cardioid a member of the limaçon family of curves, named for its resemblance to a heart its equation is given as and where convex limaҫon a type of one-loop limaçon represented by and such that dimpled limaҫon a type of one-loop limaçon represented by and such that inner-loop limaçon a polar curve similar to the cardioid, but with an inner loop passes through the pole twice represented by and where lemniscate a polar curve resembling a figure 8 and given by the equation and one-loop limaҫon a polar curve represented by and such that and may be dimpled or convex does not pass through the pole polar equation an equation describing a curve on the polar grid. Glossary Archimedes’ spiral a polar curve given by When multiplied by a constant, the equation appears as As the curve continues to widen in a spiral path over the domain. The formula that produces the graph of an Archimedes’ spiral is given by See (Figure).The formulas that produce the graphs of rose curves are given by and where if is even, there are petals, and if is odd, there are petals.The formulas that produce the graphs of a lemniscates are given by and where See (Figure).The formulas that produce the graphs of an inner-loop limaçon are given by and for and See (Figure).The formulas that produce the graphs of a one-loop limaçon are given by and for See (Figure).The formulas that produce the graphs of a cardioid are given by and for and See (Figure).Some formulas that produce the graph of a circle in polar coordinates are given by and See (Figure).The zeros of a polar equation are found by setting and solving for See (Figure).The maximum value of a polar equation is found by substituting the value that leads to the maximum value of the trigonometric expression.Polar equations may be graphed by making a table of values for and.If an equation fails a symmetry test, the graph may or may not exhibit symmetry.
There are three symmetry tests that indicate whether the graph of a polar equation will exhibit symmetry.It is easier to graph polar equations if we can test the equations for symmetry with respect to the line the polar axis, or the pole.For example, suppose we are given the equation We replace with to determine if the new equation is equivalent to the original equation. In the first test, we consider symmetry with respect to the line ( y-axis).
Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of to determine the graph of a polar equation. By performing three tests, we will see how to apply the properties of symmetry to polar equations.
If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. Symmetry is a property that helps us recognize and plot the graph of any equation. Recall that the coordinate pair indicates that we move counterclockwise from the polar axis (positive x-axis) by an angle of and extend a ray from the pole (origin) units in the direction of All points that satisfy the polar equation are on the graph. Just as a rectangular equation such as describes the relationship between and on a Cartesian grid, a polar equation describes a relationship between and on a polar grid.