A blog of Python-related topics and code.
Truchet Tiles
Posted by: christian on 5 Nov 2018
Truchet Tiles are decorated squares which can tile a plane to create a pattern. For example, the four tiles:
Reaching Orbit
Posted by: christian on 22 Oct 2018
The following code illustrates the effect of the initial velocity on the dynamics of an object released in a gravitational field. A very simple numerical integration of the equation of motion gives the trajectory, which is plotted below for four different initial speeds for a rocket released at 200 km altitude parallel to the Earth's surface. At this altitude the speed needed for a circular orbit is about 7.8 km/s. You can read more about this kind of simulation at the Wikipedia page for Newton's cannonball.
The bounding box of a rotated object
Posted by: christian on 4 Oct 2018
The following code plots a two-dimensional object and its bounding box for several rotations about an arbitrary point. Three types of bounding box are considered: (1) the original bounding box, rotated by the same amount as the object, (2) the bounding box to that bounding box (such that its sides remain parallel to the axes), and (3) the bounding box to the rotated object with sides parallel to the axes).
Moiré patterns in a pair of hexagonal lattices
Posted by: christian on 25 Sep 2018
A Moiré pattern is an interference pattern that occurs when two grids of repeating lines or shapes are rotated by a small amount relative to one another (oblig. xkcd).
Gyromotion of a charged particle in a magnetic field
Posted by: christian in Featured on frontpage on 11 Sep 2018
A charged particle of mass $m$ and charge $q$ moving with a velocity $\boldsymbol{v}$ in an an electric field $\boldsymbol{E}$ and a magnetic field $\boldsymbol{B}$ is subject to a Lorentz force, $\boldsymbol{F}$, given by
$$
\boldsymbol{F} = q(\boldsymbol{E} + \boldsymbol{v}\times\boldsymbol{B}).
$$
The equation of motion for a single particle is therefore given by Newton's second law as
$$
\boldsymbol{\ddot{r}} = \frac{q}{m}(\boldsymbol{E} + \boldsymbol{v}\times\boldsymbol{B}).
$$
Here we will consider a uniform magnetic field, $\boldsymbol{B} = (0,0,B)$ and zero electric field, $E=0$. In this case, the trajectory of the particle can be obtained by solving the equation of motion analytically, but here we integrate it numerically using SciPy's integrate.odeint method. Assuming the particle starts off with non-zero components of its velocity parallel ($v_\parallel$) and perpendicular ($v_\perp$) to the magnetic field, it moves in a helix, with radius given by
$$
\rho = \frac{mv_\perp}{|q|B},
$$
known as the Larmor or cyclotron radius (or gyroradius).