Hypocycloids ,Hypotrochoids and Polar Cuves Speaker: Tuyetdong Phan-Yamada Glendale College CMC3-South 2015 Annual Spring Conference A hypocycloid is a curve traced out by a point A on one (red) circle (B, b) as it rolls around another fixed (green) circle (O, a). æ a - b ö x = (a - b)cost + bcosç t÷ è b ø a - b y = (a - b)sint - bsin( t) b CMC3-South 2015 Annual Spring Conference The Meaning of b/a If we simplify b/a=r/s, then 1. s is the number of cusps of the hypocycloids. 2. 2πr is the period of the hypocycloids CMC3-South 2015 Annual Spring Conference An Epicycloid is a Hypocycloid. If b<0 or b>a, it is an epicycloid. a + c x = (a + c)cost - ccos( t) c a + c y = (a + c)sint - csin( t) c CMC3-South 2015 Annual Spring Conference Co-ratios If b’/a=r’/s, b/a=r/s and r’+ r =s, then they generate the same graph. CMC3-South 2015 Annual Spring Conference The Double Generation Theorem In hypocycloids they generate the same graph but their directions are reversed. CMC3-South 2015 Annual Spring Conference Circulant Graphs if s is a prime Direction(0246135) If b/a=r/s, this will naturally produce a circulant graph with s vertices, where vertices i and (i+r) mod s are adjacent. CMC3-South 2015 Annual Spring Conference Circulant Graphs A complete circulant graph with 7 vertices CMC3-South 2015 Annual Spring Conference Simple Hypo/Epicycloids If b/a= 1/s or b/a=(s-1)/s, then it is an s-cusped simple hypo/epicycloid. CMC3-South 2015 Annual Spring Conference Astroids x2 y2 The family of ellipses + =1 m2 (a - m)2 is inscribed in the astroid [a,b,t] for 0<m<a CMC3-South 2015 Annual Spring Conference Non-Simple Hypo/Epicycloids All cusps lie at even πr/ s rays from the positive x-axis whereas the intersections between arcs are on odd πr/ s rays from the positive x-axis. CMC3-South 2015 Annual Spring Conference Hypotrochoids æ a - b ö x = (a - b)cost + d cosç t÷ è b ø æ a - b ö y = (a - b)sint - dsinç t÷ è b ø 0<b<a, d is the distance from B to A. CMC3-South 2015 Annual Spring Conference Epitrochoids æ a + b ö x = (a + b)cost - d cosç t÷ è b ø æ a + b ö y = (a + b)sint - d sinç t÷ è b ø An epitrochoid is a hypotrochoid with b<0 or b>a CMC3-South 2015 Annual Spring Conference Epitrochoids Motion of Jupiter (in the middle loop) vs Epitrochoids CMC3-South 2015 Annual Spring Conference Polar Curves For any hypotrochoids [a,b,d,t], if d=a-b, it is a polar curve. æ a - b ö x = (a - b)cost + d cosç t÷ è b ø æ a - b ö y = (a - b)sint - d sinç t÷ è b ø Replace d=a-b and apply some trig. identities. 2b - a Let w = t 2b æ a ö x = 2(a - b)cosç w÷cosw è 2b - a ø æ a ö y = 2(a - b)cosç w÷sinw è 2b - a ø CMC3-South 2015 Annual Spring Conference Polar curves For any hypotrochoid of the form æ a ö æ s ö x = 2(a - b)cosç w÷cosw = 2(a - b)cosç w÷cosw è 2b - a ø è 2r - s ø b r where = æ a ö æ s ö y = 2(a - b)cosç w÷sinw = 2(a - b)cosç w÷sinw a s è 2b - a ø è 2r - s ø 1. Let n=s/(2r-s), then it is a rose curve with n petals if n is odd and 2n petals if n is even. 2. Otherwise, the curve is a rosette with all the intersection lying on concentric circles (O, Ri) for i=1, 2, 3, …N | 2r - s | -1 N = if s is odd. 2 | 2r - s | N = -1 f s is even. 2 æ ip ö R = 2(a - b)cosç ÷ i è 2r - s ø CMC3-South 2015 Annual Spring Conference Rose Curves s a n = = 2r - s 2b - a CMC3-South 2015 Annual Spring Conference Rosettes Rosette with a hypotrochoid Rosette with an epitrochoid CMC3-South 2015 Annual Spring Conference “3-D” Polar Curve x=r(t)cos(t); y=r(t)sin(t) CMC3-South 2015 Annual Spring Conference References 1. T. Phan-Yamada and E. Gwin "Hypocycloids and Hypotrochoids," AMATYC Journal, 09/2014 1. T. Phan-Yamada, W. Yamada III, "Exploring Polar Curves with GeoGebra," Mathematics Teacher, 10/2012 1. Hall, L. (1992) Trochoids, Roses, and Thorns – Beyond the Spirograph, The College Mathematical Journal, 23(1), 20-35 1. Davis, D.M. (1993). The Nature and Power of Mathematics, (pp. 55-58) New York: Dover Publications. 1. Weisstein, E. W. "Circulant Graph." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CirculantGraph.html 2. Weisstein, E. W. "Hypocycloid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Hypocycloid.html CMC3-South 2015 Annual Spring Conference.