Stereographic Projection
References:
Dexter Perkins, 2002, Mineralogy, 2nd edition. Prentice Hall, New
Jersey, 483 p.
Bloss, F.D., 1971, Crystallography and Crystal Chemistry: Holt,
Reinhardt, and Winston, New York, 545 p.
Klein, C., and Hurlbut, C.S.Jr., 1993, Manual of Mineralogy (after
James Dana), 21st edition: John Wiley & Sons, New York, 681 p.
Stereographic Projection
 A convenient way of looking at the symmetry of a
crystal
 Projection of 3D orientation data and symmetry of a
crystal into 2D using spherical projection
 Projection lowers the Euclidian dimension of the
object by 1, i.e., planes become lines, and lines
become point!
 The poles (normals) of crystal faces are projected
onto the inside surface of a sphere as points, and
then reprojected onto its equatorial plane
 In mineralogy, it involves projection of faces,
edges, mirror planes, and rotation axes onto a flat
equatorial plane of a sphere, in correct angular
relationships
 In mineralogy, stereograms have no geographic
significance, and cannot show shape of crystal
faces!
Why need projection?
 Projection of all crystal faces of a crystal leads to
many great circles or poles to these great circles
 These great circles and poles allow one to
determine the exact angular relationship, and
symmetry relationships
 for example, between crystal faces and rotation
axes, or between axes and mirror planes
Stereonets
 Wulff net (Equal angle)

Used in mineralogy & structural geology
when angles are meant to be preserved
 e.g., for crystallography and core analysis
 Schmidt net (Equal area)

Used in structural geology for orientation
analysis when area is meant to be preserved
for statistical analysis
Wulff net
Wulff Stereonet (Equal Angle net)
 Shows the projection of great circles and small circles
 Great circle: line of intersection of a plane, that passes
through the center of the sphere, with the surface of the
sphere (like lines of longitude on Earth)
 NOTE: Angular relationships between points can only be
measured on great circles (not small circles)!
 Small circle: loci of all positions of a point on the surface of
the sphere when rotated about an axis such as the North pole
(like lines of latitude on Earth)
 In mineralogy the upper and lower hemispheres are used
 In structural geology only the lower hemisphere is used
Great & Small circles View From a Pole
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
Preparing to Plot
 Mount your stereonet on a cardboard. Laminate it.
Pass a thumbtack through the center.
 Secure the thumbtack with a masking tape
 Place a sheet of tracing paper on the stereonet
 Put a scotch tape at the center of the tracing paper,
and pierce the paper and tape through the pin
 The tracing paper can now rotate around the thumb
tack without enlarging the hole (because of scotch
tape)
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
Projection of the pole to a face!
 Visualize that one of the faces of a crystal, positioned at the center
of a sphere, dips down to the right (from the horizontal)
 Imagine that when the pole of this face is extended up to inner
surface of the upper hemisphere, a bulb turns on at point P!
 Also imagine that you are watching the bulb from the South pole
of the sphere (i.e., your eyes are at the S pole)
 The ray of light, coming off point ‘P‘ to reach your eyes at point ‘S’,
intersects the equatorial plane at point p’
 Point p’is the stereographic projection of point ‘P’, i.e., of the pole
of the crystal face
How Projection is done (Figure 9.29)
 The pole (i.e., normal) to each crystal face is
extended until it intersects the inner surface
of the hemisphere (upper or lower) say at
point ‘P’
 If this point is in the upper hemisphere,
connect it to the S-pole (no geographic
significance!)
