tilted cavatappo surfaces:
cavatappo 2.0
Inspired by Chris Rorres' talk on
Archimedes screw
strakes versus helicoidal designs (September 28, 2012 at Villanova University),
and having already had a first go around with corkscrew pasta (cavatappo
1.0),
bob was
encouraged to consider a generalization of
George Legendre's cavatappo
design (Pasta by Design,
CBS Sunday
morning spot) by tilting the vertical circle undergoing the
screw rotation. In particular, tilting it back exactly by the upward
tilt of the helical center path makes this circle move perpendicular to
itself, to give the pleasing shape shown here (just a fixed radius
tube or tubular surface of a helix). George is an architect, so it is not clear how he decided
on his representation of the cavatappo other than it was the easiest
construction to consider. However, by adding a tilt parameter, one
allows the family to specialize to the vertical cylinder and hence
capture in one continuous family all the simplest special curved
surfaces ranging from the sphere to the torus to helical
tube and back to the even simpler cylinder. The real pasta shape
in fact is unclear and has ends which are neither vertical as in
George's model or orthogonal to the central helix as in this new shape
shown to the right (cavatappo 2.0),
but then mathematical models are usually simpler than the real world
system they model.
While the vertical circular crosssections (meridians) are not
geodesics in George's model, they are in the new model with orthogonal
tilting (black curves) as is the case for surfaces of revolution, and they admit simple orthogonal trajectories \(
\displaystyle v=v_0 + \frac{8}{17} u\), but which are distinct from the
"parallels", unlike the case for surfaces of revolution. Note the stunning surface graphics of Maple 16!
More to come... stay tuned. (8,15,17) is the fourth
Pythagorean
triple. Math by MathJax. \[
\displaystyle
\left[ \begin {array}{c} x\\ y\\ z\end {array} \right]
=
\left[ \begin {array}{c} \left( \frac32+\cos \left( v \right) \right) \cos \left( u \right) +{\frac
{8}{17}}\,\sin \left( v \right) \sin \left( u \right) \\
\left( \frac32+\cos \left( v \right) \right) \sin \left( u \right) {\frac {8}{17}}\,\sin \left( v \right) \cos
\left( u \right) \\ {\frac {15}{17}}\,\sin \left( v \right) +\frac45\,u\end {array}
\right]
\]
\[u=0..2\pi,\ v=0..5\pi \]
\[
ds^2 = (dv\frac{8}{17}du)^2 + \left(\left(\frac32+\cos(v)\right)^2  \left(\frac{8}{17}\right)^2 \cos^2(v)\right) du^2
\]
Serendipity factor: George picked a frequency \( \displaystyle\frac{5}{2\pi}\approx
0.7958 \sim 0.80 = \frac{4}{5}\) for the central helix so that
after one azimuthal revolution around the symmetry axis of the helix at
radius \(\displaystyle\frac{3}{2}\), the vertical rise is 5. Using \(\displaystyle\frac{4}{5}\)
instead leads to the fourth Pythagorean triple right triangle to
describe the helical angle of inclination \(\displaystyle\arctan\left(\frac{4}{5}/\frac{3}{2}\right)
= \arctan\left(\frac{8}{15}\right) \). I also rescaled his millimeter
units down by a factor of 2 for the mathematical surface, to have
simpler numbers. [Sol
Lederman in his blog (404) published the model developed in
Sander Huisman's blog (404) which used this inclination angle for his cellentani,
but a smaller \(b/a=1/2\) ratio than the value \(2/3\) used by George.]
The three black circles are meridians. The red and green curves are
the outer and inner equator (maximum positive and minimum positive
curvature), while the "parallel" white curves are
equally spaced "parallels" (orbits of the screwsymmetry group,
lines of constant curvature) between
them, spaced at angular intervals of \(\pi/4\). Halfway between the
equators are the northern and southern polar circles of zero curvature,
separating the two halves of positive (outer) and negative (inner)
curvature.
Scroll down below your window for the coordinate/parameter grid
plots.
