The possibility of using homogeneous (projective) coordinates in 2D measurement exercises Možnost uporabe homogenih (projektivnih) koordinat v dvodimenzionalnih merskih nalogah Aleksandar Ganič ', Milivoj Vulič2, Franc Runovc2, Tina Habe3 University of Belgrade, Faculty of Mining and Geology, Bušina 7, 11000 Belgrade, Serbia: E-mail: aganic@rgf.bg.ac.yu 2University of Ljubljana, Faculty of Natural Sciences and Engineering, Aškerčeva cesta 12, SI-1000 Ljubljana, Slovenia; E-mail: milivoj.vulic@ntf.uni-lj.si, franc.runovc@ntf.uni-lj.si 3Geoinvest d.o.o., Dimičeva ulica 16, SI-1000 Ljubljana, Slovenia; E-mail: tinahabe@gmail.com Received: November 29, 2007 Accepted: January 12, 2008 Abstract: This work's intention is to present the basic characteristics of projective geometry and the use of homogeneous (projective) coordinates in two-dimensional (further denoted as 2D) measurement exercises. The concept of a projective plane originates from the Euclidean plane, assuming all our given points are ideal and lie upon an ideal line verging towards infinity. The term "ideal point" is taken to mean an intersection of all lines that are parallel in the finite space. By introducing these so-called ideal points and ideal lines, the calculations in 2D measurement exercises - ones that are usually carried out under the rules of Euclidean trigonometry - have been simplified, as the calculations of directional angles and lengths are no longer necessary. As a practical example of the use of projective coordinates, an intersection is presented, such that it can also be used for the Collins Method of Resection, seeing as it is based upon using said intersection twice. Izvleček: Predstavljene so osnovne značilnosti projektivne geometrije in s tem uporaba homogenih (projektivnih) koordinat v dvodimenzionalnih merskih nalogah. Projektivno ravnino dobimo iz evklidske ravnine, če privzamemo točke in premico v neskončnosti na kateri ležijo vse točke v neskončnosti. Neskončna točka predstavlja presečišče vseh premic, ki so v končnosti med seboj vzporedne. Z uvedbo neskončnih točk in neskončne premice se izračuni v dvodimenzionalnih merskih nalogah, ki se navadno vršijo po pravilih evklidske trigonometrije, poenostavijo saj računanje smernih kotov in dolžin ni potrebno. Kot praktični primer uporabe projektivnih koor- dinat je prikazan zunanji urez, ki ga je možno uporabiti tudi pri Collinsovi metodi notranjega ureza, saj temelji na dvakratni uporabi zunanjega ureza. Key words: homogeneous coordinates of a point, homogeneous coordinates of a line, incidence relation, principle of duality, ideal point, ideal line, intersection, resection Ključne besede: homogene koordinate točke, homogene koordinate premice, relacija incidence, princip dualnosti, neskončna točka, premica v neskončnosti, zunanji urez, notranji urez Introduction In 2D measurement exercises, all calculations are usually carried out under the rule of Euclidean geometry - where the points, lines and their relationships are defined differently than in a projective plane - in which the coordinates of an unknown point are established through the aid the calculation of so-called lengths and angles of our site. As a reaction to the latter, the article at hand is meant to present the basic relations between points and lines of the projective plane and depict their use in 2D measurement exercises. A practical example of using said projective coordinates would be an intersection, in which the coordinates of a new point are calculated from the measured angles of given points of an existing triangulation network. In what follows, the coordinates of a point and the formulas of lines on a Euclidean plane shall be marked using upper case, whereas the coordinates of points and formulas of lines on the projective plane will be denoted using lower case letters. Point and line In projective geometry, a point is defined as a set of three coordinates that equal the set (y x a) and therefore