Notes on Spherical Trigonometry

3.60 Symmetry, Structure and Tensor Properties of Materials NOTES ON SPHERICAL TRIGONOMETRY Spherical Trigonometry differs from Euclidean Plane Geometry in that all of the action takes place on the surface of a sphere which, for convenience, we can take as having unit radius. A GREAT CIRCLE is defined as the circle of intersection (unit radius) of the sphere and any plane passed through the center of the sphere. A SMALL CIRCLE (k< 1) is the circle of intersection with the sphere of any plane which intersects the sphere but does not pass through its center. DISTANCE BETWEEN TWO POINTS: Consider two points A & B on the surface of the sphere. The "distance" between A & B is not the line connecting them (as in plane geometry). As we must remain on the surface of the sphere, we define A & B and O, the center of the sphere, as points which intersect the sphere. The length of the arc connecting the points A & B, between A & B, and at the center of the sphere. (Note that this definition makes lengths independent of the radius of the reference sphere. Moreover, as lengths are measured in angles, one can speak of trigonometric functions of lengths, mind-boggling as this might be in plane geometry). POLE of AN ARC or of A GREAT CIRCLE: We define as the pole of arc AB the point of intersection with the sphere of a perpendicular line constructed at the center of the sphere and normal to the plane of the great circle which defines AB. (eg. The Earth's North Pole is the pole of the Equator.) A property of a pole is that it is a "distance" of 90° away from any point on arc AB. Spherical Angles We define as the measure of the spherical angle < BAC in a spherical triangle, the dihedral angle between the planes which define the great circles of arcs BA and AC. Polar Triangles [Diagram: A spherical triangle ABC with points A, B, and C marked on the sphere.] CONSIDER A SPHERICAL TRIANGLE ABC. FIND: A', the pole of arc BC B', the pole of arc AC C', the pole of arc AB Now, consider A', B', C' A'B'C' is said to be the polar triangle of ABC An amusing feature of these triangles is that they are mutually polar - that is, if A'B'C' is the polar-triangle of ABC, then ABC is also the polar triangle of A'B'C'. Proof: B' is pole of AC, so B' is 90° away from any point on AC, in particular point A. C' is pole of AB, so C' is 90° away from any point on AB, in particular point A. Thus point A is now established as 90° from B' and 90° from C'. Thus A must be the pole of B'C'. Similar arguments show that B is the pole of A'C', and C is the pole of A'B'. QED. In two polar triangles, the angle in one is the supplement of the opposite side in the other. CONSIDER A PAIR OF POLAR TRIANGLES, ABC AND A'B'C'. LET SIDE ∠BAC = α SIDE B'C' = a' This theorem then states: α' + α = 180° Proof: Extend the two great circles which define AB and AC until they intersect B'C' at points P and Q', respectively. B' is pole of AC, so B'P' = 90° C' is pole of AB, so C'Q' = 90° Thus B'C' = (B'P' + P'Q') + (Q'C') = α' + P'Q' But the length of P'Q' is numerically equal to the angle used to measure α (see definition of spherical angles, above) α' + α = 180° QED. WHAT OPERATION ARISES which RELATES ① DIRECTLY TO ②? WE MAY ANSWER THE QUESTION BY THE PROCESS of ELIMINATION. ONLY 4 OPERATIONS ARE POSSIBLE IN A 3-DIMENSIONAL SPACE: TRANSLATION, REFLECTION, ROTATION, AND INVERSION. REFLECTION AND INVERSION CHANGE "HANDEDNESS" AND WON'T DO THE JOB. TRANSLATION PRODUCES AN OBJECT OF THE SAME HANDEDNESS", BUT MOTIONS AND ARE NOT PARALLEL. MOREOVER, THE DISTANCE BETWEEN AND DEPENDS ON EXACTLY WHERE THE INITIAL MOTI is PLACED RELATIVE TO AX TRANSLATION WON'T WORK EITHER. THEREFORE, BY ELIMINATION, THE THIES OPERATION MUST BE A ROTATION; Bp . Ax = C WE CAN FIND THE LOCATION of Axis C BY FINDING THE SMALL CIRCLE which CONNECTS AND AND PASSING A PERPENDICULAR THROUGH THE CENTER of the sphere. NOTE THAT IS THE SENSE of ROTATION of Ax AND Bp IS THE SAME, Axis C ALSO ACTS IN THE SAME DIRECTION. TO ANSWER OUR QUESTION QUANTITATIVELY, WE ALSO NEED TO KNOW THE MAGNITUDE of Y AND ALSO THE ORIENTATION of C RELATIVE TO THE FIRST TWO AXES. IN OTHER WORDS, FOR A AND Bp COMBINED AT A PARTICULAR ANGLE C wish To Know THE VALUES of Y AND THE TWO REMAINING INTER-AXIAL Angles a AND b. CLEARLY, The Values of a, b AND & DEPEND ON THE VALUES SELECTED for a, β ALD C CRYSTALLOGRAPHIC CONSTRAINTS WE HAVE SHOWN THAT THE PATTERNS of ATOMS IN CRYSTALS MUST, BY DEFINITION, BE TRANSLATIONALLY PERIODIC AND THAT ONLY 1-folo, 2-folo, 3-folo, 4-folo AND 6-FOLD ROTATION AXES MAY COEXIST With A TRANSLATION. ACCORDINGLY, IF WE RESTRICT OUR ATTENTION TO TO THE ANGULAR Theours of 1,2,3,4 OR 6. THERE IS NOW A RATHER SEVERE CONSTRAINT On The Aile Car which A AND & ARE COMBINED & AISO CANNOT BE ANY OID VALVE. IT MUST ALSO TURN OUT TO BE THE ANGULAR Throw of ErthER 1, 2, 3, 4 or 6. CRYSTALLOGRAPHIC ROTATION PATES, & & & CANNOT BE AWY DO VAIVES, BUT WAI BE TO EULER'S CONSTRUCTION IT IS NOT EXACTLY OBVIOUS HOW ONE OSTAINS a b & c IN TERMS of a, β AND C. THE REACIONSHIP MAY BE CATAINED FROM A CONSTRUCTION DUE TO THE MATHEMATICIAN LEONHARD EUIER (1707-1785). CONSIDER THE POINTS A AND B WHERE AXES AX AND BB EMERGE FROM THE REFERENCE SPHERE. CONNECT AB. BY A GREAT CIRCLE. The LENGTH of the ARC AB is thus EqUAL To e, The Aught was which AR AND B3 ARE COMBINED