Symmetry and Properties of rystals (MSE638) Spherical Trigonometry Somnath howmick Materials Science and Engineering, IIT Kanpur March 7, 2018
Spherical Triangle Great circle: circle (unit radius) of intersection of the sphere and a plane passing through it s center. Spherical triangle- arcs made by 3 great circles intersecting at. Sides of the spherical triangle are denoted by lower case a, b, c. Spherical length of a side - draw a great circle passing through two points (say ) and measure the angle at center (a = O). onvention: take the shorter arc, i.e., spherical length always < π. oth vertices and angles at vertices denoted by upper case,,. Spherical angle : dihedral between the planes defining the great circles of arcs and. onvention: spherical angles are always < π. 2 / 12
Polar Triangle Pole of arc - through origin, draw a perpendicular to the plane of great circle. is the pole of, is the pole of, is the pole of. Note that, is located in the same half as. Note that, is located in the same half as. Note that, is located in the same half as. Polar triangle - spherical triangle made by joining the poles. is the polar triangle of the spherical triangle. 3 / 12
Theorem Suppose l is a spherical line and P is a point not located on l. P π/2 π/2 l 1 If P is a pole of l, for any on l, spherical distance P = π 2. P O = π = spherical distance P = π. 2 2 P O = π = spherical distance P = π. 2 2 2 For, on l, if P = P = π 2, then P is pole of l. OP is perpendicular to the great circle passing through. 3 P is pole of l &, on l. Spherical distance = P = P. = O = P. P 4 / 12
Theorem 1 If is polar triangle to, then they are mutually polar, i.e., is also polar triangle to. : pole of spherical length = π 2. : pole of spherical length = π 2. This implies that is the pole of. 5 / 12
Theorem D E 1 If is polar triangle to, then + = π. : pole of E spherical length E = π 2. : pole of D spherical length D = π 2. E + D = D + DE + E }{{} +DE = + DE = π and are mutually polar. Since is pole to DE, spherical length DE =. Thus, + = π. Similarly, + = π and + = π. 6 / 12
Polar Triangle b c a π- ' π-a ' π-c π- is polar triangle to Spherical length = π Spherical length = π Spherical length = π is polar triangle to Spherical length = π = π a Spherical length = π = π b Spherical length = π = π c Spherical angles and spherical lengths in a polar triangle can be expressed in terms of angles and lengths of. π- π-b ' 7 / 12
ombination of Three Rotation xes γ α β L L L O β α = γ. α, β, γ are crystallographic rotation axes. 8 / 12
ombination of Three Rotation xes O γ α β b a c b α b c a β a ' O: origin of the sphere,, and are the points where rotation axis α, β and γ crosses the surface of the sphere.,, are arcs, part of three great circles. a: angle between β and γ. b: angle between γ and α. c: angle between α and β. pply rotation α to the axis γ moves to. Follow this by applying rotation β moves to. 9 / 12 '
ombination of Three Rotation xes b a b γ/2 a b α c β a α/2 β/2 c ' pply rotation α to the axis γ moves to. Follow this by applying rotation β moves to. Rotation preserves length as well as angle between two vectors. Spherical distance = = b and = = a. Thus, spherical triangles and are isosceles. Thus, spherical angle = α 2, = β 2 and = γ 2. 10 / 12
pply Law of osine γ/2 b a α/2 β/2 c π-β/2 ' π-a ' π-c π-γ/2 π-α/2 π-b ' cos c = cos a cos b + sin a sin b cos γ 2 Note that, we know α, β & γ and the above equation does not help to solve a, b, c!! Let s work with the, polar triangle to. cos(π γ 2 ) = cos(π α 2 ) cos(π β 2 ) + sin(π α 2 ) sin(π β 2 ) cos(π c) Simplifying the above equation: cos c = cos γ 2 +cos α 2 cos β 2 sin α 2 sin β 2 Similarly: cos a = cos α 2 +cos β 2 cos γ 2 sin β 2 sin γ 2 Similarly: cos b = cos β 2 +cos α 2 cos γ 2 sin α 2 sin γ 2 11 / 12
Summary We derived how three rotation axes α, β and γ, intersecting at a point in space, are oriented with respect to each other. cos a = cos α 2 +cos β 2 cos γ 2 sin β 2 sin γ 2 cos b = cos β 2 +cos α 2 cos γ 2 sin α 2 sin γ 2, a is the angle between β and γ., b is the angle between α and γ. cos c = cos γ 2 +cos α 2 cos β 2, c is the angle between sin α 2 sin β α and β. 2 We are going to use these relations to derive the crystallographic 3D point groups. 12 / 12