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Hypocycloids and Hypotrochoids Hypocycloids and Hypotrochoids Tuyetdong Phan-Yamada Elysus Gwin Glendale Community College Los Medanos College Introduction This paper compiles several theorems relating to hypocycloids and epicy- cloids with the aim of providing a project for calculus students interested in a further understanding of polar curves. Experiments lead to finding connec- tions between hypocycloids, hypotrochoids, and polar curves. Throughout, there is a direct connection between characteristics of the parametric equations and easily identifiable visual aspects of the corresponding curves. GeoGebra is used extensively. Relationships with circulant graphs and ancient Greek models of planetary orbits are also described. Readers can download a set of ready-made GeoGebra files (hypocycloid-trochoid.ggb and hypo-epicycloid.ggb) to explore the concepts discussed in this article. See Phan-Yamada (n.d.). These files along with their step-by-step instruc- tions can be used as a project for calculus students. Hypocycloids A hypocycloid [a, b, t] is a curve traced out by a point P on one (red) circle (B, b) as it rolls around inside another fixed (green) circle (O,a ) for a > 0. If the initial position of P is (a, 0) and the parameter t is chosen as in Figure 1, Figure 1a . a = 3, b = 1, b/a = 1/3 it can be shown that the parametric equations of the hypocycloid are ab− ab− x=−+( ab )cos tb cos t and y=−−( ab )sin tb sin t , where b b a and b are real numbers and a > b > 0. Figures 1a, 1b, and 1c depict some hypocycloids with given values. If the red circle is rolled about the exterior of the fixed green circle, the resulting curve is called an epicycloid. The standard parametric equations of ac+ epicycloids are x=+−( ac )cos tc cos t and c ac+ y=+−( ac )sin tc sin t , where a and c are real numbers. Here, c c represents the radius of the outer circle that is being moved. Returning to hypocycloids, if we loosen the restrictions on b, allowing it to be greater than a or negative, then the resulting curve is an epicycloid. We notate hypocycloids as [a, b, t], which is an adaptation of the notation presented in Hall (1992). Theorem 1: Figure 1b. a = 2, b = 1 .2, b/a = 3/5 The hypocycloid [a, b, t] is an epicycloid for b < 0 and b > a. www.amatyc.org 5 By reversing the argument in theorem 1, it follows that all epicycloids are hypocycloids, hence, they inherit all properties, as found in the following discus- sion of hypocycloids. Properties of Hypocycloids Theorem 2: ab− Let [a, b, t] be the hypocycloid x=−+( ab )cos tb cos t and b ab− y=−−( ab )sin tb sin t that is traced by a fixed pointP on the circum- b ference of the circle (B, |b|) rolling around a circle (O, a), where a > 0 and b ≠ 0 are real numbers with a rational ratio. If b/a = r/s, where s and r are relatively prime natural numbers, then a) The hypo/epicycloid intersects the circle (O, a) exactly s times. These intersections are called cusps of the hypo/epicycloid. Figure 1c .a = 7, b = 2, b/a = 2/7 b) The period of the hypocycloid is 2πr. The Figure 1 set illustrates examples of 3-, 5- and 7-cusped hypocycloids with s = 3, 5, and 7, respectively. There are no 1- or 2-cusped hypocycloids. If a = 2b, meaning b/a = 1/2, then a – b = b. Thus, the parametric equations of hypocycloid become x = 2bcos t and y = 0. This means point P traces along the x-axis with the interval [–a, a]. If a = b, then a – b = 0. Thus, the parametric equations of the hypocycloid be- come x = a and y = 0, so the graph of g is a point P(a, 0). However, we have 1-cusped epicycloids for b/a = –1 and 2-cusped epicy- cloid for b/a = –1/2, as shown in Figures 2a, 2b, and 2c. Figures 3a and 3b shows incomplete and complete curves for the case b/a = 3/5, meaning r = 3. Using graphing software such as GeoGebra, we observe that with a = 4 hypocycloids, b/a = 2/5 and b/a = 3/5 are exactly the same graphs as shown in Figure 2a . b/a = 1/2; segment Figures 4a and 4b. In general, if b’/a= r’/s and b/a = r/s and r’ + r =s, then they generate the same graph. Such ratios are called co-ratios of s-cusped hypocycloids. However, their directions are opposite. If they both start at (a, 0), one of them will roll clockwise and the other will roll counterclockwise, as shown in Figures 5a, 5b, and 5c. In