 If this point is in the lower hemisphere,
connect it to the N-pole
Figure 9.29 cont’d
 For faces below the equator (when using lower hemisphere),
place an open circle symbol (◯) where the line intersects the
equatorial plane (this is the stereographic projection of the face)
 For faces above the equator, using the upper hemisphere, place a
solid circle symbol (●) where the line intersects the equator
 Use a bull’s-eye (◉) to show a point above the page that
coincides with one directly below it (using both hemispheres)
 Reorient the stereogram such that lines of symmetry are north-
south or east-west
Projection of Planar Elements
 Crystals have faces and mirror planes which are
planes, so they intersect the surface of the sphere
along lines
 These elements can be represented either as:
 Planes, which become great circle after
projection
 Poles (normals) to the planes, which become
points after projection
Projection of Linear Elements
 We can show all of the symmetry elements of a crystal
and their relative positions stereographically
 Edges, pole to crystal faces, and rotation or
rotoinversion axes are lines
 When extended through the origin of the sphere, lines
intersect the surface of the sphere as points
 Each of these points, when connected to the North or
South pole of the sphere, is projected onto the
equatorial plane, and depicted as a polygon symbol
with the same number of sides as the ‘fold’ of the axis
Mirror and Polygon Symbols
 To plot symmetry axes on the stereonet, use the
following conventions:
Mirror plane:
― (solid line)
Special cases of planes
 Stereographic projection of a horizontal face or mirror
plane is along the perimeter of the equatorial plane
 This plane is called the ‘primitive’
 Stereographic projection of a vertical face or mirror plane is
along the diameters of the equatorial plane, i.e.,
 They pass through the center
 They are straight ‘great circles’
 Inclined faces and mirror planes plot along curved
great circles that do not pass through the center
Special cases of lines
 Vertical lines (e.g., rotation axes, edges) plot at the
center of the equatorial plane
 Horizontal lines plot on the primitive
 Inclined lines plot between the primitive and the
center
Plotting the rotation axes
 Vertical axes (normal to page) will only have one
polygon
 Horizontal axes (in the plane of page) intersect the
primitive twice, hence they have two polygons
 Inclined axes will have one polygon
 An open circle in the middle of the polygon shows
there is a center of symmetry
Mirror and Polygon Symbols
 To plot symmetry axes on the stereonet, use the
following conventions:
Mirror plane:
― (solid line)
Measuring angle between faces
 Three cases:
1. On the primitive, the angle is read directly on the
circumference of the net
2. On a diameter, the paper is rotated until the zone is
coincident with the vertical diameter and the angle
measured by the diameter
3. On a great circle (an inclined zone), rotate the paper
until the zone coincides with a great circle on the net;
read the angle along the great circle
Going from one hemisphere to another
 During rotation of the pole to a face by a certain angle,
we may reach the primitive before we are finished with
the rotation
 In this case we are moving from one hemisphere to
another
 Move the pole back away from the primitive along that
same small circle you followed out to the primitive,
until it has been moved the correct total number of
degrees
 Then note its new position with the point symbol for
the new hemisphere
How to find reflection of a point
 Having a symmetry (mirror) plane and a point p,
find the reflection of point p (i.e., p’) across the
mirror:
 Align the mirror along a great circle
 Rotate point p along a small circle to the mirror
plane
 Count an equal angle beyond the mirror plane to
find point p’
 If the primitive is reached before p’,then count
inward along the same great circle
Convention
 By convention (Klein and Hurlbut, p.62), we
place the crystal at the center of the sphere such
that the:
 c-axis (normal of face 001) is the vertical axis
 b-axis (normal of face 010) is east-west
 a-axis (normal of face 100) is north-south
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
3-D Symmetry Conventions
http://www.kean.edu/~csmart/Mineralogy/Lectures
http://www.kean.edu/~csmart/Mineralogy/Lectures
http://www.kean.edu/~csmart/Mineralogy/Lectures
Crystallographic Angles
 Interfacial angle: between two crystal faces is the
angle between poles to the two faces.
 The interfacial angle can be measured
with a contact goniometer
 These angles are plotted
on the stereonet
Making a stereographic projection of a crystal face pole
Use a contact goniometer to measure the interfacial
angles (also measures poles)
The ρ and φ angles
 Generally, it is the angles of the spherical
projection, ρ and φ, that are given for each
face of a crystal (measured with goniometer)
 If these are known, then the actual angles
between any two faces can easily be obtained
through trigonometry, or by use of the
stereonet
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
The ρ angle
 The ρ angle, is between the c axis and the pole to
the crystal face, measured downward from the
North pole of the sphere
 A crystal face has a ρ angle measured in the
vertical plane containing the axis of the sphere and
the face pole, and a φ angle measured in the
horizontal equatorial plane
 Note that the (010) face has a ρ angle of 90o
Plotting ρ and φ
Suppose you measured r = 60o and F = 30o
These angular measurements are
similar to those we use for latitude
and longitude to plot positions of
points on the Earth's surface
For the Earth, longitude is similar to
the φ angle, except longitude is
measured from the Greenwich
Meridian, defined as φ = 0o
Latitude is measured in the vertical
plane, up from the equator, shown as
the angle θ. Thus, the ρ angle is what
is called the colatitude
(90o - latitude).
