征求内摆线，外摆线的解

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内外摆线的参数方程和极坐标方程怎么求，图像怎么画？

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Imagine going on a bicycle ride through the country. The tires stay in contact with the road and rotate in a predictable pattern. Now suppose a very determined ant is tired after a long day and wants to get home. So he hangs onto the side of the tire and gets a free ride. The path that this ant travels down a straight road is called a cycloid. A cycloid generated by a circle (or bicycle wheel) of radius $a$ is given by the parametric equations

$$x\left(t\right)=a\left(t-\text{sin}t\right),y\left(t\right)=a\left(1-\text{cos}t\right).$$To see why this is true, consider the path that the center of the wheel takes. The center moves along the $x$-axis at a constant height equal to the radius of the wheel. If the radius is $a$, then the coordinates of the center can be given by the equations$$x\left(t\right)=at,y\left(t\right)=a$$for any value of $t$. Next, consider the ant, which rotates around the center along a circular path. If the bicycle is moving from left to right then the wheels are rotating in a clockwise direction. A possible parameterization of the circular motion of the ant (relative to the center of the wheel) is given by$$x\left(t\right)=\text{−}a\text{sin}t,y\left(t\right)=\text{−}a\text{cos}t.$$(The negative sign is needed to reverse the orientation of the curve. If the negative sign were not there, we would have to imagine the wheel rotating counterclockwise.) Adding these equations together gives the equations for the cycloid.$$x\left(t\right)=a\left(t-\text{sin}t\right),y\left(t\right)=a\left(1-\text{cos}t\right).$$
A wheel traveling along a road without slipping; the point on the edge of the wheel traces out a cycloid.Now suppose that the bicycle wheel doesn’t travel along a straight road but instead moves along the inside of a larger wheel. In this graph, the green circle is traveling around the blue circle in a counterclockwise direction. A point on the edge of the green circle traces out the red graph, which is called a hypocycloid.
Graph of the hypocycloid described by the parametric equations shown.($a=4$ and $b=1.$The result is a hypocycloid with four cusps.) The general parametric equations for a hypocycloid are$$\begin{array}{}\\ \\ x\left(t\right)=\left(a-b\right)\text{cos}t+b\text{cos}\left(\frac{a-b}{b}\right)t\hfill \\ y\left(t\right)=\left(a-b\right)\text{sin}t-b\text{sin}\left(\frac{a-b}{b}\right)t.\hfill \end{array}$$

These equations are a bit more complicated, but the derivation is somewhat similar to the equations for the cycloid. In this case we assume the radius of the larger circle is a and the radius of the smaller circle is b. Then the center of the wheel travels along a circle of radius $a-b.$ This fact explains the first term in each equation above. The period of the second trigonometric function in both $x\left(t\right)$ and $y\left(t\right)$ is equal to $\frac{2\pi b}{a-b}.$

The ratio $\frac{a}{b}$ is related to the number of cusps on the graph (cusps are the corners or pointed ends of the graph). This ratio can lead to some very interesting graphs, depending on whether or not the ratio is rational. The last two hypocycloids have irrational values for $\frac{a}{b}.$ In these cases the hypocycloids have an infinite number of cusps, so they never return to their starting point. These are examples of what are known as space-filling curves.

Graph of various hypocycloids corresponding to different values of $a/b$ .Travels with My Ant: The Curtate and Prolate Cycloids
Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid, called the curtate and prolate cycloids.
First, let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire.

As we have discussed, we have a lot of flexibility when parameterizing a curve. In this case we let our parameter t represent the angle the tire has rotated through. We see that after the tire has rotated through an angle of t, the position of the center of the wheel, $C=\left(x_C,y_C\right),$ is given by

$$x_C=at\text{and}{y}_C=a.$$

Furthermore, letting $A=\left(x_A,y_A\right)$ denote the position of the ant, we note that

$$x_C-x_A=a\text{sin}t\text{and}{y}_C-y_A=a\text{cos}t.$$Then$$\begin{array}{c}{x}_A=x_C-a\text{sin}t=at-a\text{sin}t=a(t-\text{sin}t)\hfill \\ y_A=y_C-a\text{cos}t=a-a\text{cos}t=a(1-\text{cos}t).\hfill \end{array}$$
(a) The ant clings to the edge of the bicycle tire as the tire rolls along the ground.
(b) Using geometry to determine the position of the ant after the tire has rotated through an angle of $t$.

Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable $t$.

After a while the ant is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel, the ant has changed his path of motion. The new path has less up-and-down motion and is called a curtate cycloid. As shown in the figure, we let b denote the distance along the spoke from the center of the wheel to the ant. As before, we let t represent the angle the tire has rotated through. Additionally, we let $C=\left(x_C,y_C\right)$ represent the position of the center of the wheel and $A=\left(x_A,y_A\right)$ represent the position of the ant.

(a) The ant climbs up one of the spokes toward the center of the wheel.
(b) The ant’s path of motion after he climbs closer to the center of the wheel. This is called a curtate cycloid.
(c) The new setup, now that the ant has moved closer to the center of the wheel.

1. What is the position of the center of the wheel after the tire has rotated through an angle of t?
2. Use geometry to find expressions for $x_C-x_A$ and for $y_C-y_A.$
3. On the basis of your answers to parts 1 and 2, what are the parametric equations representing the curtate cycloid?
   Once the ant’s head clears, he realizes that the bicyclist has made a turn, and is now traveling away from his home. So he drops off the bicycle tire and looks around. Fortunately, there is a set of train tracks nearby, headed back in the right direction. So the ant heads over to the train tracks to wait. After a while, a train goes by, heading in the right direction, and he manages to jump up and just catch the edge of the train wheel (without getting squished!).
   The ant is still worried about getting dizzy, but the train wheel is slippery and has no spokes to climb, so he decides to just hang on to the edge of the wheel and hope for the best. Now, train wheels have a flange to keep the wheel running on the tracks. So, in this case, since the ant is hanging on to the very edge of the flange, the distance from the center of the wheel to the ant is actually greater than the radius of the wheel.
   The setup here is essentially the same as when the ant climbed up the spoke on the bicycle wheel. We let b denote the distance from the center of the wheel to the ant, and we let t represent the angle the tire has rotated through. Additionally, we let $C=\left(x_C,y_C\right)$ represent the position of the center of the wheel and $A=\left(x_A,y_A\right)$ represent the position of the ant.
   When the distance from the center of the wheel to the ant is greater than the radius of the wheel, his path of motion is called a prolate cycloid. A graph of a prolate cycloid is shown in the figure.
   
   (a) The ant is hanging onto the flange of the train wheel.
   (b) The new setup, now that the ant has jumped onto the train wheel.
   (c) The ant travels along a prolate cycloid.
4. Using the same approach you used in parts 1– 3, find the parametric equations for the path of motion of the ant.
5. What do you notice about your answer to part 3 and your answer to part 4?
   Notice that the ant is actually traveling backward at times (the “loops” in the graph), even though the train continues to move forward. He is probably going to be really dizzy by the time he gets home!

Notice that the ant is actually traveling backward at times (the “loops” in the graph), even though the train continues to move forward.