As we predicted, the curve maintains its elliptical shape and the only effect of the offset angle Α is rotation
Nina del Ser - Mathematics
about the origin which, had we set up our coordinate system taking into account Α, would not have occurred.
Α
octopus
| 9
obvious
By investigating further we can show that the angle by which the ellipse is rotated is 2 . The most
way to do this is to find an expression for the magnitude of the seat’s position vector, which will then enable
us to find the time t when |r| reaches its maximum value. Since we know that for the case when Α 0 the
ellipse’s semi-major axis coincides with the x-axis (i.e t 0), the value of the new time t at which |r| is at a
maximum will also correspond to the rotation of the original ellipse about the origin. From figure 5:
r x cost bcos Α t
r
2
r y cost bsin Α t
cost b cos Α t 2 cost b sin Α t 2
r 2 cos 2 t sin 2 t b 2 cos 2 Α t b 2 sin 2 Α t 2bcostcos Α t 2bsintsin Α t
r 2 1 b 2 2bcos Α 2t
1 b 2 2 bcos Α 2 t
r
(3)
From (3) we can conclude that the maximum values of |r| occur when cos(Α-2t) 1, at t
its minimum values occur when cos(Α-2t) -1, at t
Α
2
Π
2
and
Α
2
3 Π
.
2
Α
2
and
Α
2
Π, whilst
Also, because of the oscillating nature
of the cosine function, we know that |r| will decrease and increase at exactly the same rate from one maxi-
mum to the next (since it is periodic with a time period of Π). These ideas suggests that a path generated by a
Α
Π
seat with any offset angle Α will have two axes of symmetry at t and t Α 2 + 2 respectively, which corre-
2
sponds with the image generated in figure 6. We can now use a rotational matrix which rotates the elliptical
Α
shape by clockwise (thus making the x and y axes its axes of symmetry, as when Α 0) to check that the
2
shape generated really will be an ellipse:
Α
2
Α
sin 2
cos
sin
cos
Α
2
Α
2
cos 2 cost bcos Α t sin 2 sint bsin Α t
cost bcos Α t
·
Α
Α
sint bsin Α t
sin 2 cost bcos Α t cos 2 sint bsin Α t
Α
Α
cos t bcos 2 t
Α
2
Α
sin
2
LHS
Α
t bsin 2 t
Α
1 b cos t
Α
2
Α
1 b sin
2
LHS
t
(4)
(4) can be now be converted into a Cartesian equation using the same method as what was used to obtain (2):
x 1 b cos Α t
y 1 b sin Α t
2
x 2
1 b 2
Α
cos 2 t
y 2
2
x 2
1 b 2
y 2
1 b 2
1 b 2
2
Α
sin 2 t
2
cos 2 2 t + sin 2 2 t
x 2
1 b 2
Α
y 2
1 b 2
Α
1
This confirms our belief that the shape observed in figure 6 is, indeed, an ellipse.
Conclusion
In this section, it has been shown that changing both the ratio of the rotor lengths (except when the lengths are
euqal) and the size of the offset angle Α has very little influence on the basic elliptical shape produced by a
seat on the octopus ride. In the next section we will therefore assume that these parameters are relatively
insignificant and start looking instead at whether changing the ratio of the speeds of the primary and sec-
ondary rotor produces any interesting effects.
113