Hypotrochoid (or more commonly known as a spirograph)A model of the famous toy known as a spirograph. A circle of radius r (see blue circle below) is allowed to roll around the inside of a circle of radius R (see black circle below). We assume that the circle rolls without slipping. If we pick a point inside or on the blue circle (see red point), at a distance d from the circles center, and watch the path this point traces out as the cirlce rolls then the resulting curve is called an hypotrochoid. In the particular case when d=r the curve is called an hypocycloid. You can change the radius of the two circles below the applet. Dragging the sliders (labeled "Move Me") in the applet will rotate the blue circle and change the position of the red point.
Explorations:1. Compare the shapes of the curves when the radii R and r are (R, r) = (5,3), (10, 6), (15, 9) and (100, 60). What can you conclude about the shape of the curve and how it is related to the numbers R and r? 2. With R = 15 and r = 7 drag the first slider to the right until t = 7. Notice the red curve has joind back up with itself again (we say the curve is now closed). This means it took 7 full revolutions around the black circle for the blue circle to return to its exact original position. With (R, r) = (17, 6) or (23, 8) how many revolutions will it take before the curve closes iteself? Do you observe a pattern? Can you make a conjecture (guess) as to how the number of revolutions until the curve closes is related to R and r? 3. Test your conjecture from question 2 for the numbers (R, r) = (21, 7), (15, 9), and (16, 8). What do you observe now? Make a new conjecture based on your new obeservations. (Question 1 may help here.) 4. Move the second slider to d = 1. This makes the curve touch the black circle. Notice for (R, r) = (15, 7) the curve touches the black circle 15 times. Determine how many times the curve touches the circle when (R, r) = (17, 6) or (23, 8). Make a conjecture and test your conjecture on (R, r) = (21, 7), (15, 9), and (16, 8). If your conjecture doesn't hold then adjust it and try again. (Question 1 may help here.)
Jamie Mulholland |