In these cases the hypocycloids have an infinite number of cusps, so they never return to their starting point. The last two hypocycloids have irrational values for a b. Figure 7.11 shows some other possibilities. The result is a hypocycloid with four cusps. Figure 7.10 corresponds to a = 4 a = 4 and b = 1. This ratio can lead to some very interesting graphs, depending on whether or not the ratio is rational. The ratio a b a b is related to the number of cusps on the graph (cusps are the corners or pointed ends of the graph), as illustrated in Figure 7.11. The period of the second trigonometric function in both x ( t ) x ( t ) and y ( t ) y ( t ) is equal to 2 π b a − b.

This fact explains the first term in each equation above. Then the center of the wheel travels along a circle of radius a − b. In this case we assume the radius of the larger circle is a and the radius of the smaller circle is b. These equations are a bit more complicated, but the derivation is somewhat similar to the equations for the cycloid. x ( t ) = ( a − b ) cos t + b cos ( a − b b ) t y ( t ) = ( a − b ) sin t − b sin ( a − b b ) t. X ( t ) = ( a − b ) cos t + b cos ( a − b b ) t y ( t ) = ( a − b ) sin t − b sin ( a − b b ) t. We study this idea in more detail in Conic Sections. According to Kepler’s laws of planetary motion, the shape of the orbit is elliptical, with the Sun at one focus of the ellipse. After one full year, we are back where we started, and a new year begins. As Earth revolves around the Sun, its physical location changes relative to the Sun. The number of the day in a year can be considered a variable that determines Earth’s position in its orbit. Then, for example, day 31 is January 31, day 59 is February 28, and so on. On January 1 of each year, the physical location of Earth with respect to the Sun is nearly the same, except for leap years, when the lag introduced by the extra 1 4 1 4 day of orbiting time is built into the calendar. Our year lasts approximately 365.25 days, but for this discussion we will use 365 days. Parametric Equations and Their GraphsĬonsider the orbit of Earth around the Sun. Then x and y are defined as functions of time, and ( x ( t ), y ( t ) ) ( x ( t ), y ( t ) ) can describe the position in the plane of a given object as it moves along a curved path. For example, if the parameter is t (a common choice), then t might represent time. The parameter is an independent variable that both x and y depend on, and as the parameter increases, the values of x and y trace out a path along a plane curve. In the two-dimensional coordinate system, parametric equations are useful for describing curves that are not necessarily functions. In this section we examine parametric equations and their graphs. 7.1.4 Recognize the parametric equations of a cycloid.7.1.3 Recognize the parametric equations of basic curves, such as a line and a circle.7.1.2 Convert the parametric equations of a curve into the form y = f ( x ).7.1.1 Plot a curve described by parametric equations.

Derivative of parametric equations