Stereographic Projection Pole Calculator
Turn any (hkl) plane into its stereographic pole: polar angle rho, azimuth phi, projected coordinates and a plotted stereogram.
🎯 What is a Stereographic Projection Pole Calculator?
A stereographic projection pole is the flat-page representation of a crystal plane. Instead of drawing the plane itself, crystallographers draw the direction of its normal: imagine the crystal sitting at the centre of a reference sphere, extend the plane normal until it pierces the sphere's surface, then project that piercing point down onto the equatorial plane from the opposite pole of the sphere. Every plane in the crystal collapses to a single dot, and the whole three-dimensional orientation of the crystal can be read from one circle. This calculator performs that construction numerically for any (hkl) plane in a cubic, tetragonal or orthorhombic unit cell, returning the polar angle ρ, the azimuth φ, the projected radius r/R and the Cartesian coordinates of the pole.
The projection turns up wherever crystal orientation matters. A metallurgist reading a Laue back-reflection pattern indexes the spots by matching them against a stereogram of the expected orientation. A materials scientist studying rolling texture in sheet steel plots thousands of grain orientations as a pole figure, which is nothing more than a density map of stereographic poles. A structural geologist measuring bedding and joint orientations in the field plots each measurement as a pole on the same net. A thin-film researcher checking epitaxial alignment compares the substrate stereogram with the film stereogram to see whether the two lattices share a common pole.
Two points cause most of the confusion. First, the projected radius is r/R = tan(ρ/2), not a linear function of ρ, so distances on the page are stretched near the primitive circle. Second, the plane normal is only parallel to the direction [hkl] when the cell is cubic. In a tetragonal or orthorhombic cell the normal follows the reciprocal vector (h/a, k/b, l/c), so plotting [hkl] instead of the true normal quietly puts the pole in the wrong place. This calculator always uses the reciprocal-vector normal, so the same input works for all three orthogonal crystal systems.
Because a plane and its inverse describe the same physical object, the tool takes whichever sense of the normal points into the upper hemisphere and tells you when it has flipped your input. It also generates the full set of symmetry-equivalent poles allowed by the orthogonal holohedry, adding index permutations when the cell edges are equal, and plots them alongside the pole you asked for. That makes the diagram a real stereogram of the {hkl} form rather than a single isolated dot.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — The (111) Pole of Aluminium
Face-centred cubic aluminium, a = b = c = 4.05 Å, plane (111)
Example 2 — A (110) Pole on the Primitive Circle
Same cubic cell, a = b = c = 4.05 Å, plane (110)
Example 3 — A Tetragonal (101) Pole
Tetragonal cell with a = b = 3.99 Å and c = 4.04 Å, plane (101)
Example 4 — A General Orthorhombic (213) Pole
Orthorhombic cell a = 4 Å, b = 5 Å, c = 6 Å, plane (213)
❓ Frequently Asked Questions
🔗 Related Calculators
What is a stereographic projection pole?
A pole is the point where the normal to a crystal plane pierces a reference sphere, projected down onto the equatorial plane. Each (hkl) plane becomes a single dot instead of a plane, so the angular relationships between many planes can be read off one flat circle at a glance.
How do you calculate the position of a pole on a stereogram?
Find the unit plane normal, take the polar angle rho = arccos(n_z) from the projection axis and the azimuth phi = atan2(n_y, n_x) from the a axis, then place the point at radius r/R = tan(rho/2) and angle phi. A pole at rho = 0 sits at the centre, a pole at rho = 90 degrees sits on the primitive circle.
Why is the projected radius tan(rho/2) and not rho?
Because the stereographic projection is drawn from the opposite pole of the reference sphere. The construction line from the south pole through the point on the sphere crosses the equatorial plane at a distance R tan(rho/2). That is what makes the projection conformal, it preserves the angles between poles.
What is the plane normal for a non-cubic crystal?
For any orthogonal cell the normal to (hkl) is parallel to the reciprocal vector (h/a, k/b, l/c). Only when a = b = c does that reduce to the direction [hkl]. Using [hkl] directly on a tetragonal or orthorhombic cell puts the pole in the wrong place, which this calculator avoids.
What do the open circles on the stereogram mean?
Open circles mark poles whose normals point into the lower hemisphere. Standard crystallographic practice plots upper-hemisphere poles as filled dots and lower-hemisphere poles as open circles on the same diagram, so both halves of the sphere can be shown on one figure without confusion.
How many symmetry-equivalent poles does {111} have in a cubic crystal?
Eight, the four body diagonals taken in both senses. Four of them appear in the upper hemisphere at rho = 54.7356 degrees and azimuths of 45, 135, 225 and 315 degrees. The cubic {110} form has twelve members and {100} has six.
What is the difference between a Wulff net and a Schmidt net?
The Wulff (equal-angle) net is the stereographic projection used here, with radius tan(rho/2). It preserves angles, which suits single-crystal orientation work. The Schmidt (equal-area) net uses radius proportional to sin(rho/2) and preserves area, which suits density contouring in texture and structural geology.
Can I use this calculator for hexagonal or trigonal crystals?
Not directly. This tool assumes the three cell edges are mutually perpendicular, which covers cubic, tetragonal and orthorhombic cells. Hexagonal cells have a 120 degree angle between a and b, so the normal formula picks up extra cross terms and the projected azimuth would be wrong.
What does rho = 0 mean physically?
The plane normal is parallel to the c axis, so the plane itself is perpendicular to c. That is exactly the (001) plane in an orthogonal cell, and its pole plots at the very centre of the stereogram with a projected radius of zero.
Why are stereographic projections still used in materials science?
Because angles between poles map to measurable distances on the net, so orientation relationships, twin laws, slip systems and epitaxial alignments can be checked by eye. Electron backscatter diffraction, Laue back-reflection indexing and texture pole figures all present their output on the same projection.
Does the pole position depend on the absolute size of the unit cell?
No, only on the ratios a:b:c. Scaling every edge by the same factor scales the normal by that factor too, and normalising removes it. A cubic cell of 3 angstroms and one of 6 angstroms give identical stereograms for the same (hkl).
How do I read the angle between two poles from the stereogram?
Rotate a Wulff net under the plot until both poles lie on the same great circle, then count the degrees between them along that circle. For a numeric answer without a net, use an interplanar angle calculator, which applies the dot product of the two plane normals directly.