Pseudo-ellipse parameterisation

To extend to the next level of motional complexity above the isotropic cone models, an anisotropic cone can be modelled. This cone is defined via the ball and socket joint pivoted mechanics with an angular restriction in all three angles. The simplest anisotropic distribution would be to create an ellipse using the standard quadric surface formula for an elliptic cone

$\displaystyle {\frac{{x^2}}{{a^2}}}$ + $\displaystyle {\frac{{y^2}}{{b^2}}}$ - $\displaystyle {\frac{{z^2}}{{c^2}}}$ = 0. (16.38)

Let the two cone opening half-angles of the ellipse be θx and θy. For a sphere of radius z = 1 and using the tilt angles θ and φ, a boundary polar angle θmax can be modelled as

$\displaystyle {\frac{{1}}{{\theta_{\textrm{max}}^2}}}$ = $\displaystyle {\frac{{\cos^2\phi}}{{\theta_x ^2}}}$ + $\displaystyle {\frac{{\sin^2\phi}}{{\theta_y ^2}}}$. (16.39)

As the quadric constants a, b and c are angles rather than axis lengths, this is not a true ellipse. It will therefore instead be called a pseudo-ellipse. The form of this pseudo-elliptic cone is shown in figure 16.13.

Figure 16.13: The pseudo-elliptic cone. The top three representations are for θx = 30o and θy = 50o. The bottom three representations are for θx = 20o and θy = 160o.
\includegraphics[width=0.7\textwidth]{images/pseudo_elliptic_cone}

The model consists of the average domain position, a single pivot point, the full motional eigenframe, and the maximum cone opening and torsion half-angles

\begin{subequations}\begin{align}\mathfrak{M}&= \mathfrak{P}+ \mathfrak{E}+ \mat...
...eta_x , \theta_y , \sigma_{\textrm{max}}\right\} , \end{align}\end{subequations}

where Pi are the average domain position translations and rotations, Ei are the Euler angles defining the motional eigenframe, pi are the coordinates of the pivot point, θx and θy are the maximum cone opening half-angles, and σmax is the torsion half-angle.

The relax user manual (PDF), created 2016-10-28.