Nina del Ser - Mathematics
octopus |
15
Loops, spikes, or smooth curvatures?
The velocity vector
So far, the investigation we have conducted has led us to a position in which we are able to predict how many
axes of symmetry any curve generated by the Octopus will have, given the value of Ω (see the “Changing the
angular speeds of the rotors” section above for further reference). We know, for instance that when Ω is set to
5 rad/s (and Β 1), the shape generated should have five axes of symmetry and look somewhat like this:
figure 33
However, how can we be sure whether the shape will have loops, spikes or a smooth curvature? By this, I
mean the following conditions respectively:
figure 34: loops
figure 35: spikes
figure 36: smooth curvature
One way to answer this question is to look at the velocity vectors of the individual primary and secondary
rotors, whose positions vectors can been worked as shown in the “�
Solving the original problem” section for a
given set of rotor lengths and speeds. Since the individual rotors both sweep out circles we know that the
velocity vectors are perpendicular to their respective position vectors. In addition, their magnitudes may
simply be worked out via the relationship |v| rΩ, where r is the magnitude of the position vector (in this case,
simply the length of the rotor), whilst Ω is the angular speed of the individual rotor vectors. Note that the
angular speed for the secondary rotor vector, v s , is the sum of the angular speeds of the primary and sec-
ondary rotors (see figure 2 for reference). Figure 37 illustrates this concept with a set-up in which both rotors
are rotating anti-clockwise; the position vectors are shown as solid lines, whilst the velocity vectors are shown
as dashed:
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