![]() ![]() This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. This spiral describes the shell shape of the chambered nautilus. 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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