Graphical representation of quadratic forms in 2 or 3 dimentions

Question

Consider a quadratic form in 2 or 3 dimensions. For example in 2 dimensions it would look like this:

$$ a_{11} x_1^2 + a_{22} x_2^2 + a_{33} x_3^2 + a_{12} x_1 x_2 + a_{13} x_1 x_3 + a_{23} x_2 x_3 $$

Such quadratic forms can be represented graphically as ellipses (in 2D) or ellipsoids (in 3D). For example in the field of crystallography it is very useful to describe the anisotropic and harmonic atomic thermal motion as ellipsoids in three dimensions. However, after some more of less extensive search I could not find any derivation of how the coefficients $a_{ij}$ relate to the type curve or surface formed. I know that if we form a symmetric matrix with $a_{ij}$'s and compute its determinant we can say what would the type of the curve/surface. But how to derive such conditions saying: "OK, your $a_{ij}$'s are like that, this will be an ellipse"?

Answer 1

The quadratic form is actually

$ a_{11} x^2 + a_{22} y^2 + a_{33} z^2 + 2 a_{12} x y + 2 a_{13} x z + 2 a_{23} yz $

and can be written compactly as

$ p^T A p $

where

$ p = [x, y, z]^T $

and

$ A = \begin{bmatrix} a_{11} && a_{12} && a_{13} \\ a_{12} && a_{22} && a_{23} \\ a_{13} && a_{23} && a_{33} \end{bmatrix} $

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Now, if

$ p^T A p = c $

where $c \ge 0 $ is constant, then we have a surface.

To determine what the type of surface and its orientation, diagonalize $A$, i.e. decompose $A$ into

$ A = R D R^T $

where $R$ is an orthogonal matrix, and $D$ is diagonal. From the signs of the diagonal entries of $D$ one can determine the type of surface. For example,

1. If $ D_{11} , D_{22}, D_{33} \gt 0 $ then you have an ellipsoid if $c \gt 0$, otherwise you have a single point when $ c = 0 $.

2. If $ D_{11}, D_{22} \gt 0 $ and $ D_{33} \lt 0 $ then you have a hyperboloid of one sheet provided that $ c \gt 0$, otherwise if $c = 0 $ then you have an elliptical cone.

3. If $ D_{11} , D_{22} \lt 0 $ and $D_{33} \gt 0 $ then you have a hyperboloid of two sheets provided that $ c \gt 0 $, otherwise if $c = 0$ then you have an elliptical cone.

4. If $D_{11} , D_{22} \gt 0 $ and $ D_{33} = 0 $ then you have an elliptical cylinder provided that $ c\gt 0 $, otherwise if $ c= 0 $ then you have a straight line.

5. If $ D_{11} \gt 0 $ , $D_{22} \lt 0 $ and $D_{33} = 0 $ then you have a hyperbolic cylinder provided that $ c \gt 0 $, otherwise if $c = 0$ then you have two intersecting planes.

6. If $ D_{11} \gt 0 $ and $ D_{22} = D_{33} = 0 $ then you have a pair of parallel planes given that $ c \gt 0 $, otherwise if $c = 0$ then you have a single plane.

Etc.