Nina del Ser - Mathematics
octopus |
v s v p v s v p v s v p
figure 38 figure 39 figure 40
figure 41 figure 42 figure 43
figure 44 figure 45 figure 46
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Convexity
The acceleration vector
Just like the velocity vectors discussed above, the acceleration vectors of the Octopus’ two rotors play an
important role in defining the shapes of its curves. The net acceleration vector, a (the sum of the two individ-
ual acceleration vectors a p and a s ) defines whether the curve generated will be concave or convex. Figures 47
and 48 illustrate what I mean by concavity and convexity:
figure 47: y x 3 concave
figure 48: y x 2 convex
Convexity is defined by the fact that the second derivative is always either positive or negative, but cannot be
vary between the two. The moment at which an Octopus curve stops being concave and becomes convex is
defined in a similar way: the net acceleration vector a must be greater than or equal to 0 at all times (it cannot
be negative because there is only one direction-towards the center- in which an octopus can be convex, unlike
figure 48, since y x 2 is just as convex as y x 2 ). We can show that this occurs when |a p | |a s | by compar-
ing vectors physically, just like in the section on velocity vectors, however, there is another way which uses
the net |a| derived from equation (5):
Let us call the lengths of the primary and secondary rotors B and b,
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