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
Zone plotting
 Zone: two or more faces whose edges of intersection are parallel
to a specific linear direction in a crystal.
 This direction is called the zone axis. A zone is indicated by a
symbol similar to that for Miller Indices of faces, the generalized
expression for a zone is [uvw], e.g., all faces parallel to the c axis
in an orthorhombic crystal are said to lie in the [001] zone
 All faces in a zone lie on a great circle
 Zone is constructed by aligning the poles to these faces on a great
circle
 The zone pole is normal (i.e., 90o) to this great circle
 On the stereogram, the lower hemisphere part of the zone circle
is dashed, while the upper is solid
 Lower-hemisphere faces are represented by open circles
The following rules are applied:
 All crystal faces are plotted as poles (lines perpendicular to the
crystal face. Thus, angles between crystal faces are really
angles between poles to crystal faces
 The b crystallographic axis is taken as the starting point. Such
an axis will be perpendicular to the (010) crystal face in any
crystal system. The [010] axis (note the zone symbol) or (010)
crystal face will therefore plot at φ = 0o and ρ = 90o
 Positive φ angles will be measured clockwise on the stereonet,
and negative φ angles will be measured counter-clockwise on
the stereonet
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
 The pole to a (010) face will coincide with the b
crystallographic axis, and will impinge on the
inside of the sphere at the equator, i.e., on the
primitive
 We define this face (010) as having a φ angle of 0o
 For any other face, the φ angle will be measured
from the b axis in a clockwise sense in the plane of
the equator
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
(111) (100) (111)
(011) (100) all
coplanar (= zone)
Thus all poles in a
zone are on the
same great circle
Rules cont’d
 Crystal faces that are on the top of the crystal (ρ < 90o) will
be plotted as closed circles (), and crystal faces on the
bottom of the crystal (ρ > 90o) will be plotted as "◯" signs.
 Place a sheet of tracing paper on the stereonet and trace the
outermost great circle. Make a reference mark on the right
side of the circle (East)
 To plot a face, first measure the φ angle along the
outermost great circle, and make a mark on your tracing
paper. Next rotate the tracing paper so that the mark lies at
the end of the E-W axis of the stereonet
Rules cont’d
 Measure the ρ angle out from the center of the
stereonet along the E-W axis of the stereonet
 Note that angles can only be measured along great
circles. These include the primitive circle, and the
E-W and N-S axis of the stereonet.
 Any two faces on the same great circle are in the same
zone. Zones can be shown as lines running through
the great circle containing faces in that zone
 The zone axis can be found by setting two faces in the
zone on the same great circle, and counting 90o away
from the intersection of the great circle along the E-W
axis
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
As an example, the ρ and φ angles for the (111) crystal
face in a crystal model is shown here. Note again that the
ρ angle is measured in the vertical plane containing the c
axis and the pole to the face, and the φ angle is measured
in the horizontal plane, clockwise from the b axis.
D and E are spherical
projections, i.e., where
the pole to the faces
intersect the inside of the
sphere
D' and E' are
stereographic projections,
when DS and ES
intersect the equator
(when projected to the
south pole)
Distance GD' = f(ρ)
as ρ  90 D’  G
as ρ  0 D’  O
Fig 6.3
Example: Stereogram in next slide
Projection of the
upper faces to
the south pole
http://super.gsnu.ac.kr/lecture/wulff/wulff-1.html
Stereographic projection for faces of an isometric
crystal
Note how the ρ angle is measured as the distance from the center
of the projection to the position where the crystal face plots
The φ angle is measured around the circumference of the circle,
in a clockwise direction away from the b crystallographic axis, or
the plotting position of the (010) crystal face.
