Transcript Hypotrochoid
Thomas Wood Math 50C
- A curve traced by a point P fixed to a circle with radius r rolling along the inside of a larger, stationary circle with radius R at a constant rate without slipping. -The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop.
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P
Sir Isaac Newton – English Mathematician (1643-1727) Philippe de la Hire – French Mathematician (1640-1718) Girard Resargues – French Mathematician (1591-1661) Gottfried Wilhelm von Liebniz – German (1646-1716) Mathematician
Wankel Rotary Engine Spirograph
First I found equations for the center of the small circle as it makes its motion around the inside of the large circle. I found that the center point C of the small circle traces out a circle as it rolls along the inside of the circumference of the large circle.
As the point C travels through an angle theta, its x-coordinate is defined as (Rcos ϴ - rcosϴ) and its y coordinate is defined as (Rsin ϴ - rsinϴ) . The radius of the circle created by the center point is (R-r).
The more difficult part is to find equations for a point P around the center.
As the small circle goes in a circular path from zero to 2 π , it travels in a counter-clockwise path around the inside of the large circle. However, the point P on the small circle rotates in a clockwise path around the center point. As the center rotates through an angle theta, the point P rotates through an angle phi in the opposite direction. The point P travels in a circular path about the center of the small circle and therefore has the parametric equations of a circle. However, since phi goes clockwise, x=dcos ϕ y=-dsin ϕ . and
Inner circle Adding these equations to the equations for the center of the inner circle gives the parametric equations x=Rcos ϴ-rcosϴ +dcos ϕ y=Rsin ϴ-rsinϴ-dsin ϕ for a hypotrochoid.
Get phi in terms of theta Since the inner circle rolls along the inside of the stationary circle without slipping, the arc length r ϕ must be equal to the arc length R ϴ.
r ϕ=R ϴ ϕ=R ϴ/r However, since the point P rotates about the circle traced by the center of the small circle, which has radius (R-r), ϕ is equal to (R-r) ϴ r
Therefore, the equations for a hypotrochoid are
x
R
cos
r
cos
d
cos(
R r
r
)
y
R
sin
r
sin
d
sin(
R
r
)
r
-2 -3 -4 -5 5 4 3 0 -1 2 1 -6
Properties and Special Cases
When r=(R-1), the hypotrochoid draws R loops and has to go from 0 to 2 π*r radians to complete the curve. As d increases, the size of the loop decreases. If d ≥ r , there are no longer loops, they become points.
For example, R=13, r=12, d=5 R=6, r=5, d=5 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -4 -2 0 x-axis 2 4 6 -4 -2 0 x-axis 2 4 6
2 1 4 3 -3 -4 0 -1 -2 If d=r, the point P is on the circumference of the inner circle and this is a special case of the hypotrochoid called the hypocycloid. For a hypocycloid, if r (which is equal to d) and R are not both even or both odd and R is not divisible by r, the hypocycloid traces a star with R points.
R=5, r=2, d=2 R=20, r=7, d=7 -10 -15 0 -5 15 10 5 -4 -2 0 x-axis 2 4 6 -25 -20 -15 -10 -5 0 x-axis 5 10 15 20 25
R=12, r=6, d=3 0 -2 -4 -6 6 4 2 -8 -6 -4 -2 R=2r 0 x-axis 2 4 6 8
R=5, r=7, d=2 0 -1 -2 -3 3 2 1 -4 -3 -2 r>R -1 0 x-axis 1 2 3 4 5
Butler, Bill. “Hypotrochoid.” Durango Bill’s Epitrochoids and Hypotrochoids. 26 Nov, 2008.
“Hypotrochoid.” 1997. 6 Dec, 2008.
“Spirograph.” Wikipedia. 2008. 7 Dec, 2008.
Wassenaar, Jan. “Hypotrochoid.” 2dcurves.com. 2005. 6 Dec, 2008.