PolarCurves：极坐标曲线

Polar Curves
1. Flowers: Verify that the polar graph of r(t)=sin3t, for t in the interval [0, 2π], is shaped
like a flower with three petals.
a) Would a domain smaller than [0, 2π] produce the same graph? Explain.
b) What values of t correspond to the petal in the first quadrant?
c) Plot the polar graph of r(t)=sin2t for t in the interval [0, 2π]. How many petals do you
see?
d) Would a domain smaller than [0, 2π] produce the same graph? Explain.
e) Experiment with the polar graphs of r(t)=sin kt for other integer values of k. Make a
conjecture about the connection between the integer k and the number of petals in the polar graph. Also, conjecture about the smallest domain needed to draw the complete flowe r for each value of k.
2. Limaçons: Polar graphs of the form r(t)=1+k sin t are called limaçons, French for snails,
possibly because of their vague resemblance to the shape of these animals for certain values of k.
a) See what you think of the resemblance by sketching the polar graph of r(t)=1+2sin t,
for t in the interval [0, 2π].
b) Your sketch for k = 2 should show a smaller loop inside a larger loop. Plot the
rectangular graph of r(t)=1+2sin t.
c) Note where this rectangular curve crosses the x-axis. For what values of x is the curve
below the x-axis?
d) What portion of the polar graph corresponds to the part of the rectangular graph that is
below the x-axis?
e) Sketch the limaçons for k = 0.75 and k = 0.25 to see other possible shapes. One should
be dented, the other egg-like. Sketch the limaçons for several additional values of k. Besides being looped, dented, or egg-like, did you find any other shapes occurring as limaçons?
Describe what happens to the shape of the limaçons as k changes.
3. Spirals: If r(t) is positive and is strictly increasing as a function of t, the polar graph if r will
be shaped like a spiral. For the spirals considered here, we will be mainly interested in their behavior as they cross the positive x-axis, for positive values of t.
a) Plot both the polar and rectangular graphs of r(t)=ln t. For which values of t does the
polar curve cross the positive x-axis?
b) What are the values of r at those points?
c) Find a function r(t) whose polar graph is a spiral which meets the positive x-axis at
precisely the integer values: 1, 2, 3, ... . Begin your search by listing the values of t for which the spiral will cross the positive x-axis, and then consider what the value of r will need to be at these points. Only when you think you’ve got the right function, sketch the graph for values of t in the interval [0, 8π] to be sure. What was your function r(t)?
d) Find a function r(t)whose polar graph meets the positive x-axis at precisely the powers of
2: 1, 2, 4, 8, ... . Proceed as you did with arithmetic spirals, listing the values of t for which the spiral crosses the positive x-axis, and then considering what the value for r will need to be
at those points. Sketch your graph for values of t in the interval [0, 8π] to confirm you have the right function. What was your function r(t)?
4. a) In problem 1 above, you conjectured the number of petals on various flowers. Which
numbers of petals never occur in a polar graph of r(t)=sin kt?
b) Create polar graphs that have the number of petals that can never occur in a graph of
r(t)=sin kt. How did you do this?
This lab is an adaptation of Lab 24 in the book Learning by Discovery: A Lab Manual for Calculus, Anita Solow, Editor.。