obviously also as an ordered set of three numbers (y x a) - which do not all equal zero at the same time, since then (ky kx ka) would be the same point for any given k ^ 0. For example, (2 3 6) is our exemplary point, and '/2 1) is another of the numerous ways to mark that exact same point, bearing in mind the principle that an unlimited number of sets of three numbers (y x a>) may correspond to each point, but only one point may correspond to each ordered set. Furthermore, from non-homogeneous coordinates of any given point, we beget an infinite number of sets of homogeneous coordinates of that same point: (1) And from homogeneous coordinates of that certain point, we can get one single ordered pair of numbers: 0=>X CO v y ^Xy1 Xx Afci, \ J M (i \ Ky X<£> Xx Xx )M Xdi X (9) (10) The ideal point and line The parallels (coefficient of site k = kt = k2) l and l2 with formulas Y = kjX + n and Y = k2X + n2 are given. For X, we enter values X = -§■ , and for Y values Y = m , then multiplying the equations with m, which in turn gives us the formulas of lines using homogeneous coordinates: y = kx + and y = kx + n2m (11) Considering that the lines are parallel, we are interested in the set of three, (v u w), such that it must correspond to both formulas. By subtracting the equations we get: rn(n1 - n2) = 0. As nj f n2, then m = 0 and the equations y = kx + n¡rn and y = kx + n2a> are reduced into y = xk. Since we are dealing with homogeneous coordinates, we can say that x = 1. From this, we come to the conclusion that y = k. Thus, we obtain a set of three, (k 1 0), which does indeed correspond to both equations. If the lines lj and l2 are parallel to the y-axis, then the formulas of the lines in homogeneous coordinates have the form x = mx1 and x = mx2. In this case, the set of three (0 1 0) corresponds to both formulas. To summarize, the set of three, (k 1 0), corresponds to a formula of the forms y = kx + np and y = kx + n2m only when k = kj = k2, or when the lines are parallel and the coefficient of site equals k. The set of three (0 1 0) corresponds to all formulas of the form x = mxj that describe the parallels of the y-axis. A bouquet of parallel lines (all parallel lines are of the same class and form organised heaps of parallel lines denoted as "bouquets") defines a point Px in projective plane that has been defined as an ideal point. The bouquet P consists of all lines that are parallel to a certain line l. The equation pertaining to line l is Y = kX + n, or X = Xp if it is parallel to the y rather than the x-axis. The line l belongs to bouquet Px exactly when the set of three (k 1 0) corresponds to the equation of line l in its homogeneous coordinates, and to pencil Px exactly when what corresponds to this equation is this set of three: (0 1 0). Consequentially, we can have the set of three (k 1 0) in the former case, and the set of three (0 1 0) in the latter case for homogeneous coordinates of the ideal point Px. Since we may multiply homogeneous coordinates with any number that is different than zero, we may say that the set of three (y x 0) represents the homogeneous coordinates of one ideal point, where y and x are any given elements different from zero. In this way, we have adjusted our homogeneous coordinates (y x a) so that they befit each and every point of our projective plane. The point with such coordinates also lies in the Euclidean plane if a f 0, and is an ideal point when a = 0 [4]. Two ideal points define the ideal line l q The solution of the system are the very coordinates of our ideal line l o or, if: (12) (13) (14) The set of three (0 0 1) represents the coordinates of an ideal line such that all ideal points lie on it. The ideal line and angle of site An ideal point represents the intersection of a group of all lines that are parallel to one another in finite space. A specific ideal point upon an ideal line belongs to each group of parallel lines that in finite space represent a so-called »angle of site« between the lines of a certain class and the positive end of the x-axis. The coordinates of the line that goes through the points Pl(yl xl ml) and P2(yl + d cos 9 xl + d sin 9 m2), where 9 is the so-called angle of site are enclosed within the line PlP2 and the positive side of x-axis, and d represents the distance between the points Pl and P2, which would be: (v u w) = i (xl(m2 - m) - d cos 9®l y(ml - m2) + d sin 9m 2 d(yl cos 9 - xl sin 9)) (15) If : m2 = m = l => (v u w) = ¡1 (-d cos 9 d sin 9 d(yl cos 9 - xl sin 9)) (16) or, when: (m2= ml= l A d f 0 A d f A ¡1 =d) => (v u w) = (- cos9 sin9 (yl cos9 - xl sin9)) (17) The set (v u w) = (- cos 9 sin 9 (yl cos 9 - xl sin 9)) represents the coordinates of the line that is notated using polar coordinates. The intersection of the line given with polar coordinates and the ideal line (0 0 1) is the ideal line, now denoted using polar coordinates: *in

(y x m ) = (sin 9 cos 9 0) (19) The notation y xm mj = (sin 9 cos 9 0) at the same time alos represents standardised coordinates of said ideal line, for which the following is valid: (20) where: Should the lines lj in l2 enclose the angle a , and the line l} is defined through the ideal point (sin cos 9; 0), then the ideal point of line l2 is defined as (sin (9^+a) cos (9^ + a) 0), where 9;+a is the angle of site of the line l2: sjVfl, Figure 1. The angle between our two lines and angles of site Slika 1. Kot med premicama in smerna kota Intersection An intersection is both a measurement and calculation method, with the aid of which the coordinates of a new point can be calculated from measured angles or (outer) directions upon given points of the existing triangulation network. A given new point is determined as an intersection of several outer directions that are controllably oriented at each standpoint. The coordinates of points L(Yl Xl) and R(Yr Xr) are given. Observation: We are observing the direction from two given points (L and R) towards the new point M. Angle a is observed from the point L between points M-left and R-right, and angle P from point R between points L-left and M-right. Based on the given and observed information, we must establish the coordinates of the unknown point M, which would translate into us looking for XM and YM: Figure 2. Intersection Slika 2. Zunanji urez The connecting line LM denotes the line left, RM the line right, and LR the line left-right. The lines left and left-right are bisected via one another in point L and enclose the angle a, while the lines right and left-right do the same in point R, therefore enclosing the angle /. By establishing the ideal point Lm of the line left and the ideal point Rm of the line right, the lines left and right have been accurately established. Thus, we have also established their intersection M, which represents the coordinates of the point we are seeking. L denotes the ideal point on the line left,, R the ideal point on the line right and LR the ot a ^ 7 ot a c-7 ot ideal point on the line left-right. Points Lot and Rot are obtained with the help of the ideal point LRot, which is established through the bisection of line left-right with the ideal line. When the point LRot is standardised (transformed into a format such as (sin 9 cos 9 0)), we can, with the help of the latter, as well as with the help of angles a and /, determine the points L and R on the line l . a ot ot ot 1. Establishing the line left-right - The coordinates of point L are (yL = YL xL = XL 1) - The coordinates of point R are (y. = YR = Yr + Xr + d sin u R xL = X = Xr + d cos uR 1) Therefore, the following is true: The formula of the line left-right is: XL+d • cosv, 1 or, its coordinates: YL+d-sinv," XL + d • COSV f 1 Yl 1 Yl + (/ sinv/ Yl X, j • R J " a-sinv,. a cosvL 2. Establishing the ideal point on the line left-right The line left-right is bisected with the ideal point (0 0 1): The coordinates of the ideal point are denoted as follows: 0 1 y*. = i r£ 1 Y, +d ■ sinv ? 