Lemniscate The curve described in polar coordinates by r2 = cos(2θ) is called a lemniscate. a) For what values of θ does there exist such a point (r, θ)? b) For what values of θ is r at its minimum length? c) For what values of θ is r at its maximum length? d) Use the information you have gathered to sketch a rough graph of this curve. Solution It is helpful to have a graph of cos(2θ) to look at while working this problem. a) For what values of θ does there exist such a point (r, θ)? We can’t take the square root of a negative number (in this class), so r is undeﬁned anywhere cos(2θ) is negative. The function f (θ) = cos(θ) has negative outputs on the interval π/2 < θ < 3π/2 and in general on all intervals of width π centered on an odd multiple of π. The graph of the function g(θ) = cos(2θ) is a horizontally compressed copy of the graph of f (θ), so cos(2θ) is zero on the intervals of width π/2 centered on odd multiples of π/2; on the intervals π/4 < θ < 3π/4 and 5π/4 < θ < 7π/4, etc. �� For all other values of θ, the points cos(2θ), θ and − cos(2θ), θ are points on the lemniscate. b) For what values of θ is r at its minimum length? When cos(2θ) = 0, r(θ) = 0. This is the smallest value r can attain (because r is deﬁned in terms of a square root function). Considering the graph cos(2θ), we observe that cos(2θ) = 0 when 2θ is an odd multiple of π/2; i.e. when θ is an odd multiple of π/4. The radius r has length 0 when θ = ..., −3π/4, −π/4, π/4, 3π/4, ... c) For what values of θ is r at its maximum length? When� cos(2θ) = 1, r = ±1. The value of cos(2θ) is never greater than 1, so r = ± cos(2θ) is never longer than 1 unit. Going back to our graph of cos(2θ), we see that the radius r is greatest when θ is a multiple of π: θ = ..., −2π, −π, 0, π, 2π, ... 1 d) Use the information you have gathered to sketch a rough graph of this curve. We know that the curve (r, θ) sweeps out a path for the intervals kπ − π/4 < θ < kπ + π/4 and does not exist outside of those intervals. On each of these intervals, r starts at its minimum value of 0, increases to its maximum 1, then decreases again. We could try to ﬁnd coordinates for the points (r, θ) when θ = kπ ± π/8, or we could make a guess at the path the curve follows between extremes. Our end result should look a little bit like the ﬁgure below. 2

Lemniscate

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