![]() This is referred to as an Archimedean spiral, after the Greek mathematician Archimedes.Īnother type of spiral is the logarithmic spiral, described by the function r = a ![]() Then the equation for the spiral becomes r = a + k θ r = a + k θ for arbitrary constants a a and k. We can remove this restriction by adding a constant to the equation. Note that when θ = 0 θ = 0 we also have r = 0, r = 0, so the spiral emanates from the origin. Therefore the equation for the spiral becomes r = k θ. In particular, d ( P, O ) = r, d ( P, O ) = r, and θ θ is the second coordinate. However, if we use polar coordinates, the equation becomes much simpler. Īlthough this equation describes the spiral, it is not possible to solve it directly for either x or y. d ( P, O ) = k θ ( x − 0 ) 2 + ( y − 0 ) 2 = k arctan ( y x ) x 2 + y 2 = k arctan ( y x ) arctan ( y x ) = x 2 + y 2 k y = x tan ( x 2 + y 2 k ). Next use the formulasĭ ( P, O ) = k θ ( x − 0 ) 2 + ( y − 0 ) 2 = k arctan ( y x ) x 2 + y 2 = k arctan ( y x ) arctan ( y x ) = x 2 + y 2 k y = x tan ( x 2 + y 2 k ). This leads to r 2 = 6 r cos θ − 8 r sin θ.
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