Miscellaneous mathematical stuff that I found interesting at one point
or another.
This calculator takes a standard film developing time and adjusts it
for temperature. Obtain the nominal developing time and temperature
from a data sheet or from the
Massive Dev Chart,
then enter those values together with the actual temperature.
The calculation follows the exponential temperature correction used on
the original Bandicoot calculator:
\[
t_{\mathrm{actual}} =
t_{\mathrm{nominal}} e^{\alpha (T_{\mathrm{nominal}} - T_{\mathrm{actual}})}.
\]Here \(T\) is measured in Celsius for the calculation, and
\(\alpha = 0.1\). Constant agitation applies the same 10% reduction
used by the original calculator. The result is intended as an
approximate guide for Ilford black-and-white film/developer processes;
other combinations may vary.
Leslie Matrices:
Leslie Matrices are used to model growth (and decline) of
age-structured populations. In the model named after Patrick H. Leslie
(1945), we have \(N\) age classes, and record how many individuals are
in each. Each time period, individuals either age into the next class,
or die. The survival rate describes the proportion that moves on to the
next age class, and the birth rate, or fecundity, describes the rate
per capita of births arising from each age category.
Where there is math writen on these pages it is constructed using
MathJax, a lovely little Javascript library just for this.
To get to play with it, I borrowed the scripts to set up a little
dynamic LaTeX box. Have a look here http://bandicoot.maths.adelaide.edu.au/MathJax/test/sample-dynamic-2.html.
A Reuleaux triangle
is the shape you get, starting from an equilateral triangle, and
adding three circular arcs, drawn from each corner. It’s not actually
a triangle, but what’s in a name?
The interesting thing is that a Reuleaux triangle is a shape of
constant width. That means it can be used as a roller (but not a
wheel). The animation below shows this – we can see that although the
centre of the triangle moves up and down, the top surface of the
rotating triangle is always level.
Follow the link below to see more, get links to 3D printable versions,
and Matlab code to play with it.
A couple of little tools for playing with Catenaries can be found at:
http://bandicoot.maths.adelaide.edu.au:3838/catWorkshop/,
and http://bandicoot.maths.adelaide.edu.au:3838/catenary/. They
aren’t really documented yet – they’re just a toy to play with Shiny – but more will come
later.
Leslie Matrices are a tool for modelling population demographic
dynamics. Wikipedia
does a decent job of explanation of these. My megre contribution
is a little online
Leslie Matrix Calculator.
A Log-azimuthal map of the world (as seen from Adelaide, Australia,
my home town) using log-distances, is a map that shows places close
by are displayed in detail, and those further away, have less.
There are a large variety of sea-shells (and land shells) that can be
simply described by rotating a ellipse around a log-spiral that has
been projected onto a cone. The image below was generated using this
approach:
A Roulette is a curve
derived by rolling one curve against another. In the image below, the
blue point on the blue ellipse is rolled along a straight line (the
x-axis) to generate the purple curve (which is called an
undularly).
Included here is some Matlab code to generate roulette curves, with a
fair degree of generality, but also showing how to generate specific
instances such as cycloids, trochoids, the Cissoid of Dioclese, and
the undularly and nodary.
