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A spiral staircase in the Cathedral of St. John the Divine.Several helical curves in the staircase project to hyperbolic spirals in its photograph.. A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals or decreasing angles of Archimedean spirals.
The Diagram shows representative examples of the different curves. The centre is marked by ‘O’ and the radius from O to the curve is shown when θ is zero. The value of ε is zero unless shown. The first and third forms are Poinsot's spirals; the second is the equiangular spiral; the fourth is the hyperbolic spiral; the fifth is the epispiral.
Download QR code; Print/export ... Archimedean spiral. Cornu spiral. Fermat's spiral. Hyperbolic spiral. Lituus. Logarithmic spiral. Piecewise constructions
For <, spiral-ring pattern; =, regular spiral; >, loose spiral. R is the distance of spiral starting point (0, R) to the center. R is the distance of spiral starting point (0, R) to the center. The calculated x and y have to be rotated backward by ( − θ {\displaystyle -\theta } ) for plotting.
It is the locus of the center of a hyperbolic spiral rolling (without skidding) on a straight line. It is the involute of the catenary function, which describes a fully flexible, inelastic, homogeneous string attached to two points that is subjected to a gravitational field. The catenary has the equation y(x) = a cosh x / a .
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
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