral:
The length of the arc of a hyperbolic spiral between can be calculated by the integral:
The area of a sector (see diagram above) of a hyperbolic spiral with equation is:
The inversion at the unit circle has in polar coordinates the simple description: .
For the two curves intersect at a fixpoint on the unit circle.
The osculating circle of the Archimedean spiral at the origin has radius (see Archimedean spiral) and the center . The image of this circle is the line (see circle inversion). Hence:
The diagram shows an example with .
Consider the central projection from point onto the image plane . This will map a point to the point
The image under this projection of the helix with parametric representation
is the curve with the polar equation
which describes a hyperbolic spiral.
For parameter the hyperbolic spiral has a pole and the helix intersects the plane at a point . One can check by calculation that the image of the helix as it approaches is the asymptote of the hyperbolic spiral.
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