Crystallography- Stereographic Projection Flashcards
The two consecutive transformations for display
Project all typical directions (e.g normal planes) of the unit cell onto a unit sphere. Each direction is a point on the sphere (θ,ψ).
Project the sphere (one hemisphere at a time) on to a plane sheet of paper (screen) using stereographic projection
Where is the light source and screen for stereographic projection?
The screen is through the equator of the sphere. If there is a point on the surface of the northern hemisphere, the light source is from the South Pole and vice versa.
How to construct the stereogram for a point
Draw a line from the light source to the point on the sphere in the opposite hemisphere. Where this line crosses the screen (equator) is where it is shown on the 2D circle fro the stereogram.
Angles on a stereogram
For 2 points on the stereogram, if a line is drawn from the origin to each, ψ is the angle between these lines. When constructing the stereogram, the angle from the vertical central axis to the point on the sphere is θ. The line between the light source and the point on the sphere is θ/2 from the vertical central axis. The length to the translated point on the screen is rTan(θ/2).
Lines and dots on sterograms
Draw one (or 2, 3) thin lines to assist eye in orientation (no mirrors). Put a dot on a random place on the stereogram (not on mirror plane or special axis). Use symmetry elements to generate copies of the dot at other places related by symmetry. Dots in the top hemisphere are filled, below they are empty. Dots in a circle indicate top and below
How does a stereogram show 3-fold rotation inversion symmetry
Open and filled circles alternate upon application of this operation.
How can rotation axes be shown?
Treat these as the lines projected onto the sphere. Do normal stereographic projection for points where axes meet sphere. Put triangle or oval or whatever order it is as the point on the stereogram
