The representation surface

The representation surface (or representation quadric) is a geometrical representation of a second rank tensor and is useful for giving us a visual image of the tensor as well as being useful for example in calculating magnitudes of material properties described by second rank tensors.

Let us consider the equation T_ijx_ix_j = 1. Here T_ij represents a second rank tensor and x_i and x_j are coordinates.

This can be written in full as:

T₁₁x₁² + T₁₂x₁x₂ + T₁₃x₁x₃ + T₂₁x₂x₁ + T₂₂x₂² + T₂₃x₂x₃ + T₃₁x₃x₁ + T₃₂x₃x₂ + T₃₃x₃² = 1

which can be plotted to obtain a 3-dimensional graph. This graph is in fact a surface that is a complete description of T.

If we want to transform the surface to a new basis, making the substitutions x_i = r_kix_k^' and x_j = r_ljx_l^', we can write the equation of the representation surface as T_ijr_kix_k^'r_ljx_l^' = 1, or equivalently as T_kl^'x_k^'x_l^' = 1.
This means that: T_kl^' = r_klr_ijT_ij

Therefore any transformations of the tensor result in identical transformations of the 3D plot.

For symmetric tensors, the quadric equation for the representation surface simplifies to:

T₁₁x₁² + T₂₂x₂² + T₃₃x₃² + 2T₁₂x₁x₂ + 2T₁₃x₁x₃ + 2T₂₃x₂x₃ = 1, which is the equation of an ellipsoid.

For non-symmetric tensors and those with negative principal values, the representation surfaces are more complex. As with the tensor itself, the representation surface has its simplest form when referred to the principal axes (when the basis vectors are in line with the radii of the ellipsoid), where the equation becomes: T₁x₁² + T₂x₂² + T₃x₃² = 1 (where T₁, T₂ and T₃ are the principal values of the tensor).