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Hypocycloids
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A hypocycloid of four cusps: And Astroid.
[a = 4] and [b = 1]
x = 3cos(t) + cos(3t)
y = 3sin(t) - sin (3t)
(shown in blue)
so 3cos(t) + cos(3t) = 4cos^3(t)
and 3sin(t) - sin(3t) = 4sin^3(t)
[a = 5] and [b = 1]
x = 4cos(t) + cos(4t)
y = 4sin(t) - sin (4t)
(shown in red)
[a = 6] and [b = 1]
x = 5cos(t) + cos(5t)
y = 5sin(t) - sin (5t)
(shown in green)
t {-20, 20}
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More Hypocycloids: [a = n/(n-1)] and [b = 1] [a = n] and [b = 1]
[a = 3/2] and [b = 1]
x = (1/2)cos(t)+cos((1/2)t)
y = (1/2)sin(t)-sin((1/2)t)
(shown in purple)
[a = 3] and [b = 1]
x = (2)cos(t)+cos(2t)
y = (2)sin(t)-sin(2t)
(shown in orange)
[r = a = 3]
x = 3cos(t)
y = 3sin(t)
link to this graph
t {-20, 20}
[a = √2/(√2-1)] and [b = 1]
x = (1/(√2-1))cos(t)+cos(1/(√2-1)t) = (1/0.41)cos(t)+cos((1/0.41)t)
y = (1/(√2-1))sin(t)-sin(1/(√2-1)t) = (1/0.41)sin(t)-sin((1/0.41)t)
(shown in purple)
[a = √2] and [b = 1]
x = (√2-1)cos(t)+cos((√2-1)t) = (0.41)cos(t)+cos((0.41)t)
y = (√2-1)sin(t)-sin((√2-1)t) = (0.41)sin(t)-sin((0.41)t)
(shown in orange)
[r = a = √2]
x = (√2)cos(t) = (1.41)cos(t)
y = (√2)sin(t) = (1.41)sin(t)
(shown in teal)
link to this graph
t {-50, 50}
***Notice that the hypocycloid gets more and more complicated at the T values increase; it never retraces its steps because a's an irrational number.
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[a = -7/6] and [b = 1]
x = (-1/6)cos(t)+cos((-1/6)t)
y = (-1/6)sin(t)-sin((-1/6)t)
(shown in purple)
[a = -7] and [b = 1]
x = (-6)cos(t)+cos(-6t)
y = (-6)sin(t)-sin(-6t)
(shown in orange)
[r = 5]
x = 5cos(t)
y = 5sin(t)
(shown in teal)
link to this graph
t {-20, 20}
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[a = -4/3] and [b = 1]
x = (-1/3)cos(t)+cos((-1/3)t)
y = (-1/3)sin(t)-sin((-1/3)t)
(shown in purple)
[a = -4] and [b = 1]
x = (-3)cos(t)+cos(-3t)
y = (-3)sin(t)-sin(-3t)
(shown in orange)
[r = 2]
x = 2cos(t)
y = 2sin(t)
(shown in teal)
link to this graph
t {-20, 20}
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[a = -3/2] and [b = 1]
x = (-1/2)cos(t)+cos((-1/2)t)
y = (-1/2)sin(t)-sin((-1/2)t)
(shown in purple)
[a = -3] and [b = 1]
x = (-2)cos(t)+cos(-2t)
y = (-2)sin(t)-sin(-2t)
(shown in orange)
[r = 1]
x = 1cos(t)
y = 1sin(t)
(shown in teal)
link to this graph
t {-20, 20}
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x = (1/2)cos(t)+cos((1/2)t)
y = (1/2)sin(t)-sin((1/2)t)
(shown in purple)
[a = 3] and [b = 1]
x = (2)cos(t)+cos(2t)
y = (2)sin(t)-sin(2t)
(shown in orange)
[r = a = 3]
x = 3cos(t)
y = 3sin(t)
t {-20, 20}
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Hypocycloids: [a = irrational] and [b = 1]
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\[a = √2/(√2-1)] and [b = 1]
x = (1/(√2-1))cos(t)+cos(1/(√2-1)t) = (1/0.41)cos(t)+cos((1/0.41)t)
y = (1/(√2-1))sin(t)-sin(1/(√2-1)t) = (1/0.41)sin(t)-sin((1/0.41)t)
(shown in purple)
[a = √2] and [b = 1]
x = (√2-1)cos(t)+cos((√2-1)t) = (0.41)cos(t)+cos((0.41)t)
y = (√2-1)sin(t)-sin((√2-1)t) = (0.41)sin(t)-sin((0.41)t)
(shown in orange)
[r = a = √2]
x = (√2)cos(t) = (1.41)cos(t)
y = (√2)sin(t) = (1.41)sin(t)
(shown in teal)
link to this graph
t {-50, 50}
***Notice that the hypocycloid gets more and more complicated at the T values increase; it never retraces its steps because a's an irrational number.
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Epicycloids
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\Epicycloids: [a = -n/(n-1)] and [b = 1]
[a = -n] and [b = 1]
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\[a = -7/6] and [b = 1]
x = (-1/6)cos(t)+cos((-1/6)t)
y = (-1/6)sin(t)-sin((-1/6)t)
(shown in purple)
[a = -7] and [b = 1]
x = (-6)cos(t)+cos(-6t)
y = (-6)sin(t)-sin(-6t)
(shown in orange)
[r = 5]
x = 5cos(t)
y = 5sin(t)
(shown in teal)
t {-20, 20}
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[a = -4/3] and [b = 1]
x = (-1/3)cos(t)+cos((-1/3)t)
y = (-1/3)sin(t)-sin((-1/3)t)
(shown in purple)
[a = -4] and [b = 1]
x = (-3)cos(t)+cos(-3t)
y = (-3)sin(t)-sin(-3t)
(shown in orange)
[r = 2]
x = 2cos(t)
y = 2sin(t)
(shown in teal)
t {-20, 20}
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[a = -3/2] and [b = 1]
x = (-1/2)cos(t)+cos((-1/2)t)
y = (-1/2)sin(t)-sin((-1/2)t)
(shown in purple)
[a = -3] and [b = 1]
x = (-2)cos(t)+cos(-2t)
y = (-2)sin(t)-sin(-2t)
(shown in orange)
[r = 1]
x = 1cos(t)
y = 1sin(t)
(shown in teal)
t {-20, 20}
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