Cyclographic and stereographic projection

What is important in the representation of a crystal ?

What is important in the representation of a crystal ?
It is not its shape
It is not its extension

What is important in the representation of a crystal ?
It is its orientation,
especially with respect to the other faces

What is important in the representation of a crystal ?
The interfacial angle is measured with the goniometer. It is
also available by geometric construction when considering
the normals of the two faces.
A line perpendicular to a face is called a pole of the crystal
face. It is named Φ.

How to flatten the Earth ?
Illustration by Rubens for
"Opticorum libri sex philosophis juxta ac mathematicis utiles",
by François d'Aiguillon.
He demonstrated how the projection is computed.

How to proceed ?
The crystal is imagined at the center of a sphere
Lines perpendicular to the faces are drawn.
Each face is represented by the intersection between the
sphere and the normal to the face. It’s the polar projection.

Cyclographic projection
The angle between the normals of two faces are found on
the sphere

Systematic way to define crystallographic angles
The pole of an hypothetical face (010) is put parallel to the b axis. It impinges inside the sphere at the equator. The Φ angle is equal 0°.
2) For any other face, the angle will be different of 0° (except for // faces.
It will be measured from the b axis in a clockwise sense in the plane of the equator.

Systematic way to define crystallographic angles
A ρ angle is defined between the c axis and the pole of the crystal face.
This is measured downward from the North pole of the sphere
The ρ angle of the face (010) is equal to 90°.
Considering an other face, (101).
The ρ angle is measured in a vertical plane containing the axis c of the sphere and the pole of the plane (101).
The Φ angle is equal to 90°. It is measured in the equatorial horizontal plane from b.

Ressemblance with the earth projection system
Nelson
The Φ angle measured in
the equatorial plane
Corresponds to the longitude.
It is measured from a
referential plane, the
Greenwich meridian, defined
as Φ = 0°.
2) The ρ angle is measured in
the vertical plane as the latitude.
However, the latitude (Θ) is measured up from the equator.
So the ρ angle (90°- Θ) is called the colatitude.

Example for a crystal - face (111)
The Φ angle measured in
the equatorial plane,
clockwise from the b axis.
2) The ρ angle is measured in
the vertical plane containing
the c axis and the pole of the
Face (111).
To project a face on a stereographic projection, we need only
to know the Φ angle and ρ angle.
Knowledge of these data for two faces permits to calculate
the interfacial angle between the two faces, either by
trigonometry or by stereography.
And later on, the mineral symmetry

Spherical coordinate system for a sphere

Projection on a plane -
Stereographic projection
2) The second step is to imagine a
system of projection on a plane of the
polar faces.
It’s a similar problem as for the projection
of the earth. 3 types of projections occur:
Orthogonal
gnomic
stereographic

Why to use a stereographic projection in crystallography?

Why to use a stereographic projection in crystallography?
The projection of a circle is maintained as a circle
The angles between great circles are also maintained.
Gnomic projection is used when drawing crystals
with perspective.
Stereographic projection is used in numerous geologic
aspects : structural geology but also applied geology

Stereographic projection
Projections are done on the equatorial plane which is also
called projection circle. Points within the north (upper)
hemisphere are projected within the equatorial plane;
those from the lower hemisphere are projected outside the
equatorial plane.
View point

Direct and reciprocal projections
N
S

Stereographic projection of crystal faces
c
b
a
View from the south pole
[example face (011) of a cube]
We put the crystal at the center of the sphere, with the (001) axis parallel to c, and the (010) parallel to the b axis.
Face (011) intersects the sphere in P.
We draw a line from P to the south pole of the sphere.
When the line intersects the equatorial plane, we plot the projected point. The stereographic projection is on the equatorial plane.
P

Stereographic projection of crystal faces
[example face (011) of a cube]

c
Projection in a vertical plane
Projection in an horizontal plane : the equatorial plane

c
Note : Face (001) horizontal ---> angle ρ = 0°
Face (010) vertical ---> angle ρ = 90°
Face (011) same intercept on b and c axes
---> angle ρ = 45°(see vertical section)
On equatorial projection, project point P’ is in intermediate
position between the center and the great circle, along the
E-W diameter. The angle ρ is measured at this distance
between the center and the projected point P’.

c
The angle Φ is measured around the circumference of the
circle on the stereographic projection, in a clockwise direction
from the East position, which coincide with the b axis or the
pole of the face (010).
What we called view from the North and South poles are visible
on the vertical projection. The north and south pole are
respectively above and below the equatorial plane
corresponding to the stereographic projection of crystals.

Correspondance with the stereographic projection of the earth and that of tectonic planes

The stereonet - Wulf net
- Components of the net (drawn with 2 intervals)
Primitive circle
Great circles (N-S connexion)
Small circles are the highly curves lines that curve upward and downward on the stereonet
Note that the primitive circle, the N-S and E-W axes are also
great circles

The stereonet - Wulf net
- Components of the net (drawn with 2 intervals)
Primitive circle
Great circles (N-S connexion)
Small circles
Angle Φ is measured
from the great circle
Angle ρ is measured
on the E-W line,
ounce angle Φ is
achieved

Significance of the great and small circles
Great circles
Small circles

Use of the stereonet - (1)
All crystal faces are plotted as poles (lines perpendicular
to the crystal face.
----> angles between crystal faces are angles between poles
The b crystallographic axis is taken as the starting point.
This axis is perpendicular to face (010) in any crystal system.
The [010] axis (notation as zone symbol) or (010) crystal face
will therefore plot at Φ =0° and ρ = 90°.
Positive Φ angles will be measured clockwise and negative
angles, counter- clockwise on the stereonet.
Crystal faces on top of the crystal (ρ < 90°) are plotted as
open circles and crystal faces on the bottom (ρ > 90°) as
« + » signs.

Use of the stereonet - practical uses (2)
To plot a face:
First measure the value of the
angle on the primitive circle.
Write a mark
Rotate the sheet of paper to lie
the mark on the E position.
Measure the ρ angle out from
the center on the E-W axis of the
Stereonet.
Place a sheet of tracing paper
on the stereonet and trace the
outermost great circle (primitive
circle). Make a reference mark
on the right side (east position)
Note: angles can only be measured along great circles

Use of the stereonet (2)

Standard coordinates for the crystal axes
in the different crystal systems
Isometric
Tetragonal
Triclinic
Hexagonal
Orthorhombic
Monoclinic

Cyclographic and stereographic projection

Spherical projection of a monoclinic crystal

Zone plane - zone circle
Reciprocal projection of the a axe

Zone circles of axes a, b and c

Cyclographic and stereographic projection

Cyclographic and stereographic projection

Symmetry related pole faces
1 - Plane of symmetry

Symmetry related pole faces
2 - Rotation axe:
Exp.: 3-fold rotational axe

Symmetry related pole faces
3 - Center of Symmetry

Inversion axes

Gnomonic projection