The stereographic projection is used in Crystallography and other scientific disciplines to represent, in a 2D space, the orientation of planes and directions located in a 3D space. XtereO is based on the application of the 2D stereographic projection of the geometric shapes of 3D polyhedrons, taking into account their symmetry.

The stereographic projection of a polyhedron requires to imagine that it is contained in a sphere and that its center coincides with the center of the sphere. In this sphere, three axes are defined: vertical axis c, with the positive side up; transversal axis b, with the positive side to the right, and the front-to-back axis a, with the positive side facing forward.

The stereographic projection is a projection of points from the surface of a sphere onto its equatorial plane. We must imagine that normals to each faces intersect the surface of the sphere at points called face poles. The face normals can be considered to radiate from a single point at the centre. Faces whose normals intersect the upper hemisphere are projected onto the equatorial plane by drawing a line from the face pole to the “south pole” of the stereographic sphere. The intersection of this line with the equatorial plane of the sphere is the stereographic projection of the face. Those which intersect onto the lower hemisphere are projected by drawing a line to the “north pole”.

The vertical faces, parallel to the c axis, are projected on the equatorial circumference; a horizontal face is projected in the center and the other faces are projected in the interior in a position that depends on its orientation with respect to the axes. Symbol X is used for faces parallel to the c axis and those that cut it on the positive side and symbol O is used for the faces that cut the c axis on the negative side. The stereographic projection represents both the faces of the polyhedron and the elements of symmetry that it present. Symmetry elements

Rotation axes are imaginary lines that cross a polyhedron passing through its center. When the polyhedron rotate around a rotation axis, all its geometric elements (faces, edges and vertices) are repeated at equal and fixed angular intervals that depend on the type of axis and the symmetry. XtereO includes symmetries with seven possible types of rotation axes:

- The 2-fold rotation axes, which operate by turning 180° in the 3D space. They can coincide with the c axis or be perpendicular to it. - The 4-fold rotation axes, which operate by turning 90° in the 3D space and coincide with the c axis. - The 3-fold rotation axes, which operate by turning 120° in the 3D space and coincide with the c axis. - The 6-fold rotation axes, which operate by turning 60° in the 3D space and coincide with the c axis. - The 4-fold rotoinversion axes, which operate by turning 90° in the 3D space followed by inversion through the center of the polyhedron. They coincide with the c axis. - The 3-fold rotoinversion axes, which operate by turning 120° in the 3D space followed by inversion through the center of the polyhedron. They coincide with the c axis. - The 6-fold rotoinversion axes, which operate by turning 60° in the 3D space followed by inversion through the center of the polyhedron. They coincide with the c axis.

A plane of symmetry is an imaginary plane that bisects a polyhedron into halves that are mirror images of each other. They are represented with the letter “m” (for “mirror”). Symmetry planes in XtereO can be perpendicular to the c axis or be parallel to it (contain it). A plane of symmetry is drawn in a stereographic projection with a continuous line, while the surfaces in which there are no planes of symmetry are represented in a dashed line.

Point Groups. The crystalline matter has a periodic and symmetrical internal structure that manifests externally generating polyhedral bodies. Up to 32 types of different combinations of rotation axes and symmetry planes can be identified in the polyhedra that represent crystals. These 32 combinations are called "Point Groups" and are grouped into 7 crystalline systems. All the symmetry elements of a Point Group coincide in a point, exactly the center of the crystal and, in stereographic projection, the center of the stereogram.