Try these other examples to investigate this assumption. How?įrom our previous examples, the number of leaves is twice nwhen n is even. What does the 8 in the equation do?īy now, you should realize that the number of leaves is determined by n. For example, the polargraph of the equation What is the effect of a and n on the graph?įirst, a determines the length of each petal. Next we want to explore polar equations that produce graphs similar to thatof roses. Similarly,as |b| decreases the size of the cardioid gets smaller. Try graphing these polar equations to test our assumption.Īs |a| increases the size of the cardioid increases. With further investigation and exploration of various a and b,we discover that if |a| = |b|, the cardiod will increase or decreasein size depending on the value of a and b. Hint: note the relationship between a and b. What do we have to change to make our cardiod larger or smaller? Hence, changing a to itsopposite has no effect on the graph of a limacon.Ĭhanging + b to - b has the same effect on the cardiod aswith the other limacons that is a reflection occurs.įor example, the following polar equations are reflections or mirror imagesof each other. Thisis true for the other types of limacons. The following graphs are severalexamples of cardiods.Ĭhanging the +1 to a -1 had no effect on the graph of the cardiod. The graph seems to flatten on one side of the limacon.ĬardioidsCardioids are the fourth type of limacon. The following polar equations and their graphs are examples of convexlimacons. The change from positiveto negative has the same effect as in the previous graphs. The dimpled limacons have the shape of an apple. As b decreases, the limacon gets smaller.ĭimpled LimaconsSeveral examples of dimpled limacons are shown along with their associatedpolar equations. The graphs for the following polar equations are shown.Īs b grows larger, the limacon also gets bigger. Lastly, we want to explore what happens as b grows larger. The other change, making a smaller, creates a larger inner loop. What if we change the positive sign to a negative sign and make asmaller?Ĭhanging the positive sign to a negative sign has the effect of reflectingthe limacon and producing a mirror image of itself. Notice the change from cos to sin rotates the limacon 90 degrees in theclockwise direction. The following graphs show examples ofeach type of limacon.

The next type of interesting graphs that we will explore are limacons.Limacon ('LEE-ma-sahn') is Old French for 'snail.' Youwill see why the name is appropriate when we graph the limacons. Notice as a grows larger, the spiral becomes tighter and smaller.In addition, each spiral originates from the same point. The following graph represents logarithmic spirals for a=1, 3, 10, 17,and 32. More specifically, consider the following example of a logarithmic spiral. In nature, you may have noticed that shells of some sea creatures are shapedlike logarithmic spirals particularly the nautilus.Ĭlick herefor more information regarding logarithmic spirals particularly the chamberednautilus. Notice the distance between the successive coilsis greater as the spiral grows. The distance between successive coils of a logarithmic spiral is notconstant as with the spirals of Archimedes. In general, logarithmic spirals have equations in the form We let a=.5, 1, 4, 10, 25 and graph each associated polar equation.Īs a grows larger, the spiral becomes larger and r is greater. We want to explore this polar equation as a grows larger. We can classify any polar equation that has the form Ĭonsideranother example of a Spiral of Archimedes: They have theappearance of a coil of rope or hose with a constant distance between successivecoils. Spirals of ArchimedesPolar graphs of the form r = at + b where a is positive and b is nonnegative are called Spirals of Archimedes. Note: In the following polar equations, ø=t. This investigation is going to explore several interesting graphs and theirassociated polar equations.

Then each point P can be located by assigning to it a polar coordinatepair (r, ø), in which the first number, r, gives the directed distancefrom O to P and the second number, ø, gives the directed angle fromthe initial ray to the segment OP: To define polar coordinates, we first fix an origin O and an initialray from O. The spirals described on shells, and called concho-spirals,are such as would result from winding plane logarithmic spirals on cones.

Archimedes spiral equation