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Math 112 HW8: Parametric Art due Thursday, August 1 1 Lissajous Curves Named after Jules Antoine Lissajous (1822–1880), Lissajous curves are generated by the para- metric equations x = A cos(at), y = B sin(bt), where varying the amplitudes, A and B, and the frequencies, a and b, gives a huge variety of figures. Note that the only effect of A and B is to stretch or compress the figure in the x- and y-directions, respectively. Without much loss of generality, we take A = B = 1 and focus on the role of a and b, which we take to be integers. 1. Consider the equations x = cos(3t), y = sin(2t). Before using a graphing utility, it’s advisable to find an interval for the parameter values that generates the complete curve. Answer the following questions. (a) What is the period of cos(3t)? (b) What is the period of sin(2t)? (c) Explain why an interval that generates the complete curve is [0, 2π], but not [0, π], even though [0, π] is the larger of the two individual periods. (d) Does the interval [π, π], which also has length 2π, also generate the complete curve? 2. Experiment with different intervals for the parameter using Matlab to confirm that the complete Lissajous curve x = cos(3t), y = sin(2t) is shown below. The Matlab file art.m is provided as a template to show you how to plot parametric equations with Matlab. When you turn in the assignment, answer the following questions after your experimentation. (a) What if you chose a parameter interval of 0 ≤ t ≤ 4π. Do you get the full curve? (b) What if you chose a parameter interval of 0 ≤ t ≤ 7. Do you get the full curve? (c) What if you chose a parameter interval of 0 ≤ t ≤ 6. Do you get the full curve? (d) Why should we use 0 ≤ t ≤ 2π rather than the other intervals above? 1 3. Print a copy of the Lissajous curve with a = 3 and b = 2 using 0 ≤ t ≤ 2π. Mark the point (x, y) where the Lissajous curve starts. Trace the points on the curve as t increases from t = 0 to t = 2π. Draw a direction arrow indicating the motion in t. As you walk along the curve count the number of times that x = cos(3t) and y = sin(2t) complete a full period. How are these counts related to the frequencies a = 3 and b = 2? 4. A nice family of Lissajous curves is generated by taking a to be an odd integer and b = a±1. Match the following parametric equations with the curves in Figure 2. Make your choices without plotting them yourself. (You can confirm your matches with Matlab if you’d like.) A: x = cos(5t), y = sin(4t) B: x = cos(3t), y = sin(4t) C: x = cos(5t), y = sin(6t) D: x = cos(9t), y = sin(8t) 5. Without graphing, describe the curve produced by the equations x = cos(10t), y = sin(10t). Confirm or revise your answer by plotting. Explain why the graph is the way it is. 6. What is the effect of introducing a phase angle in the equations? For example, plot and describe the curve x = cos(5t+π/4), y = sin(4t), in which the x-coordinate has the phase angle π/4. How does this differ from the regular Lissajous curve with a = 5 and b = 4? What about other phase shifts? 7. Graph the curve described by the equations x = cos(2t), y = sin(t). This one doesn’t look like the other curves. Use the double-angle identity and the pythagorean identity to eliminate the parameter t and find an equation relating x and y. You will find the form is x = f(y). 2 2 Hypocycloids Imagine a circle of radius r rolling on the inside of a larger circle of radius R, where R = kr with k > 1. A visual example of this can be seen in this link: hypocycloid video. The path traced out by a point on the circumference of the smaller circle is a hypocycloid, described by the parametric equations x(t) = (k − 1)r ( cos(t) + cos((k − 1)t) k − 1 ) , y(t) = (k − 1)r ( sin(t) − sin((k − 1)t) k − 1 ) . 8. Verify that with r = 1 and k = 3, the result is the deltoid shown below. What is the length of the smallest interval in t that generates the complete curve? 9. Plot the astroid, which is the hypocycloid that results with r = 1 and k = 4. How many points (cusps) does the curve have? 10. Graph several hypocycloids with varying values of k and provide evidence for the fact that if k is an integer, then the resulting curve closes on itself and has k points (cusps). Turn in at least 4 graphs, and label the k you used for each one. 11. Graph several hypocycloids with varying values of k and provide evidence for the fact that if k = p/q, where p and q are integers with no common factors, then the resulting curve closes on itself and has p points (cusps). You may need to adjust the parameter intervals to get a full curve. Turn in at least 4 graphs, and label the k you used for each one. 12. Graph several hypocycloids with varying values of k and provide evidence for the fact that if k is irrational, then the resulting curve never closes on itself and fills a circle of radius R = kr except for a disk of radius R− r in the larger circle (a washer!). Since the curves never close up, the larger the parameter interval, the more of the graph you will see. Turn in at least 4 graphs, and label the k and the parameter interval you used for each one. 13. What value(s) of k will generate a star like the one shown in the figure below? 3 https://en.wikipedia.org/wiki/Hypocycloid#/media/File:Astroid2.gif https://en.wikipedia.org/wiki/Irrational_number#Examples Lissajous Curves Hypocycloids