“As the crow flies” is an idiom used to describe the most direct path between two points. The polar coordinate system is a useful parametrization of the plane that, rather than describing horizontal and vertical position relative to the origin in the usual way, describes a point in terms of distance from the origin and direction. The origin is also known as the pole (hence polar coordinates).
Let \(\overline{OP}\) be a line segment from the origin to a given point \(P\) in the plane. The length of \(\overline{OP}\) is the distance (or radius) \(r\) from the origin to \(P\text{.}\) The polar axis is a ray starting at the origin.
To define the "direction" of \(P\text{,}\) we form an angle \(\theta\) by letting the polar axis serve as the initial ray and \(\overrightarrow{OP}\) as the terminal ray. We will set the positive \(x\)-axis as the polar axis and assume the movement in the positive direction is counter-clockwise (as in trigonometry). Notice that, unlike in the rectangular (or Cartesian) coordinate system, the polar coordinates \((r,\theta)\) for a point are not unique, as we could turn either way to face a given point (or even spin around a number of times before facing that direction).
Furthermore, by allowing \(r\) to be negative, we can also "walk backwards" to get to a point by facing in the opposite direction. Rather than the grid lines defined by specific values for \(x\) and \(y\) in the rectangular coordinate system, specific values of \(r\) correspond to circles of radius \(r\) centered about the origin, and specific values of \(\theta\) correspond to lines going through the pole (called radial lines).
Plot the Cartesian point \(P=(x,y)=(\sqrt{3},-1)\) and draw line segments connecting the origin to \(P\text{,}\) the origin to \((x,y)=(\sqrt{3},0)\text{,}\) and \(P\) to \((x,y)=(\sqrt{3},0)\text{.}\)
If a polar graph is symmetric about the \(x\)-axis, then if the point \((r,\theta)\) lies on the graph, then the point \((r,-\theta)\) or \((-r, \pi-\theta)\) also lies on the graph.
If a polar graph is symmetric about the \(y\)-axis, then if the point \((r,\theta)\) lies on the graph, then the point \((r,\pi-\theta)\) or \((-r,-\theta)\) also lies on the graph.
If a polar graph is rotationally symmetric about the origin, then if the point \((r,\theta)\) lies on the graph, then the point \((-r,\theta)\) or \((r,\pi+\theta)\) also lies on the graph.
