Nina del Ser - Mathematics
6
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The ellipse
Influence of the length of rotor on the basic elliptical shape
When this set of parametric equations is plotted using Mathematica the following interesting shape is
observed:
figure 3
The curve traced out by a seat under the conditions set out in the original problem seems to be an ellipse. Let’s
see, therefore, if we can manipulate this set of parametric equations in order to try to arrive at the Cartesian
equation of an ellipse,
x 2
a
2
y 2
b 2
1.
x 1.5cost
2
x 2
1.5 2
2
2
y 2
cos 2 t
x 2
1.5
2
y 0.5sint
y 0.5 2 sin 2 t
x 1.5 cos t
2
0.5 2
y 2
0.5 2
x 2
1.5 2
sin 2 t
cos 2 t sin 2 t
y 2
0.5 2
1
As can be seen from the manipulations performed above, this curve is, indeed, an ellipse. In addition, we
realise that the elliptical shape is not constricted to this specific set of primary and secondary rotor lengths,
since equation (1) can be generalised to include any ratio of lengths. For the sake of simplicity, we keep the
length of the primary rotor constant at 1 meter, and only change that of the secondary, denoting its length as b
(the degree of freedom in the ratios of lengths is not diminished in any way). Therefore our generalised set of
parametric equations becomes:
x 1 b cos t
x
2
1 b
2
y 1 b sint
y 2
cos 2 t
x 2
1 b
2
x 2
1 b 2
1 b 2
y 2
1 b 2
sin 2 t
cos 2 t + sin 2 t
y 2
1 b 2
1
(2)
(2) shows that the curve will preserve its elliptic character, irrespective of the value given to b, as long as b
110
1. When b 0, it is clear that (2) generates a unit circle (which is therefore simply a special case of the ellipse,
in which the two foci lie on top of each other), centered at the origin. Figure 4 shows what happens when the