Math 133 Polar Coordinates Stewart 10.3/I,II Points in polar coordinates. The first and greatest achievement of modern mathematics was Descartes description of geometric objects b numbers, using a sstem of coordinates. In the simplest example, Cartesian or rectangular coordinates on the plane locate a point P in terms of two coordinate measurements x and : how far over and how far up the point is, moving parallel to the marked axes. We loosel sa that P is the pair (x, ), because the coordinates tell how to get there from the origin. The name P is like identifing a house as the Jones place, whereas the coordinates are like saing the third house to the right on the second street down. In this section, we learn how to locate the point P using a different pair of measurements, the polar coordinates (r, θ). The radius r is the distance from the origin. The angle θ is measured couterclockwise in radians, from the positive x-axis ra to the ra from orgin through the point P. This is like pointing to the house 500 ards in that direction. Unlike rectangular coordinates, the polar coordinates of a point are multivalent, having man equivalent versions because of the ambiguit of angles. For example, the point (x, ) = (0, 1) on the positive -axis corresponds to (r, θ) = (1, ), where θ = means a 1 4 turn counterclockwise from the positive x-axis. However, we could equall well get to this point b a 3 4 turn clockwise, giving (r, θ) = (1, 3 ). In fact, we could get to the point b 1 1 4 turns counterclockwise, 1 3 4 clockwise, etc. In general, we must consider all angles that differ b a multiple of a full turn as the same, meaning the define the same point: (r, θ) = (r, θ+n) for an integer n. It is also useful to allow negative radius: ( r, θ) means to move out along the line at angle θ, but in the opposite direction from the positive ra, along the ra θ ± ; thus: ( r, θ) = (r, θ ± ). There is even more ambiguit for the origin (x, ) = (0, 0), which can be written as (r, θ) = (0, θ) for an angle at all. Both tpes of coordinates completel locate a point, so given either (x, ) or (r, θ),
we can find the other b simple trigonometric formulas: Given (r, θ) = find (x, ) with Given (x, ) = find (r, θ) with { x = r cos(θ) = r sin(θ). { r = x + θ = arctan( x ). Here we get θ from the defining formula tan(θ) = x, and we could equall well use sin(θ) =, etc., alwas remembering we can change θ to θ+n. Also, since x + < arctan(θ) <, we must define arctan( ) =, arctan( ) = ; and we must adjust the angle b ± if the point lies left of the x-axis. Curves in polar coordinates. An geometric object in the plane is a set (collection) of points, so we can describe it b a set of coordinate pairs. For example, the unit circle C is the set of all points at distance 1 from the origin; the coordinates of these points form the set of all pairs (x, ) which satisf the Pthagorean equation x + = 1: C = {(x, ) such that x + = 1}. Again, the equalit of these sets is meant loosel: a pair of numbers like (x, ) = ( 3 5, 4 5 ) is not literall a geometric point on the circle, but it identifies a point b means of the rectangular coordinate sstem. Now, polar coordinates are speciall adapted to describe round, turn shapes centered at the origin, and the make the equation of the circle as simple as possible: C = {(r, θ) such that r = 1}. example: The line x + = 1 is not at all circular or centered at the origin, and its equation becomes complicated in polar coordinates: x + = 1 = r cos(θ) + r sin(θ) = 1 = r = 1 cos(θ)+ sin(θ) = 1 sec(θ 4 ). The last equalit follows from the identit cos(θ 4 ) = cos(θ) cos( 4 ) + sin(θ) sin( 4 ) = 1 (cos(θ) + sin(θ)). Similar reasoning gives the polar form of a general linear equation. For ax+b = 0, we get θ = α for the constant angle α = arctan( b a ). For c 0, we get: ax + b = c = r = c a cos(θ) + b sin(θ) = c a + b sec(θ α), Summarizing: ( ) ( ) { x arctan ( ) if x 0 x θ = arcsin = arccos = x + x + arctan ( ) x + sgn() if x < 0 The last formula is expressed in computer languages as atan(,x). There is no separate curve connecting the points: the curve is just all the points.