Explanation of Previous Slide
• In the previous slide, the upper faces of an isometric
crystal are plotted. These faces belong to forms {100},
{110}, and {111}
• Form: set of identical faces related by the
rotational symmetry (shown by poles/dots in
stereograms)
• Faces (111) and (110) both have a φ angle of 45o
• The ρ angle for these faces is measured along a line
from the center of the stereonet (where the (001) face
plots) toward the edge. For the (111) face the ρ angle is
45o, and for the (110) face the ρ angle is 90o
As an example all of the faces, both
upper and lower, for a crystal in the
class 4/m 2/m in the forms
{100}(hexahedron - 6 faces), {110}
(dodecahedron, 12 faces), and
{111} (octahedron, 8 faces) in the
stereogram to the right.
Rotation axes are indicated by the
symbols as discussed above
Mirror planes are shown as solid
lines and curves, and the primitive
circle represents a mirror
plane. Note how the symmetry of
the crystal can easily be observed
in the stereogram.
http://www.tulane.edu/~sanelson/eens211/stereographic_projections.htm
How rotational axes are shown
 Axes that are parallel to the page are indicated by
straight lines with proper polygons at the end
 Solid line if these are parallel to a mirror plane
 Dashed otherwise
 Oblique axes plot as polygons between center and
primitive
 The distance between the polygon and center is
proportional to the angle between the axis and
pole to the face (ρ angle )
Mirror planes (see Figure 9.20 of Perkins)
 Horizontal mirror planes (in the plane of the page) plot
as solid primitive
 Vertical mirror planes (i.e., normal to the page) plot as
solid straight line through the center
 Inclined mirror planes (inclined to the page) plot as
solid curved great circle
Axes normal to Mirror planes
 See Figure 9.21 Perkins
 For example: 1/m, 2/m, 3/m, 4/m, and 6/m
Depiction of Rotoinversion
 Involve a combination of rotation and inversion in one
operation
 See Figure 9.22 Perkins, p. 191 for the rotoinversions on
stereonet
i.e., 1-, 2-, 3-, 4-, and 6Axis
12346-
Rotation axis We use these equivalents
360
I
180
m
120
90
60
3/m
(inversion center)
Guide for the handed out exercise
Page 1: Use the interfacial angles given for the top crystal (the
bottom crystal is given as a guide!), and determine the r and 
angles for each labeled face (use the Powerpoint slides as a guide
for how to do it). The table gives the interfacial angles for all the
faces. Do the following:
 First decide on the system of the crystal. Then orient the
crystallographic axes. For example, if the top crystal is tetragonal,
then the pole to faces ‘y’ and ‘z’ (i.e., top and bottom faces) is the c
axis, and the poles to faces ‘b’ and ‘a’ are the ‘a2’ and ‘a1’ axes,
respectively, such that  and r for face ‘b’ are 0o and 90o,
respectively, and  and r for face ‘a’ are 90o and 90o, respectively,
and r for face y is 0o (has no ). So, faces ‘a’ and ‘b’ plot on the
primitive, and faces y and z plot at the center of the stereonet.
 Plot and label the faces on the Wulff net. Add the symmetry
elements. Use solid symbol for the faces whose poles intersect the
upper hemisphere, and open symbols for those with poles
intersecting the lower hemisphere.
Guide for the handed out exercise
Page 2: Use the interfacial angles given for the top crystal, and
determine the r and  angles for each labeled face.
Do the following:
 First decide on the system of the crystal. Then orient the
crystallographic axes. For example, if the crystal is
monoclinic, the ‘b’ axis must be selected at the intersection of
the principal faces, perpendicular to the mirror, and parallel to
the 2-fold axis. The zone axis is the ‘c’ axis. See slide #32.
Also read the note at the bottom of page 2!
 Having the ‘b’ axis ( = 0o and r= 90o) and the ‘a’ and ‘c’ axes,
plot all the faces on the stereonet.
 Plot and label all the faces on the Wulff net. Add the symmetry
elements. Use solid symbol for the faces whose poles intersect
the upper hemisphere, and open symbols for those with poles
intersecting the lower hemisphere.
Guide for the handed out exercise
 Page 3: The r and  angles are given for a crystal faces ‘a’
through ‘r’ of a crystal (not shown).
 Plot all the faces on the Wulff net using these angles
 Label all the points on the net to correspond with the faces
 Put the elements of symmetry (e.g., 2-fold axes, m planes)
on the net