Quaternions and Projective Geometry (January 1, 1903)

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Royal Astronomer of Ireland, Prof. Charles Jasper Joly, details his mapping of quaternions onto projective geometry. Taken from the Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character Vol. 201, (1903), pp. 223-327. Public Domain work.
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  [ 223 VIII. Quaternions and Projective Geometry. By Professor Charles Jasper, Joly, Royal Astronomer of Ireland. Communicated by Sir Robert Ball, F.R.S. Received November 27, — EeadDecember 11, 1902. Introduction. A quaternion q adequately represents apoint Q to which a determinate weight is attributed, and, conversely, when the point and its weight aregiven, thequaternion is defined without ambiguity. This is evident from the identity <? = (i + ||)sg (A), in which Qq is regarded as a weight placed at the extremity of the vector drawn from any assumed srcin o. It is sometimes convenient to employ capitals Q concurrently with italics q to represent the same point, it being understood that Q = <?;== 1 + OQ (C). Thus Q represents the point Q affected with a unit weight. The point o may be called the scalar point, for we have In order to develop the method, it becomes necessary to employ certain special symbols. With one exception these are found in Art. 365 of 'Hamilton's Elements of Quaternions/ though in quite a different connection. We write (a, b) = bSa - aS6, [a, 6] = V . VaVb .-(E); and in particular for points of unit weight, these become (a, b) = b — a, [a, b] = V.VaVb = V. Va . (b — a) . . . (F). Thus (ab) is the product of the weights SaS6 into the vector connecting the points, and [ab] is the product ofthe weights into the moment of the vector connecting the points with respect to the scalar point. The two functions (ab) and [ab] completely define the line ab. VOL. CCL— A 338. 20.6.03  224 PROFESSOR C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. Affain Hamilton writes [a 9 b,c] = (a,b,c) — [b,c]Sa — [c,a]S6 — [a,&]Sc; (a,&,c) = S[a,&, c] = SVaV&Vc . (G) or if we replace a, 6, c by (1 + a)Sa, (1 + /2)S6, (1 + y)Sc, where a, /3 and y are the vectors from the scalar point to three points a, 6 and c, we have [A, B, C] = 8ot/3y — V(/3y + ya + a/3) ; (a, B, c) = Sa/3y . . , '(H). Hence it appears that [a, 6, c] is the symbol of theplane a 9 b 9 c ; for —V [a, &, c] (a, 6,c) 1 is the reciprocal of the vector perpendicular from the scalar- point on that plane. Also (a, b, c) is the sextupied volume of the tetrahedron oabc. Again, Hamilton writes for four quaternions (abed) = S . a[bcd] ......... (I) and in terms of the vectors this is seen to bethe products of the weights into the sextupied volume of the pyramid (abcd). Other notations may ol course be employed for these five combinatorial functions oftwo, three, orfour quaternions or points, but Hamilton's use of thebrackets seems to be quite satisfactory. In the same article Hamilton gives two most useful identities connecting any five quaternions. These are a(bode) + b(cdea) + c(deab) + d(eabc) + e(abcd) = . . . . (J), and e(abcd) = [bcd]8ae — [acd]Sbe + [abd]$ce — [abc]$de . . . (K), which enable us to express any point in terms of any four given points, or in terms of any four given planes. The equation of a plane may be written in the form and thus Z, any quaternionwhatever, may beregarded as the symbol of a plane as well as of a point. On the whole, it seems most convenient to take as the auxiliary quadricthesphereof unitradius S a s = (M)whose centre is the scalar point. With this convention theplane Blq = is the polarof the point I with respect to the auxiliary quadric ; or theplane is the reciprocal of the point Z. Thus the principle ofdualityoccupies a prominent position. The formulae of reciprocation ([abc] ; [abd]) = [ah] (abed) ; [ [abc] ; [abd] ] = — (ab) (abed) . . . (N) connecting any four quaternions are worthy of notice, and are easily proved by  PROFESSOR 0. