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Spherical Trigonometry Appendix A Spherical Trigonometry In Section 2-4 we saw how efficiently geometric algebra describes relations among directions and angles in a plane. Here we turn to the study of such relations in 3-dimensional space. Our aim is to see how the traditional subjects of solid geometry and spherical trigonometry can best be handled with geometric algebra. Spherical trigonometry is useful in subjects as diverse as crystalography and celestial navigation. First, let us see how to determine the angle that a line makes with a plane from the directions of the line and the plane. The direction of a given line is represented by a unit vector while that of a plane is represented by a unit bivector The angle between and is defined by the product where i is the unit righthanded trivector, is a unit vector and The vector and trivector parts of (A.1) are Our assumption that i is the unit righthanded trivector fixes the sign of so, by (A.2b), is positive (negative) when is righthanded (left- handed). The angle α is uniquely determined by (A.1) if it is restricted to the range Equations (A.2a, b) are perfectly consistent with the conventional in- terpretation of and as components of projection and rejection, as shown in Figure A.1. Thus, according to the definition of projection by Equation (4.5b) of Section 2-4, we have We can interpret this equation as follows. First the unit vector is projected into a vector with magnitude Right multiplication by then rotates the projected vector through a right angle into the vector which can be expressed as a unit vector times its magnitude Thus, the 661 662 Appendix A product is equivalent to a projection of into the plane followed by a rotation by To interpret the exponential in (A.1), multiply (A.1) by to get This expresses the unit bivector as the product of orthogonal unit vectors and The factor has rotated into the plane. We recognize as a spinor which rotates vectors in the plane through an angle The unit vector specifies the axis of rota- tion. Now let us consider how to represent the angle between two planes algebraically. The Fig. A.1. The angle between a vec- directions of two given planes are represented tor and a bivector. by unit bivectors and We define the dihedral angle c between and by the equation Both the magnitude of the angle (with ) and the direction of of its plane are determined by (A.3). Separating (A.3) into scalar and bivector parts, we have Note that for c = 0, Equation (A.3) becomes so since The geometrical interpretation of (A.3) is indicated in Figure A.2, which shows the two planes intersecting in a line with direction The vector is therefore a com- mon factor of the bivectors and Consequently there exist unit vectors and orthogonal to such that and have the factorizations Fig. A.2. Dihedral Angle. Note that the order of factors corresponds to the orien- Note that the orientation of tations assigned in Figure A.2. The common factor is chosen so it opposes that of if is brought into vanishes when and are multiplied; coincidence with by ro- tating it through the angle c. Note that this last equality has the same form as Equation (4.9) with the unit Spherical Trigonometry 663 bivector of the plane expressed as the dual of the unit vector It should be clear now that (A.4b) expresses the fact that is a bivector determining a plane perpendicular to the line of intersection of the and planes and so intersecting these planes at right angles. Fig. A.3. A spherical triangle. Fig. A.4. Altitudes of a spherical triangle. Now we are prepared to analyze the relations among three distinct direc- tions. Three unit vectors can be regarded as vertices of a spherical triangle on a unit sphere, as shown in Figure A.3. The sides of the triangle are arcs of great circles determined by the intersection of the circle with the planes determined by each pair of vectors. The sides of the triangle have lengths A, B, C which are equal to the angles between the vectors This relation is completely described by the equations where The unit bivectors are directions for the planes determining the sides of the spherical triangle by intersection with the sphere. Of course, we have the relations as well as The angles a, b, c of the spherical triangle in Figure A.3 are dihedral angles between planes, so they are determined by equations of the form (A.3), namely, 664 Appendix A where Equations (A.5a, b, c) and (A.7a, b, c) with (A.6a, b) understood deter- mine all the relations among the sides and angles of a spherical triangle. Let us see how these equations can be used to derive the fundamental equations of spherical trigonometry. Taking the product of Equations (A.5a, b, c) and noticing that we get This equation can be solved for any one of the angles A, B, C in terms of the other two. To solve for C, multiply (A.8) by to get If the exponentials are expanded into scalar and bivector parts and (A.7a) is used in the form then (A.9) assumes the expanded form Separating this into scalar and bivector parts, we get Equation (A.10) is called the cosine law for sides in spherical trigonometry. It relates three sides and an angle of a spherical triangle and determines any one of these quantities when the other two are known. Since the value of C can be determined from A and B by (A.10), the direction is then determined by (A.11). Thus equations (A.10) and (A.11) together determine C from A and B. Nothing resembling Equation (A.11) appears in traditional spherical trigonometry, because it relates directions whereas the traditional theory is concerned only with scalar relations. Of course, all sorts of scalar relations can be generated from (A.11) by multiplying by any one of the available bivectors, but they are only of marginal interest. The great value of (A.11) is evident in our study of rotations in 3 dimensions in Section 5-3. We can analyze consequences of (A.7a, b, c) in the same way we analyzed (A.5a, b, c). Observing that Spherical Trigonometry 665 we obtain from the product of Equations (A.7a, c, b) This should be compared with (A.8). From (A.12) we get The scalar part of this equation gives us This is called the cosine law for angles in spherical trigonometry. Obviously, it determines the relation among three angles and a side of the spherical triangle. The cosine law was derived by considering products of vectors in pairs, so we may expect to find a different “law” by considering the product Inserting from (A.5b) and the corresponding relations from (A.5c, a) into we get We can find an analogous relation from the product Using (A.7b), we ascertain that Obtaining the corresponding relations from (A.7a, b), we get The ratio of (A.15) to (A.16) gives us This is called the sine law in spherical trigonometry. Obviously, it relates any two sides of a spherical triangle to the two opposing angles. We get further information about the spherical triangle by considering the product of each vector with the bivector of the opposing side. Each product has the form of Equation (A.1) which corresponds to Figure A.1. Thus, we have the equation The angles are “altitudes” of the spherical triangle with lengths as shown by Figure A.4. If the trivector parts of (A.18a, b, c) are substituted into (A.15) and (A.16) we get the corresponding equations of traditional spherical trigonometry: 666 Appendix A This completes our algebraic analysis of basic relations in spherical trigo- nometry. Exercises (A.1) Prove that Equation (A.8) is equivalent to the equations and (A.2) The spherical triangle satisfying Equations (A.5) and (A.7) determines another spherical triangle by the duality relations The triangle is called the polar triangle of the triangle because its sides are arcs of great circles with and as poles. Prove that the sides of the polar triangle are equal to the exterior angles supplementary to the interior angles of the primary triangle, and, conversely, that the sides A, B, C of the primary triangle are equal to the exterior angles supplementary to the interior angles of the polar triangle. From Equations (A.18a, b, c) prove that corresponding altitudes of the two triangles lie on the same great circle and that the distance along the great circle between and is (A.3) For the right spherical triangle with prove that (A.4) Prove that an equilateral spherical triangle is equiangular with angle a related to side A by (A.5) Assuming (A.5a, b, c), prove that Spherical Trigonometry 667 Hence, Use this to derive the sine law, and, from (A. 19), expressions for the altitudes of a spherical triangle. (A.6) Prove that Assuming (A.5a, b, c) use the identity to prove that Note that this can be used with (A. 19) to find the altitudes of a spherical triangle from the sides. (A.7) Find the surface area and volume of a parallelopiped with edges of lengths a, b, c and face angles A, B, (A.8) Establish the identity Note that the three terms differ only by a cyclic permutation of the first three vectors.
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