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

$${\frac{{x^2}}{{a^2}}} {\frac{{y^2}}{{b^2}}} {\frac{{z^2}}{{c^2}}}$$ (16.38)

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

$${\frac{{1}}{{\theta_{\textrm{max}}^2}}} {\frac{{\cos^2\phi}}{{\theta_x ^2}}} {\frac{{\sin^2\phi}}{{\theta_y ^2}}}$$ (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 θ_x = 30^o and θ_y = 50^o. The bottom three representations are for θ_x = 20^o and θ_y = 160^o.

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

where P_i are the average domain position translations and rotations, E_i are the Euler angles defining the motional eigenframe, p_i are the coordinates of the pivot point, θ_x and θ_y are the maximum cone opening half-angles, and σ_max is the torsion half-angle.