The biaxial indicatrix is similar to the uniaxial indicatrix, except now there are three principal indices of refraction instead of two. The biaxial indicatrix is constructed by plotting the principal indices along 3 mutually perpendicular axes.

again, nalpha < nbeta < ngamma
So that the length of X<Y<Z.

Indicatrix is a triaxial ellipsoid elongated along the Z axis, and flattened along the X axis.

Indicatrix has 3 principal sections, all ellipses:

Random sections through the indicatrix also form ellipses.

The uniaxial indicatrix exhibited a single circular section, a biaxial indicatrix exhibits two circular sections with radius = nbeta; the circular sections intersect along the Y indicatrix axis, which also has a radius of nbeta.

Look at the X - Z plane in the above image.

The axes of the ellipse are = nalpha & ngamma.
The radii vary from nalpha through nbeta to ngamma.
Remember that nalpha < nbeta < ngamma, so a radii = nbeta must be present on the X - Z plane.
The length of indicatrix along the Y axis is also nbeta, so the Y axis and radii nbeta in X - Z plane defines a circular section, with radius nbeta.

In the biaxial indicatrix the directions perpendicular to the circular sections define the OPTIC AXES of the biaxial mineral. Optic axes lie within the X - Z plane, and this plane is the OPTIC PLANE.

The acute angle between the optic axes is the optic or 2V angle.

The indicatrix axis, either X or Z, which bisects the 2V angle is the ACUTE BISECTRIX or Bxa.

The indicatrix axis, either X or Z, which bisects the obtuse angle between the optic axes is the OBTUSE BISECTRIX or Bxo.

The Y axis is perpendicular to the optic plane and forms the OPTIC NORMAL.