Fermat's spiral

A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral (for which this distance is invariant) and the logarithmic spiral (for which the distance between turns is proportional to the distance from the center). Fermat spirals are named after Pierre de Fermat. Their applications include curvature continuous blending of curves, modeling plant growth and the shapes of certain spiral galaxies, and the design of variable capacitors, solar power reflector arrays, and cyclotrons. The representation of the Fermat spiral in polar coordinates is given by the equation for . The two choices of sign give the two branches of the spiral, which meet smoothly at the origin. If the same variables were reinterpreted as Cartesian coordinates, this would be the equation of a parabola with horizontal axis, which again has two branches above and below the axis, meeting at the origin. The Fermat spiral with polar equation can be converted to the Cartesian coordinates by using the standard conversion formulas and . Using the polar equation for the spiral to eliminate from these conversions produces parametric equations for one branche of the curve: and the second one They generate the points of branches of the curve as the parameter ranges over the positive real numbers. For any generated in this way, dividing by cancels the parts of the parametric equations, leaving the simpler equation . From this equation, substituting by (a rearranged form of the polar equation for the spiral) and then substituting by (the conversion from Cartesian to polar) leaves an equation for the Fermat spiral in terms of only and : Because the sign of is lost when it is squared, this equation covers both branches of the curve. A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedean and hyperbolic spiral.