Stereographic projection *Any plane passing the center of the reference sphere intersects the sphere in a trace called great circle * A plane can be represented by its great circle or pole, which is the intersection of its plane normal with the reference sphere

Stereographic projection

Pole on upper sphere can also be projected to the horizontal (equatorial) plane

Projections of the two ends of a line or plane normal on the equatorial plane are symmetrical with respect to the center O.

Projections of the two ends of a line or plane normal on the equatorial plane are symmetrical with respect to the center O. U P’ X P’ P O O P L

A great circle representing a plane is divided to two half circles, one in upper reference sphere, the other in lower sphere Each half circle is projected as a trace on the equatorial plane The two traces are symmetrical with respect to their associated common diameter

N W E S

The position of pole P can be defined by two angles f and r

The position of projection P’ can be obtained by r = R tan(r/2)

The trace of each semi-great circle hinged along NS projects on WNES plane as a meridian

As the semi-great circle swings along NS, the end point of each radius draws on the upper sphere a curve which projects on WNES plane as a parallel

The weaving of meridians and parallels makes the Wulff net

Two projected poles can always be rotated along the net normal to a same meridian (not parallel) such that their intersecting angle can be counted from the net

P : a pole at (F1,1) NMS : its trace

The projection of a plane trace and pole can be found from each other by rotating the projection along net normal to the following position

Terms about zone Zone axis Zone pole: direct projection of a zone axis Zone plane : the plane ⊥ zone axis Zone circle: intersection of a zone plane with the reference circle

Zone circle and zone pole

Trace of P2’: zone circle If P2’ is the projection of a zone axis, then all poles of the planes parallel to the zone axis lie on the trace of P2’ P2’: zone pole Trace of P2’: zone circle

Rotation of a poles about NS axis by a fixed angle: the corresponding poles moving along a parallel *Pole A1 move to pole A2 *Pole B1 moves 40° to the net end then another 20° along the same parallel to B1’ corresponding to a movement on the lower half reference sphere, pole corresponding to B1’ on upper half sphere is B2

m: mirror plane F1: face 1 F2: face 2 N1: normal of F1 N2: normal of N2 N1, N2 lie on a plane which is 丄to m

A plane not passing through the center of the reference sphere intersects the sphere on a small circle which also projects as a circle, but the center of the former circle does not project as the center of the latter.

Projection of a small circle centered at Y

Rotation of a pole A1 along an inclined axis B1: B1B3  B2  B2  B3  B1 A1A1  A2  A3  A4  A4 A plane not passing through the center of the reference sphere intersects the sphere on a small circle which also projects as a circle.

Rotation of a pole A1 along an inclined axis B1:

A small circle with center C’, after projection, Small circle  circle A1, A4, D (centering at C) and C’ B1

Rotation of 3 directions along b axis

Rotation of 3 directions along b axis

Rotation of 3 directions along b axis

Standard coordinates for crystal axes

Standard coordinates for crystal axes

Standard coordinates for crystal axes

Standard coordinates for crystal axes

Projection of a monoclinic crystal -110 -1-10 -a 0-1-1 01-1 x x -b +b +C 0-11 011 +a 110 1-10

Projection of a monoclinic crystal

Projection of a monoclinic crystal

Projection of a monoclinic crystal

(a) Zone plane (stippled) (b) zone circle with zone axis ā, note [100]•[0xx]=0

Location of axes for a triclinic crystal: the circle on net has a radius of a along WE axis of the net

Zone circles corresponding to a, b, c axes of a triclinic crystal

Standard projections of cubic crystals on (a) (001), (b) (011)

d/(a/h)=cosr, d/(b/k)=coss, d/(c/l)=cost h:k:l=acosr : bcoss : ccost measure 3 angles to calculate hkl

The face poles of six faces related by -3 axis that is (a) perpendicular (b) oblique to the plane of projection

Homework assignment Textbook of Bloss 4-3 p1(f=60, =50), p2 (f=-70, =35) 4-5 A (f=-50, =45), B (f=20, =90) 4-8 p (f=50, =40)

Prob. 4-3 Answer sinsinf sincosf cos p1(f=60, =50) 0.6634 0.3830 0.6428 p2(f=-70, =35) -0.5390 0.1962 0.8192 C=AxB/|AxB| 0.1935 -0.91765 0.3471 C (zone axis) f= 168.10, =69.69 p1^p2 = 75.87

Prob. 4-5 Answer sinsinf sincosf cos A (f=-50, =45) -0.5417 0.4545 0.7071 B (f=20, =90) 0.3420 0.9397 C=AxB/|AxB| -0.6848 0.2418 C (face pole) f=-70.6, (F=109.4) =133.2 (=46.8) A^B = 76

4-8 Answer f=-106, =-30 (a) (b) f=-106,  = -30 f= -88,  = 60

*great circle WNES is projected to W’N’E’S’ *Pole P is projected to P’