> 2. DIFFERENTIABILITY AND CONTINUITY (Nicholas Thompson)
Nick: Let me be your math consultant! Taught that stuff at Caltech many years!!
The mathematicians are horn swogglin' you with mis-understood function theory!
A'course the f'n roof is continuous. If it weren't the rain would come through!
It is trivial to write a continuous function, f(x) defined for 0<x <=c and g(x)
defined for c<x<1 with f(c) = g(c), with the peak at x=c and a different slope
for x=c, than for x>c. But the function is continuous. Just like a roof ridge.
A geometric function has, at each point, some degree of continuity, denoted by
C N, where N is the order of the first discontinuous derivative. The triangular
roof frame rafter is C1, meaning continuous in ordinate, discontinuous in
slope. Smoother shapes have continuity of higher derivatives. Analytic
functions have infinite continuity (thanks to M. Cauchy!). Airfoils have to be
very smooth, but they can't be infinity smooth, since we need to tailor the
pressure distribution to control separation, and the trailing edge must usually
be sharp. Some of my airfoils of the olden days, when we did this by hand,
were C16 -- that is continuous only up to the 16th derivative. The airfoil I
designed for the Victor B Mk II(1956) is that rough, 'cause we did things on
Friden calculators in them days. But, as the RAF nuclear delivery system in the
hottest days of the Cold War, it scared the daylights out of the Ruzski. The
airfoil on the Gossamer Condor (Lissaman 7769) is much smoother than that,
although that too was pretty primitive. I did it personally using the old
(1971) TRS with punched tape inputs. I used the Radio Shack computer eksactly
as Picasso recommended: as an automated calculator to make the tiring number
crunches needed to provide answers to my questions. Incidentally, with a
trained geometric eye, which I think I have since I've been laying out airfoils
and streamline shapes since the 50's, you can "see" about 4 derivative
continuity. But the bloody air is unforgiving and wants higher smoothness than
that. It responds to curvature of curvature of curvature that you didn't even
know was there. But the computer does. Artists talk only up to C3, meaning
continuity of curvature. Art Deco derives a lot of its arresting visual tension
by deliberately exploiting discontinuities in curvature - for example a scroll
of fixed radius terminating a straight banister (C3). Art Nouveau designers
would rather die than do such thing -- for them it's all swooning smoothity!!
I'm sure this is more than you wanted to know, but I love digressing on this,
and for 20 years gave a course at Art Center on Leonardo and his art and
technology. He was not a mathematician, even by the fairly unsophisticated
standards of the High Renaissance, but how he longed to express things
mathematically!!
Peter Lissaman, Da Vinci Ventures
Expertise is not knowing everything, but knowing what to look for.
1454 Miracerros Loop South, Santa Fe, New Mexico 87505
TEL: (505) 983-7728 FAX: (505) 983-1694
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