If k is rational, the curve is closed and of finite length, but if irrational, it is not closed and is of infinite length.A polar grid with several angles labeled in degrees Moreover, when k is even, the entire rose is traced for &theta ∈ 0, 2 &pi, but when k is odd, the rose is traced for &theta ∈ 0, &pi. When c is the integer k, there are 2 k petals when k is even, and k petals when k is odd. In the rose, the parameter c does not have to be an integer as it is in Table 7.1.2. (See, for example, the Wikipedia article here. There is a lemniscate of Gerono, so the modifier "of Bernoulli" is essential when referencing the curve shown in Table 7.1.2. Table 7.1.2 lists five standard polar curves.Ĭircle with radius a and center at the origin &theta ∈ 0, 2 &pi Ĭircle with radius a / 2 and center at a / 2, 0 &theta ∈ 0, &pi Ĭircle with radius a / 2 and center at 0, a / 2 &theta ∈ 0, &pi X = r cos θ = f θ cos θ y = r sin θ = f θ sin θįortunately, Maple has efficient tools for graphing a polar curve, obviating the need for making these algebraic changes. Alternatively, the explicit form r = f &theta defines the curve parametrically in Cartesian coordinates via the equations Ĭonsequently, one way to graph a polar curve is to convert it to Cartesian coordinates, and apply a tool that graphs implicit functions. Converting the explicit form results in the equation x 2 + y 2 = f arctan y / x or the more precise x 2 + y 2 = f arctan y, x. Converting the implicit form to Cartesian coordinates results in the equation F x 2 + y 2, arctan y / x = 0, or the more precise equation F x 2 + y 2, arctan y, x = 0. Often, this equation can be rearranged to the explicit form r = f &theta. Īn equation of the form F r, &theta = 0 defines a curve in polar coordinates. When y = &pi, sin x + &pi = − sin x and cos x + &pi = − cos x. Sin x + y = sin x cos y + cos x sin yĬos x + y = cos x cos y − sin x sin y This convention is consistent with the trig formulas This introduction concludes with the following note.Īlthough the radius r = x 2 + y 2 is positive, the convention − r, &theta = r, &theta + &pi is observed. The horizontal red lines are the lines y = constant, which become the red curves on the left. On the left in Figure 7.1.4 the vertical green lines are the lines x = constant, which become the green curves on the left. P3:=display(p1,p2,scaling=constrained,labels=,view=,axis=],tickmarks=,labelfont=):įigure 7.1.4 Gridlines from the Cartesian plane mapped to the polar plane P3:=display(P1,P2,scaling=constrained,view=,labels=,labelfont=): It is rare that such an interpretation is needed in the calculus, so the figure is not an essential one in the theory under development. It shows how the Cartesian gridlines map over to the polar plane. ![]() ![]() Print(display(,k=0.18)], insequence=true, scaling=constrained)) įigure 7.1.3 The curve r = 2 1 − cos &theta morphed between the r &theta -plane and the xy -planeįigure 7.1.4 is included for the sake of completeness. P4:=textplot(, style=point, symbol=solidcircle,symbolsize=15, color=red): Table 7.1.1 Cartesian and polar coordinates ![]() Polar coordinates are related to Cartesian coordinates by the formulas in Table 7.1.1. The numbers r and &theta are the polar coordinates of the Cartesian point x, y. In a Cartesian plane, the point x, y is at a distance r from the origin, and the ray from the origin to x, y (shown in red) makes an angle &theta with the positive x -axis. Chapter 7: Additional Applications of IntegrationĪn alternative to rectangular Cartesian coordinates, polar coordinates are predicated on the right-triangle trigonometry in Figure 7.1.1.
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