# Blog

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## Non-linear least squares fitting of a two-dimensional data

The scipy.optimize.curve_fit routine can be used to fit two-dimensional data, but the fitted data (the ydata argument) must be repacked as a one-dimensional array first. The independent variable (the xdata argument) must then be an array of shape (2,M) where M is the total number of data points.

## ExB drift for an arbitrary electric potential

As described in the previous blog post charged particle moving in crossed constant magnetic and electric fields exhibits a drift velocity, $(\boldsymbol{E}\times\boldsymbol{B})/B^2$, perpendicular to both $\boldsymbol{E}$ and $\boldsymbol{B}$. The particle's trajectory in this situation can be found analytically. For an arbitrary $\boldsymbol{E}$, some kind of numerical integration of the equation of motion is usually necessary, but the force that the particle experiences at an instant is perpendicular to to the electric field and the particle therefore undergoes its gyromotion along an isocontour of electrostatic potential, $V$ (since $\boldsymbol{E} = -\nabla V$).

## Depicting a torus as an SVG image

SVG may not be the most obvious choice for depicting a 3-D object, but with some care over the perspective and ordering of the plotted points, it can be done.

## Reaching Orbit

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.

## Moiré patterns in a pair of hexagonal lattices

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).