example: Consider the Archimedean spiral, the shape of the groove on an old vinl record (solid blue line). This is defined b a point moving steadil outward as it turns around the origin: in parametric polar coordinates, (r(t), θ(t)) = (t, t) for t 0, meaning at time t the radius and angle are both t. Converting into rectangular coordinates: (x(t), (t)) = (r(t) cos θ(t), r(t) sin θ(t)) = (t cos(t), t sin(t)). Deparametrizing gives the rθ and x-equations: r = θ + n for integer n = ( x + = arctan + n x) = = x tan x +. For example, we can tell the points (x, ) = (n, 0) are on the spiral, because 0 = n tan (n) + 0. Actuall, the last equation defines the spiral together with its natural continuation back past its center point, namel the 1 turn rotation of the original spiral (dashed red line). Sketching polar graphs. Remember that a function f is just a rule taking input numbers to output numbers. It does not care what letters we use for inputs and outputs, or how we interpret those letters geometricall. We usuall illustrate the function b drawing its rectangular graph = f(x), in which f controls the height above each point on the x-axis. But another wa to illustrate this function is the polar graph r = f(θ), in which f controls the radius r along each ra θ. We can sketch the polar graph r = f(θ) b plotting points, just as for a rectangular graph. For example, consider the polar curve: r = sin(θ). We imagine the plane as a field, with us standing at the origin. We look along the positive x-axis and draw a point at radius 0, namel the origin itself. As we increase θ > 0, turning slowl to the left, we increase the radius as sin(θ) increases. The radius tops out at 1 when θ = along the positive -axis; and as we continue to turn the point comes back in to the origin when θ =. After that, as we turn toward negative directions the radius becomes negative, so we draw points behind us, in fact retracing the original curve.
Actuall, this is a computer plot to turn the qualitative stor above into an accurate graph. But we reall could do this b hand, b plotting r for some standard θ: deg 0 30 45 60 90 10 135 150 180 θ 0 6 4 3 3 3 4 5 6 r 0 0.5 0.7 0.9 1 0.9 0.7 0.5 0 As we said, the angles θ give negative radius and re-plot the same points. The computer does the same kind of plotting, but with man more points, so wh bother with our piddl hand sketches and point plotting? Because of a ver great danger for anone who uses mathematics. If ou let the computer do the thinking, not just the calculating, ou are read to accept an bizarre wrong answer without an wa to check it. Then one tpo error will escalate until our scientific paper has to be retracted, our compan s expenses are ten times what ou predicted, our bridge collapses, our rocket crashes. Don t let it happen! Before accepting the computer s answer, ou must check the expected answer qualitativel against a stor or sketch, and quantitativel b plotting sample points. From the sketch, we ma guess this curve is a circle, which we verif b converting to an x-equation, and simplifing b completing the square: r = sin(θ) = x + = x + = x + = 0 = x + ( 1 ) + ( 1 ) = ( 1 ) = x + ( 1 ) = ( 1 ). Indeed, this is a circle of radius 1 centered at (x, ) = (0, 1 ).
More sketching. We sketch the curve: r = 1 + sin(θ). This is more complicated, so instead of computing a table of θ and r values, we start b drawing the function r = f(θ) = 1 + sin(θ) in our usual wa as a rectangular graph, labeling the horizontal and vertical axes b r and θ because that is how we intend to draw them later in the polar graph. Even without precise values, we can sketch the polar graph b adjusting the radius according to the heights of the rectangular graph (dotted lines). The blue lobe is traced b θ [ 4, 3 4 ]; then the green lobe is for θ [ 3 4, 7 4 ]. Graphicall, we crush the entire horizontal θ-axis in the rectangular graph to the origin in the polar graph, spreading out the radial lines like a fan.