Representation quadric

If all three principal values are positive, the quadric surface is an ellipsoid with semiaxes a, = TfiVz, but if one or two of the principal values are negative the quadric surface is a hyperboloid. For example, the (relative) impermeability tensor 3 is defined by nfn, where k is the permittivity and n0 is the permittivity of free space. As for any symmetric 7(2) the components of 3 define the representation quadric I3ijxixj= 1, which here is called the indicatrix or optical index ellipsoid. Referred to principal axes the indicatrix is... 

At temperatures below Tc BaTi03 belongs to the tetragonal crystal class (symmetry group 4mm) it is optically uniaxial, and the optic axis is the x3 axis (nQ = 2.416, ne = 2.364). When an electric field is applied in an arbitrary direction the representation quadric for the relative impermeability is perturbed to... 

Second-Rank Symmetrical Tensors, the Representation Quadric... 

So any second-rank symmetrical tensor can be geometrically represented by a representational quadric. The equation of a quadric can be simplified into the form... 

The components of a symmetrical second-rank tensor, referred to its principal axes, transform like the three coefficients of the general equation of a second-degree surface (a quadric) referred to its principal axes (Nye, 1957). Hence, if all three of the quadric s coefficients are positive, an ellipsoid becomes the geometrical representation of a symmetrical second-rank tensor property (e.g., electrical and thermal conductivity, permittivity, permeability, dielectric and magnetic susceptibility). The ellipsoid has inherent symmetry mmm. The relevant features are that (1) it is centrosymmetric, (2) it has three mirror planes perpendicular to the... 

The schema provides for regular conic curves (e.g. circle, ellipse, etc.), and quadric surfaces (e.g. cylinder, cone, etc.). Each of these has a specified parameterisation which must be adhered to when providing parameter values for other entities (e.g. trimmed curves). This is an attempt to exclude the idea of default parameterisation, the interpretation of which may differ from one implementation to another. In addition, parametric curves and surfaces are provided for by B-spline entities. These are specified by their control points, and may be rational or non-rational, uniform or non-uniform. Bezier representation of curves and surfaces is provided as a... 

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