stereographic projections 1up

8/11/2019 Stereographic Projections 1up

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Stereographic projections

Stereographic projection is a graphical

technique for representing the angular

relationships between planes and directions in

crystals on a 2D piece of paper Can be used to calculate angles between planes

etc.

Is used in the interpretation of Laue photographsfor the orientation of crystals


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Stereographic projection 2

We can represent the

orientation of a plane using

the normal to that plane

If we inscribe a sphere aroundthe crystal of interest, the

point(s) where the normal(s)

intersect the sphere are thepoles of the planes {100} poles of a

cubic crystal


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Stereographic projections 3

The projection of a plane (trace)

passing through the origin of thecrystal onto the surface of the sphere is

a great circle

The projection of a plane that does not

pass through the origin is a small circle

We can in principle measure the angle

between two plane normals on the

surface of the sphere to find the anglebetween two planes

We make this measurement along a

great circle (MLK in figure) Great circles for

the two marked planes


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Stereographic projections 4 Making measurements on the surface of a

sphere is tricky

Project everything from the sphericalsurface onto a plane

Pick a diameter of the sphere, put planeperpendicular to diameter and in contact

with one end (or through the middle of thesphere), project from other end of diameterthrough entity to be projected onto theplane

As drawn, entities in hemisphere near B

will end up outside the basic circle. Pointson hemisphere including A will end upinside.

To avoid this problem, change projectionpoint to the other end of diameter anddistinguish points in the two hemispheresby marking them with different symbols(usuall o en versus filled in)


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Wulff net Problems involving the

stereographic projection are oftenhandled using a Wulff net

Imagine a globe with lines of latitudeand longitude marked on the surface.

Orient the globe so that the NS axis isparallel to the projection plane andproject all the lines onto the plane

The longitude lines end up as great circles

in the projection and the latitude lines assmall circles

The lines in the projection can be usedto read off angular coordinates

Just like using latitude and longitude tospecify geographical location


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Angular measurement on a Wulff net

Read off angles

between poles

along greatcircles

- Not along small

circles


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Example projection of poles for crystal faces


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Different habits for cubic crystals


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Using a Wulff net 1 A Wulff net is usually used by

drawing the stereographic projection

under study on tracing paper, placingthe tracing over the net so that theircenters coincide and putting a pinthrough their centers. Rotation of the

tracing about the pin does not changethe angular relationship between thepoles (equivalent to rotating sphereabout projection axis)

To measure an angle between twopoles, rotate the tracing until thepoles of interest lie on the same greatcircle and then read off the angular

difference


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Finding the trace of a pole

The projection of a plane

corresponding to a pole iscalled the trace of the pole.

The great circle representing

the trace can be found byrotating the projection untilthe pole lies on the equator ofthe Wulff net. The trace is

then the great circle 90 fromthe pole


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Rotation of a projection about an axis in

the projection plane

Rotate the projection about

the center until the desiredrotation axis is coincident

with the NS axis

Move points along (orparallel) to small circles

through the desired rotation

angle A1 moves to A2

B1 moves to B2


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Rotation about a direction (pole) that is

inclined to the projection plane

To rotate about the pole B1 by 40

Bring rotation axis to projection

center by

Rotation around center to bring axis

onto equator

Rotation around equator by 48

Brings B1 to B2

Brings A1 to A2 Rotate around B2 by 40

Brings A2 to A3

Move rotation axis back to original

orientation

Moves B2 to B3

Moves A3 to A4

Then rotate around projection center

to get rotation axis back to starting

B1 position

Starting positions

Final positions

Moves involved in rotation


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Standard projection A standard projection shows the angular relationships

between different poles for a given crystal orientation Useful for identifying crystal orientations

Note all reflections on a

common great circle

belong to the same zone.

The zone axis lies at 90

to the zone


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Determining Miller indices for poles Compare unknown pole to standard projection or

measure angle of projection For orthogonal cell, indices hkl for the pole obey

h:k:l = acos : bcos : ccos

Where a, b and c cell constants

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