History of the Asteroid

The year is 1674, the man is Olaus (or Ole) Roemer a Dutch Astronomer following the grand tradition of Galileo. Roemer, who is most famous for first measuring the speed of light to a reasonable degree, is a good example of the well rounded Renaissance Man. He was an astronomer, but also dabbled in physics and Geometry the Renaissance language of science. During this time machinery of ropes, pulleys and gears was a “new” area, to be dabbled in by any man interested in modern science. Roemer did his share of dabbling and in 1674 postulated that if one is trying to create a machine (say for example a large clock tower), then one should use hypocycloids (wheels running within wheels) in the gears since this creates the least amount of friction between two gears, which allows them to run more efficiently. While talking about this he happened to mention using an hypocycloid with a large wheel of radius 4 times that of the smaller wheel’s radius, which moves on the inside of the larger wheel.

Figure 1. Double Generation of the Asteroid.

The next and pretty much only other famous discovery regarding the asteroid was done by Daniel Bernoulli in 1725, when he came up with his famous double

Parameterizing the Asteroid

One advantage that we have today that the ancient Greek mathematicians didn’t have is our ability to describe a curve compactly as an algebraic or trigonometric equation or set of equations. What would have taken Euclid 4 pages to describe (for example the parabola) can today, be universally understood by four simple symbols, namely y = x². Thus it is in our best interest to, whenever possible, derive a set of equations which describe a given curve. We will now derive the parametric equations for the asteroid. We start by looking at Figure 2, in this case, like in Roemer’s original idea, we have a small circle of radius a, inside a large circle of radius 4a. To get our equations we first notice that as the small circle moves around the large circle , it’s center is always at a distance 3a from the origin. This tells that as the small circle moves it traces out another circle of radius 3a (See Figure 3). Thus we note, that the curve is actually created by the combined movement of the two circles. So, we can simplify our task by finding the parametric equations of the two circles and adding them together to get the equations of the Asteroid.

Figure 2. Setup.

Figure 3. Setup.

Since the parametric equations of a circle are x = r cos(θ), and y = r sin(θ), and here r = 3a, we see that the parametric equations of the larger, dotted circle is x = 3acos(θ), and y = 3asin(θ). Now, for the smaller circle (see Figure 4).

Figure 4. Smaller Circle Parametrization.

Figure 5. Relating the Angles.

For the X-value equation we have,

Figure 6: The Asteroid.

Cartesian Coordinates

Once you have the parametric equations of the asteroid, getting the cartesian coordinates is no trouble at all. We simply take the cube root of x and y, then square them, and add them. We get,

x2∕3  +  y2∕3 =	(4a) cos2(θ) + (4a) sin2(θ)
=	(4a) (cos2(θ) + sin2(θ))
=	(4a)

x2∕3  +  y2∕3 = R

(7)

Interesting Properties of the Asteroid

(paraphrased from Lockwood)

The evolute of the asteroid is also an asteroid, but it is rotated 45^∘, and twice the size.

The asteroid can also be drawn as the envelope of a fixed line sliding with its ends attached to the x, and y axes.

“The Asteroid is the hypocycloid formed by rolling a circle of radius 1a or 3a on the inside of one of radius 4a”

The Asteroid is the envelope of a family of ellipses where the sum of the major and minor axes is constant.

The Asteroid has arclength of 6a, and the area inside it is πa²

References

Animations of the Astroid

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TheDoubleRaduisAstroid