The formulas on the right in Table 7.1.1 comprise the forward mapping that gives the polar coordinates of the point whose Cartesian coordinates are $\left(x\,y\right)$. Looked at this way, the imposition of polar coordinates is a mapping of the Cartesian point to the polar point $\left(r\,\mathrm{\θ}\right)$ that lives in a rectangular polar plane.

The formulas on the left in Table 7.1.1 comprise the inverse mapping that gives the Cartesian coordinates of the point whose polar coordinates are $\left(r\,\mathrm{\θ}\right)$. It is customary to pull the rectangular grid lines of the polar plane back onto the Cartesian plane. From this perspective, the Cartesian point does not move, it simply gets a new name.

The right-hand side in Figure 7.1.2 shows the rectangular gridlines of the polar plane; the left-hand side, the inverse image of these gridlines pulled back to the Cartesian plane by the formulas on the left in Table 7.1.1. The red and green curves on the left in Figure 7.1.2 are generally called the coordinate curves for polar coordinates.

On the right in Figure 7.1.2 the vertical green lines are the lines $r\=\mathrm{constant}$, which become the green concentric circles on the left. The horizontal red lines are the lines $\mathrm{\θ}\=\mathrm{constant}$, which become the radial lines on the left. Hence, each Cartesian point on the left has two names, either the Cartesian name $\left(x\,y\right)$ or the polar name $\left(r\,\mathrm{\θ}\right)$. In this sense, imposing polar coordinates is a change of coordinates, that is, a change of names for the points in the Cartesian plane.

On the right in Figure 7.1.3 the curve $r\=2\left(1-\cos \left(\mathrm{\θ}\right)\right)$ is drawn in the $\mathrm{r\θ}$-plane. This plane has a rectangular grid, with horizontal grid lines $\mathrm{\θ}\=\mathrm{constant}$ and vertical grid lines $r\=\mathrm{constant}$. The curve is given as $r\=r\left(\mathrm{\θ}\right)$, but graphed as if it were given as $\mathrm{\θ}\=\mathrm{\θ}\left(r\right)$. On the left in Figure 7.1.3 the same curve is graphed in the $\mathrm{xy}$-plane on which the polar coordinate curves (circles and radial lines) have been superimposed. The animation in the middle of Figure 7.1.3 is an attempt at visualizing how the "bell" on the right becomes the "bean" on the left.$$

Figure 7.1.4 is included for the sake of completeness. It shows how the Cartesian gridlines map over to the polar plane. 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.

On the left in Figure 7.1.4 the vertical green lines are the lines $x\=\mathrm{constant}$, which become the green curves on the left. The horizontal red lines are the lines $y\=\mathrm{constant}$, which become the red curves on the left.

This introduction concludes with the following note.

This convention is consistent with the trig formulas

$\sin \left(x\+y\right)$$\=$$\sin \left(x\right)\cos \left(y\right)+\cos \left(x\right)\sin \left(y\right)$

$\cos \left(x\+y\right)$$\=$$\cos \left(x\right)\cos \left(y\right)-\sin \left(x\right)\sin \left(y\right)$

When $y\=\mathrm{\π}$, $\sin \left(x\+\mathrm{\π}\right)\=-\sin \left(x\right)$ and $\cos \left(x\+\mathrm{\π}\right)\=-\cos \left(x\right)$.

An equation of the form $F\left(r\,\mathrm{\θ}\right)\=0$ defines a curve in polar coordinates. Often, this equation can be rearranged to the explicit form $r\=f\left(\mathrm{\θ}\right)$. Converting the implicit form to Cartesian coordinates results in the equation $F\left(\sqrt{x^2\+y^2}\,\arctan \left(y\/x\right)\right)\=0$, or the more precise equation $F\left(\sqrt{x^2\+y^2}\,\arctan \left(y\,x\right)\right)\=0$. Converting the explicit form results in the equation $\sqrt{x^2\+y^2}\=f\left(\arctan \left(y\/x\right)\right)$ or the more precise $\sqrt{x^2\+y^2}\=f\left(\arctan \left(y\,x\right)\right)$.

Consequently, one way to graph a polar curve is to convert it to Cartesian coordinates, and apply a tool that graphs implicit functions. Alternatively, the explicit form $r\=f\left(\mathrm{\θ}\right)$ defines the curve parametrically in Cartesian coordinates via the equations

Fortunately, Maple has efficient tools for graphing a polar curve, obviating the need for making these algebraic changes.

Table 7.1.2 lists five standard polar curves.

Table 7.1.3 lists equations that define conic sections given in polar coordinates. Each such conic has a focus at the origin, and the directrix parallel to a coordinate axis.

The distance from a point on the conic to the directrix is given by $d$; the distance between such a point and the focus, divided by $d$ is the eccentricity, $e$. (This symbol for the eccentricity is not the exponential "e".) Both $d$ and $e$ are taken as positive numbers.

The conic is an ellipse, parabola, or hyperbola accordingly as $e<1\&comma;e\&equals;1$, or $e\&gt;1$, respectively.

For the ellipse, $a\&equals;\frac{e d}{1-e^2}\&comma;b\&equals;\frac{e d}{\sqrt{1-e^2}}\&comma;c\&equals;\sqrt{a^2-b^2}\&equals;\frac{e^{2}d}{1-e^2}$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively; and $c$ is the distance from the center of the ellipse to a focus (of which there are two). For the hyperbola, $c\&equals;\sqrt{a^2\&plus;b^2}$.