1 YL 1 0 1 Yl 1 rf sinv* y, d sinv ,R i^-cosvf = -d sinv* I Yl 1 Yl +d sinv/1 = 0 (23) (24) (25) (26) (27) (28) (29) (30) The point LRm = A( - d sin urr - d cos ur 0) is standardised, which means that it is multiplied by factor: I X = JyLK+XLK2+aLK2 Consequentially, we transform the notation LRm = X( - d sin ur- d cos ur 0) into LRa X( ± sin ur± cos ur 0). l K = - V>V +XLK2 +<0LK —=X (-¿-sinvf -rf-cosvf O) (31) K = ^(j ■ sin v * J + (d -cos v y t^>LR. =*,(-<*■ sinvf -rf cosvf 0) K = LR„=X(-d-sinv/ -rf-cosv* 0) k = = =>LR =k(-d-sinvf -d-cosvf (A 7? v > So, finally, the standardised ideal point is: or: LR, - (+ sin (v* + Jbu ) +cos ^* + kn ) o) 3. Establishing the ideal points on the lines left and right (a) Ideal point L of the line left (32) (33) (34) (35) (36) (37) The line left-right encloses, along with the positive side of x-axis, the angle uR, and with M= uR - the line left, the angle a. The angle of site of the line left is uf = u* - a and the ideal line is of the form: L = ( sin (u* - a) cos (uR - a) 0) (38) (b) Ideal point R of the line right The coefficient of site of the line right is U1 = uR + P and the ideal line is of the form: R = ( sin (u* + ß) cos (u* + ß) 0) (39) 4. Establishing the lines left and right (a) Establishing the line left: The line left goes through the points L and L: iep "M, sin v f ■ cos a - cosv * ■ si n a cosv," -cosa + sinv* -sina 0 X ^leji cosv I-cosa +3111%'"-sina 0 '■left I = 0 (40) vkfi ■ X, ¡if 0 sinv"-cosa-cosv"-sina 1 1 = cosv" ■ cosct +sinv£ -sina = cos(y"-o. (41) = -sinv" ■ cosa-cosv"'Sina = -sin(y*-a) (42) sinv, -cosa-cosv, -sina cosv, -cosa+sinv, -sina L ' i. Y Y ' left A left (43) = -Y& 'Cos(vf -a )-■ sin (v* -a ) Coordinates of the line left: (b) Establishing the line right The line right is defined by the points R and Rw: vrigln cosv£ -cos (i -sinv£ -sin p 0 I = cosv£ -cosp -sinv^ ■ sin = cos(y £ + [i^ (45) (46) 0 sinv^ - cos p +cosvj£ - sin p 1 I = -sinv^ - cosp + cosv■ sinp - -sin^v^ + p^) ^^ (48) Coordinates of the line right: 5. Establishing the unknown point M The pointM is defined as the intersection of the lines LM = L L and RM = R R: A X X = 0 (50) M = (y„ % ©„)= / «M 11- V V- v \ ".'«(ir Wrtghl *** [i] (51) Figure 3. Intersection with projective coordinates Slika 3. Zunanji urez s projektivnimi koordinatami Conclusions In projective geometry, the lines have their own coordinates inasmuch as points do. A point is defined as an ordered set of three numbers (y x m), which are not allowed to simultaneously take on the value of zero, seeing as the latter would make (ky kx km) the same point for any given k ^ 0. We can obtain an infinite number of ordered sets of coordinates of a given point from the non-homogeneous coordinates of that same point, just as well as we can obtain only one ordered pair of numbers from homogeneous coordinates. A line, much like a point, is an ordered set of three numbers, (v u w), which must not all equal zero at the same time, seeing as that would again make (pv pu pw) the same line for any given p ^ 0. The set (v u w) = (x}-x2 y-y2 y1x2-y2x1) represents the coordinates of the line that are denoted using rectangular coordinates, whilst the set (v u w) = (- cos y sin y y} cos y - x1 sin y) represents the coordinates of the line that is denoted using polar coordinates. By using projective coordinates, the establishment of the intersection R(Y X) of two lines - the line l, defined by the points At(Y; X;) and A2(Y2X2), and the line l2, on which the points B^(Y} X) and B2(Y2 X2) lie - is very simple, and is defined using the help of nine second order determinants: First, the coordinates of lines and l2 are established, for which six of secondary-order determinants need to be calculated: (52) (53) The intersection R, as the intersection of lines and l2, is established with the calcula- tion of another three secondary-order determinants: = 1 *>R = (y x «)= u, w>, i 'i W, v, V, «! V, », R=(y X <0 )= > ± [ X 1>(Y X) I (O CO (54) (55) The projective plane, as has already been mentioned, originates from the Euclidean plane when we assume ideal points and an ideal line. Thus, the point with coordinates (y x m) lies in the Euclidean plane if m f 0, and m = 0 to render our point ideal. Two different ideal points define the ideal line (0 0 1), on which all ideal points lie. The notation (sin 9 cos 9 0) represents these coordinates of our ideal point, from which we can find the angle which a given line encloses in partnership with the positive side of the x-axis. If the line ll and the positive side of the first axis enclose the angle and both ll and l2 enclose the angle a, then the ideal point of the line l2 can be determined as (sin(^±a) cos(^±a) 0), where y2 = fl±a is the angle of site of the line l2. Via the introduction of ideal points and an ideal line, we are able to avoid the process of calculation of lengths and angles of site in 2D measurement exercises (the practical example here being the one depicted using intersections). The coordinates of an unknown point may always be established as the intersection of two lines, defined by the given and ideal points. Povzetek Možnost uporabe homogenih (projektivnih) koordinat v dvodimenzionalnih merskih nalogah V projektivni geometriji imajo poleg točk tudi premice koordinate. Točka je definirana kot urejena trojka števil (y x o), ki niso vse hkrati enake nič, s tem da je (ky Xx Xo) ista točka za katerikoli X ^ 0. Iz nehomogenih koordinat neke točke dobimo neskončno urejenih trojk homogenih koordinat iste točke, iz homogenih koordinat neke točke pa lahko dobimo eno samo urejeno dvojico števil. Premica je urejena trojka števil (v u w), ki niso vse hkrati enake nič, s tem da je (^v ^u pw) ista premica za katerikoli ^ ^ 0. Trojka (v u w) = (x1-x2y1-y2yl x2- y2 x1} predstavlja koordinate premice zapisane s pravokotnimi koordinatami, trojka (v u w) = (- cos y sin y y1 cos y - x1 sin pa koordinate premice zapisane s polarnimi koordinatami. Z uporabo projektivnih koordinat je določitev presečišča R(YX) dveh premic, premice lp ki ju določata točki A1(Y1 X) in A2(Y2 X2) ter premice l2 na kateri ležita točki B1(Y1 X1) in B2(Y2X2) zelo enostavna, saj se presečišče določi s pomočjo devetih determinant drugega reda: Najprej se določijo koordinate premic l1 in l2 za kar je potrebno rešiti šest determinant drugega reda: Presečišče R, kot presek premic l1 in l2 pa je odrejeno z rešitvijo še treh determinant drugega reda: ^ II Vi? \AJ V V fJ [i = 1 => R = (y x to)= f w, w, V, V, 1/, V "2 w2 W1 v2 v2 M, R=(y x ca)= y i i = (r x i)= (k x) Projektivno ravnino dobimo iz evklidske ravnine, če privzamemo točke in premico v neskončnosti. Točka s koordinatami (y x a) leži v evklidski ravnini, če je a ^ 0 in je točka v neskončnosti, če je a = 0. Dve različni neskončni točki določata premico v neskončnosti (0 0 1) na kateri ležijo vse točke v neskončnosti. Zapis (sin 9 cos 9 0) predstavlaj koordinate neskončne točke iz katerih razberemo kot, ki ga neka premica oklepa s pozitivnim delom abscisne osi. Če oklepa premica ^ s pozitivnim delom prve osi kot p1 s premico l2 pa kot a lahko neskončno točko premice l2 določimo kot (sin (p+a) cos(p1+a) 0) pri čemer je p2 = p+a smerni kot premice l2. Z uvedbo neskončnih točk in neskončne premice se v merskih dvodimenzionalnih nalogah (praktični primer uporabe prikazan v zunanjem urezu) izognemo računanju dolžin in smernih kotov. Koordinate neznane točke vedno določimo kot presečišče dveh premic, ki jih določat dani točki ter točki v neskončnosti. References [1] Habe, T. (2007): Možnost uporabe homogenih (projektivnih) koordinat v dvodimen- zionalnih merskih nalogah: Diplomsko delo. Faculty of Natural Sciences and Engineering, Ljubljana, 77 p. [2] Ayres, F. (1967): Theory and problems ofprojective geometry. New York, US. 243 P- [3] Palman, D. (1984): Projektivna geometrija. Zagreb, Croatia, 343 p. [4] Vidav, I. (1981): Afina in projektivna geometrija. Ljubljana, Slovenia, 170 p.