Calculus 1A – Laboratory Project 1

For the exercises below, explain in your answers in your own words, so that a novice mathematician could easily follow your work with ease.  That is, narrate your equations and diagrams in a thoroughly convincing manner.

 

1.      A hypocycloid is a curve traced out by a fixed point P on a circle C of radius b as C rolls on the inside of circle with center O and radius a.  For example, the situation with a = 4 , b = 1 is shown below (assuming that P starts at ( 4,0) .)  However, in parts (a) through (d) below, assume that a and b are arbitrary.

a.       Parameterize the center point, Q, of the moving circle in terms of a, b, and θ.  Note that Q is the terminal point of the vector

b.      Express the angle φ in terms of a, b, and θ. 

c.       Express the vector  in terms of a, b, and θ. 

d.      Add vectors  and  to find the coordinates of the point P, as parameterized by θ.

2.      Consider the case where a = 5 is the radius of the fixed circle and b = 1 is the radius of the rolling circle.

a.       Complete a table for θ, x, and y where  and use it to graph the path of the hypocycloid the point P  traces in the x,y plane.

b.      Use trigonometric identities to show that part of the curve can be parameterized by .

3.      Note that dividing both a, and b by b will give parametric equations of the form   which describes a curve of the same shape on a scale normed to fit the unit circle.  Consider these parametric equations in the problems below:

a.       Sketch graphs for a = 7 and b = 1,2,…,6

b.      Sketch a graphs for  and .  What conclusions can you draw about how the values of a and b determine the shape of the graph?

c.       How is the shape of the curve different if  is irrational or rational?  Explain and provide illustrative examples.

4.      If one circle rolls outside another circle, the curve traced out by P is called an epicycloid.  Find parametric equations for the epicylcolid given by the path of a point, P, on a circle of radius b rolling around the outside of a circle of radius a.

5.      Investigate the how the parameters a and b in #4 determine the shape of the epicycolid.  Use analysis similar to that in problems 1-3.