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There are four important constructions that can be done very easily on the stereographic projection with the aid of a ruler, compasses and protractor.

(i) To plot the projection of a point at a given angle from the *N*-
pole. Consider a meridional section of the sphere through the point *P* as in
Fig. 6a. The point *P* can be inserted (using the protractor) and joined to
*S*. It cuts the equatorial section at *p*. A circle of radius *Op* can then
be transferred to the projection (Fig. 6b) and this circle describes the
projections of all points at from the *N*-pole. If the orientation of
the meridian on which *P* lies has already been fixed on the stereogram then *p*
is the point where this intersects the circle; if not already fixed by other
considerations such a meridian can be drawn in as a diameter of the primitive in
an arbitrary orientation.

(ii) To plot the projection of the opposite of a given point.

On the meridional section of the sphere through the point *P* (Fig. 7a) it is
easy to join *P* to the centre and extend it to the opposite point
(its antipodes). The projection of to the *S*-pole then gives its
projection at a distance from the centre. If *p* has
already been plotted on the stereograph (Fig. 7b) then it can be joined to the
centre *O* and this line continued a distance transferred from
Fig. 7a.

For the following construction (iii) we shall require the opposite of *P*
projected to the *S*-pole in this way. However, if, for other purposes, we
require to project the opposite of *P* to the *N*-pole then the procedure is
very simple; on this convention is simply obtained by joining *p*
to the centre of the stereogram and continuing this line for an equal distance
beyond the centre to make as in Fig. 8.

(iii) to draw the projection of a great circle through any two points. Let
the projections of the two points *P* and *Q* be *p* and *q* (Fig. 9). Since
every great circle through any point passes through the opposite of that point,
the great circle through *P* and *Q* must pass through , the
opposite of *P*. Thus the required projection is the circle through
, and is found by means of construction (ii). The
centre of this circle is at the intersection of the perpendicular bisectors of
*pq* and , and the circle can therefore be drawn.

(iv) To draw the projection of a small circle of radius from a given
point. Such a circle is the locus of all points at an angular distance
from the given point *P*, and is therefore very useful when we wish to plot a
point at some specified angle from another point. Consider the meridional
section of the sphere through *P* (Fig. 10a). Mark off two points, *A* and *B*
at angular distances at each side of *P*. Join *A*, *B* and *P* to *S*
to find their projections *a*, *b* and *p*. Transfer to the stereogram
distances *ap* and *bp* along the diameter through *p*. Then *a* and *b*
obviously lie on the required circle and its centre must be the mid-point of
*ab*. (Note that this centre is displaced outwards from *p* itself.) The
circle can therefore be drawn directly (Fig. 10b).

It is commonly required to find a point at a given angle from *p* and at some
other given angle from *q*. Such a point is at the intersection of the two
corresponding small circles. There are of course two such points of
intersection, but there will usually be some qualitative reason to show which of
these two points is the one required.

All the four procedures described above have involved a subsidiary drawing of a meridional section of the sphere; the actual construction has been performed on this section, and appropriate measurements have been transferred from it to the stereogram itself. This use of a subsidiary drawing is helpful in clarifying the principles involved, but it is not necessary in practice. If the meridional section is imagined to be rotated by 90 about its equatorial diameter then it coincides with the plane of the stereogram, so that the construction lines can be drawn on the stereogram itself. If they are erased afterwards then the same result is obtained directly without having to transfer a measurement. This way of carrying out construction (ii) is shown by broken lines in Fig. 11, and the method is equally applicable to all the constructions.

**Copyright © 1984, 1997 International Union of
Crystallography**