If you sweep the angle one or more complete revolutions in either direction, you get to the same place! Polar coordinates: once the polar axis is positioned, each point in the plane has infinitely many coordinate pairs $\,(r,\theta)\,$ that can represent it.Rectangular coordinates: once coordinate axes are positioned, each point in the plane corresponds to a unique pair of real numbers $\,(x,y)\,$.įor rectangular coordinates, there is a one-to-one correspondence between points in the plane and coordinate pairs $\,(x,y)\,$.įor rectangular coordinates, it is as if the points in the plane and the pairs of real numbers $\,(x,y)\,$ are tied together with strings!.However, when going from a point in a plane to a coordinate pair representing itrectangular coordinates You wouldn't want a coordinate pair specifying (say) one point on Monday, and a different point on Tuesday! Polar coordinates: once the polar axis is positioned, each coordinate $\,(r,\theta)\,$ specifies a unique point in the plane.Rectangular coordinates: once coordinate axes are positioned, each coordinate $\,(x,y)\,$ specifies a unique point in the plane.In both rectangular and polar coordinates, a given coordinate pair specifies a unique point in the plane: move $\,2\,$ units in the direction of the rotated ray.rotate the polar axis $\,90^\circ\,$ counterclockwise.
a different placement of the polar axis.the point with polar coordinates $\,(r,\theta)\,$ is shown in green.the red circle has radius $\,2\,$ its center is the pole.Of course, any unit of length can be used.Įxamples: Plotting Points in Polar Coordinates You can agree that $\,r = 3\,$ means that $\,r\,$ is (say) $\,3\,$ centimeters.In this case, the polar axis acts like (half of) a number line. You can agree that the polar axis ‘carries’ the units.That is, $\,r\,$ is $\,3\,$ WHAT? Here are two approaches you can take: When you say something like ‘$\,r = 3\,$’ there is an implied understanding of One easy way (without introducing any new notation) is to always say something like ‘$\,(x,y) = (1,2)\,$’ versus If you're using both rectangular and polar coordinates in the same discussion, then you need a way to distinguish between the two. Distinguish between rectangular and polar coordinates:.In other words, if the radius is zero, thenregardless of the angleyou get the endpoint of the polar axis. For $\,r \lt 0\,$, move in the opposite direction of the rotated ray.įor any angle $\,\theta\,$, the polar coordinates $\,(0,\theta)\,$ gives the pole.For $\,r \gt 0\,$, move in the direction of the rotated ray.Rotate the polar axis about the pole by an amount $\,|\theta|\,$:įor $\,\theta \gt 0\,$, rotate counterclockwise ( green ray at right).įor $\,\theta = 0\,$, there is no rotation.įor $\,\theta < 0\,$, rotate clockwise ( blue ray at right).įor convenience, call this the rotated ray.
To determine the position of the point with polar coordinates $\,(r,\theta)\,$: Other units (like degrees) can also be used for $\,\theta\,$.
If $\,\theta\,$ has no units, then it is the radian measure of an angle. Let $\,r\,$ (the radius) and $\,\theta\,$ (the angle) be real numbers. the endpoint of the ray is called the poleĬonventionally, the ray is horizontal, pointing to the right.Rectangular/Cartesian/$\,xy\,$ Coordinate SystemĪ polar coordinate system is determined by: It's fun (and instructive) to compare the ways the plane is ‘divided up’ for rectangular versus polar coordinates.Ī more precise discussion of polar coordinates follows. ‘Graph Paper’ for the Rectangular versus Polar Coordinate Systems Than to the (more familiar and widely-used) rules for how to move from the starting placeĭepending on what you're doing, one type of coordinate system might be much simpler to use than another!Īs you'll see, applications involving concentric circles (circles with a common center)Īre probably going to be much better suited to polar coordinates (this section).a starting place (i.e., some sort of ‘origin’).‘Where is this point?’ or ‘How do you get to this point?’ A coordinate system always involves: (Think of a dot on a piece of paper.)Ī coordinate system is used to answer these questions:
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