4 Due February 2009 Name:
Maple Project 1: Parametric Equations
Math 2720(Mackey)
Please print clearly
Example: Consider the ellipse x2 + 4y 2 = 4. This curve can be parametrized by
x = ( x(t), y(t) ) ,
where x(t) = 2 cos(t),
y(t) = sin(t) for t [0, 2].
Does this parametrization make sense? Check it out by substituing for x and y into the equation for the ellipse. The plot command in Maple can be used to draw this curve: plot( [2*cos(t), sin(t), t = 0..2*Pi] ); Note that no spaces are allowed between the dots in a plot command. The resulting graph is somewhat distorted, because Maple did not use the same vertical and horizontal scales. You can fix this by using plot( [2*cos(t), sin(t), t = 0..2*Pi], scaling = constrained);
Assignment: The curve traced by a point P on a circle of radius r that rolls along the inside of another circle of larger radius R is called a hypocycloid. This curve can be parametrized by
x = ( x(t), y(t) )
where
x(t) = (R - r) cos
(r ) (( r) ) t + r cos 1 - t R R (r ) (( r) ) y(t) = (R - r) sin t - r sin 1 - t R R
This formula is derived in Example 1.9.4 on p.80 of our text. See Figs 1.9.3, 1.9.4. Take the first two non-zero, distinct digits of your WIN number. Set r to be the smaller of these two numbers, and R to be the larger of them. Record them here, with along your simplified equations for x(t) and y(t): R r x(t) y(t)
1. Use the equations of your hypocycloid, and basic knowledge of trigonometric functions to (a) predict the range of values attained by the x and y coordinates of points on your hypocycloid. (3 pts) (b) predict the smallest range of t (starting with t = 0) that should be used to get a complete picture of your hypocycloid. (3 pts) Give clear and complete mathematical justification for your answers. You may experiment in Maple, to provide evidence or confirmation for your answer. But experimentation does NOT explain WHY what you observe is happening, not will it substitute as an answer for this part. 2. Plot your hypocycloid in Maple. Turn in a hard copy of your plot along with the Maple command(s) used to gene...