If r is the curve parametrised by arc length (i.e.| r'(s) | = 1; see natural parametrization) then the center of curvature at s is
Such parametrisation is usually between difficult and impossible, but it's still feasible to access r". If x is any (reasonably differentiable) parametrisation, and s gives arc length over the same parameter, then the desired r would give r(s(t)) = x(t) which if differentiated twice gives
r'(s(t))s'(t) = x'(t)
r''(s(t))s'(t)2 + r'(s(t))s''(t) = x''(t)
which we rearrange to
Recognising that
s'(t) = | x'(t) |
eliminates the need to know s itself, thus eliminating the integration in which the analytic impossibilities lie.
However, the mechanisms of evolution are less well understood, and it is these mechanisms that are described by several theories of evolution.
Evolution does not proceed from any basic randomness, although genetic changes are not coupled to selection and may be characterised as "random" relative to selection pressures, nor do they anticipate the needs of a species.
One of the favorite anti-evolutionist challenges to the existence of transitionalfossils is the supposed lack of transitional forms in the evolution of the whales.
What they don't appreciate is that this rate of evolution is all that is required to produce the diversity of all living things from a common ancestor.
Evolution makes predictions about what we would expect to see in the fossil record, comparative anatomy, genetic sequences, geographical distribution of species, etc., and these predictions have been verified many times over.
Evolution is supported by a wide range of observations throughout the fields of genetics, anatomy, ecology, animal behavior, paleontology, and others.
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