fact, both the circle radius b and the circle radius (a – b) rolling on the inside of a fixed circle (O,a ) generate the same hypocycloid, but their directions are reversed. In other words, they have the same x-coordinates but their y-coor- dinates are opposite of each other. This property, known as the double generation theorem, was discovered by Daniel Bernoulli in 1725. Theorem 3: The Double Generation Theorem For any hypocycloids [a, b, t] and [a, b’, w], where a, b, b’ are positive integers, if b’ + b = a then the two hypocycloids coincide. We note in passing that certain kinds of hypocycloids can be made to cor- Figure 2b. b/a = –1 or b/a = 2; cardioid (1-cusped ) respond to cyclic circulant graphs (Weisstein, n.d.a.). In particular, cusps may be labeled 0, 1, 2, …, (s – 1) counterclockwise about the hypocycloid with each representing a vertex. Then two vertices are adjacent if the corresponding cusps 6 MathAMATYC Educator ~ Vol. 6, No. 1 ~ September 2014 Figure 2c . b/a = –1/2 or b/a = 3/2; nephroid (2-cusped) Figure 3a. Incomplete curve g as t ∈ (0, 2π ) Figure 3b. Complete curve g as t ∈ (0, 6π ) Figure 4a . b/a = 2/5 www.amatyc.org 7 Figure 4b. b/a = 3/5 Figure 5a . b/a = 1/7; direction: (0123456) Figure 5b. b/a = 6/7; direction: (0654321) Figure 5c. Red: b/a = 1/7 or 6/7; blue: b/a = 2/7 or 5/7; black: b/a = 3/7 or 4/7 8 MathAMATYC Educator ~ Vol. 6, No. 1 ~ September 2014 are connected by the curve. This will naturally produce a circulant graph with s vertices, where vertices i and (i + r) mod s are adjacent, i = 0, 1, 2, …, (s – 1). For example, if b/a = 2/7 then r = 2 and s = 7. Hence, the hypocycloid has 7 points and the corresponding graph will have 7 vertices. As t grows, the curve traces out a path through the cusps 2 to 4 to 6 and so on. We abbreviate this as (0246135). The hypocycloid generated by b/a = 5/7 will have the same shape but is traced out in the other direction, with the order of vertices being (0531642). (See the blue curve in Figure 5c.) By superimposing hypocycloids with the same cusps but different r values, we can further represent every other circulant graph on s vertices. Up to isomor- phism, there are (s – 1)/2 such hypocycloids for each prime s. Hypocycloids or epicycloids without intersections of any arcs are called s-cusped simple hypocycloids (or epicycloids). Weisstein (n.d.b.) gives the names for some special hypocycloids (or epicycloids), as shown in Figure 6a, Figure 6a . b/a = 1/3 or b/a = 2/3; 6b, and 6c. deltoid: 3-cusped hypocycloid Theorem 4: Let [a, b, t] be a hypocycloid (or epicycloid). If b/a = ±1/s, then the curve is an s-cusped simple hypocycloid (or epicycloid). b 1 We will see the curve cross itself when ≠± , as shown in Figures 7a, as 7b, and 7c. Note that all cusps lie on green lines an even π/s from the positive x-axis whereas the intersections between arcs are on either blue lines an odd π/s from the positive x-axis. Hypotrochoids A hypotrochoid [a, b, d, t] is a curve traced out by a point D on the line that coincides with a diameter of the circle (B, b) as it rolls around along another (fixed) circle (O,a ). Let the distance between D and B be d, and let a, b, and d Figure 6b. b/a = 1/4 or b/a = 3/4; astroid: 4-cusped hypocycloid be non-zero rational numbers such that a > 0 and d > 0. The parametric equa- ab− tions for a hypotrochoid are x=−+( ab )cos td cos t and b ab− y=−−( ab )sin td sin t . As with hypocycloids, an epitrochoid is a b hypotrochoid when b < 0 or b > a. Figures 8a, 8b, and 8c show some examples hypocycloids or epicycloid (blue curve) and hypotrochoids or epitrochoids (black curve). Davis (1993) includes a discussion of the use of cycloids to describe the motion of planets by ancient Greek mathematicians (pp. 55–58). Briefly, Ptolemy used the epicycle (literally, on the circle in Greek) to explain the motion of planets around the Earth. Archimedes and Plutarch relate an earlier theory of planetary motion around the sun put forward by Aristarchus. Orbits centered on the sun conform to epitrochoids. Figure 9a shows the motion of Saturn (the outer curve), Jupiter (the middle), and Mars (the inner) as viewed from the Earth and assuming a circular orbit. Jupiter’s path has 11 loops, curled outward with no intersections between the big loops.
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