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 225 replacing thequaternions by 1 + a > 1 + A 1 + y, and 1 + §respectively. In complicated relations it may be saferto separate thequaternions as in theseformulas by semi-colons, but generally the commas or semi-colons may be omitted without causing any ambiguity. These new interpretations are not in the least inconsistent with any principle of the calculus of quaternions. We are still at libertyto regard a quaternion as the separable sum ofa vector and a scalar, oras the ratio or product of two vectors, oras an operator, as well as a symbol ofa point or of a plane. In particular, in addition to Hamilton's definition of a vector as aright line of given direction and of given magnitude, and in addition to his subsequent interpre-tations ofa vector as the ratio or product of two mutually rectangular vectors,or as a versor, we may now consider a vector as denoting the point at infinity in its direction, or theplane through the centre of reciprocation. For the vector OQ of equation (B) becomes infinitely long if Sq = 0, and the plane Blq = passes through the scalar point if 81 = 0. We may also observe that the difference of two unit points A — b is the vector from one point B to the other A, and this again is in agreement with the opening sections of the   Lectures. Additional illustrations and examples may be found in a paper on   The Interpre- tationof a Quaternion as a Point-symbol,   ' Trans. Roy. Irish Acad./ vol. 32, The only other symbols peculiar to this method are the symbols for quaternion arrays. The five functions (a&),[a&], [a&c],(a&c), and (abed) areparticularcases of arrays, being, in fact, arrays of one row. In general thearray of m rows and n columns CO} C'o eta ... (%% 1 £ S ããã *-'u < ã » ã » ã ã s (O) I PiP% Ps ãã ã Pti ) may be defined as a function of mn quaternion constituents, which vanishes if, and only if, the groups of the constituents composing the rows were connected by linearrelations with the same set of scalarmultipliers. In other words, the array vanishes if scalars t l9 t 2 . . . i n can be found tosatisfy the m equations t l h [ + t^h 4 ããã H wii := 0? aãã hPi + hPz+ ã ã ã +t»pH = o. The expansion of arrays is considered in a paperon   Quaternion Arrays/' c Trans. Roy. Irish Acad./ vol. 32, pp. 17-30. vol. cci. a, 2 G  226 PEOFESSOE C. J. JOLY ON QUATERNIONS AND PEOJECTIYE GEOMETEY. SECTION I. Fundamental Geometrical Properties of a LinearQuaternion Function. Art. Page 1. Definition of a linear quaternion function 226 2. The general linear transformation effected by a linear function . 226 3. Specification of a function by four quaternions or five points and their deriveds . .. 227 4. The transformation of planes effected by theinverse of the conjugate function . .. 227 5. Geometrical interpretation of Hamilton's method of inversion 227 6. Geometrical illustration of the relations connecting Hamilton's auxiliary functions . 228 7. The united points of a linear transformation. .............. 229 8. Eelations connecting the united points of a function/ with those of its conjugate/' . 230 9. Introduction of the functions /o = | (/+/'), f, = \ (/-/') ã 231 10.Scjf Q q = and &$/,$> — representthegeneral quadric surface and the general linear \J V-IJ-J-i- L/XC-A. f « ff ã *s ãa a© sb 8D a aa«e eããas « iwl) 1 11. The pole of a planeS#5 = to thequadric is/o 1 ^; and thepoint of concourse oflinesof the complex in the plane is/ 1 /;. . .. ... 232 12. The united points of/ form a quadrilateral on the sphere of reciprocation .... 232 1. The quaternion equation f(p + q)=fp+fq ......... (1), may be regarded as a definition of the nature ofa linear quaternion function f thequaternions p and q being perfectly arbitrary. As a corollary, if x is any scalar, and on resolving fq in terms of any four arbitrary quaternions a 1; a. 2 , a 3 , a 4 . we must have an expression ofthe form fq = a 1 S6 1 g f + oc^Bb^q + a 3 S& 3 g r + aJ8&4# ... .. . (3), because the coefficients of the four quaternions a must be scalar and distributive functionsof 5. Sixteen constantsenter into thecomposition of the function/, being four for each of thequaternions 6. 2. When a quaternion is regarded as the symbol of a point, the operation of the function f produces a linear transformation of the most general kind. The equations f(xa + yb) = xfa + yfb ; f(xa + yb + 2c) = ,r/a + ?//& + zfc . . (4), show that the right line a, b is converted into the right line fa, fb 9 and the plane containing three points a, 6, c into theplane containing their correspondents, fa,fb andyb. The homographic character of thetransformation is alsoclearly exhibited.
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