APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The erth s rottion cuses mss redistribution such tht the equtoril rdius is lrger thn the polr rdius resulting in n ellipsoidl shpe. The other importnt ppliction of the ellipse to modern geodesy rises from the fct tht plnetry nd rtificil erth stellite orbits re ellipticl. The mthemtics of the ellipse re reviewed to provide ll the definitions importnt in geodesy. Some definitions for stellite orbit mechnics re provided where confusion with geodetic nottion often occurs. The historicl usge of ellipse terminology hs been developed in severl different fields, resulting in multiple wys to define the ellipse. This hs led to confusion of symbols, or nottion. In some cses the sme nottion is used for different quntities. Eqully confusing, the sme nottion is sometimes used whether the xes origin is locted t either the center of the ellipse or t one focus of the ellipse. The reltionships between the nottions of stndrd mthemtic textbooks, of geodesy, nd of stellite pplictions re provided in this ppendix. Within geodesy, the nottion sometimes vries, nd this too is noted. This presenttion of the different nottions is to ssist the user to identify the context, nd to enble the user to be ble to shift between these contexts. There is no officil ellipse definition since it cn be defined in so mny wys. Some of these definitions re illustrted in this ppendix without rigorous development of the mthemtics. A common nottion is used in ll the exmples in order to illustrte the connections between the different wys in which the ellipse my be formed nd defined. I. Ellipse symbols This tble is compiltion of the symbols used for the prmeters importnt for defining the ellipse. The symbol most commonly used, or best relted to geodesy, for ech prmeter is listed in the left hnd column. These re the symbols used in the exmples of this ppendix. Other frequently used symbols re included in the right hnd column. In 1
ddition, there re double entries for the symbols used to signify different prmeters. These prmeters re explined in this ppendix. Tble A-1 Ellipse Terminology Symbol Prmeter Other symbols A b Point of pogee semimjor xis semiminor xis c hlf focl seprtion e, ε E Eccentric, Prmetric, or reduced ngle or eccentric nomly e, t, u, β e (first) eccentricity ε e Second eccentricity ε F f foci (first) flttening (or ellipticity) f Second flttening M P Men nomly Point of perigee P(x, y) Points on the ellipse Q, mny p Semiltus rectum R Rdil distnce from focus r R M Rdius of curvture in meridin direction M R N rdius of curvture in prime verticl N, ν, R ν r S α rdil distnce from center Distnce from focus to ellipse ngulr eccentricity ε Liner eccentricity E Θ True nomly f, θ, ν, ψ θ centrl or geocentric ngle φ φ Geodetic ltitude 2
II. Ellipse components nd definitions A. Conceptul ellipse A simple wy to illustrte the ellipse is to picture piece of string with ech end fstened to fixed points clled focus points or foci (Figure A1). The string length is rbitrrily set to 2. If pencil is used to pull the string tight nd is then moved round the foci, the resulting shpe will be n ellipse. The length of string remins constnt t 2, but the distnce (S 1 nd S 2 ) from the pencil to ech focus will chnge t ech point. The foci re locted t F 1 nd F 2, nd A nd B re two rbitrry points on the ellipse. All the points on the closed curve defined by the ellipse re represented by the set of x nd y points, P(x,y). Figure A1. Outline (light dshed line) of the ellipse formed by pencil stretching out piece of string. Two positions of the string re indicted, A nd B, to illustrte tht while the distnce from ech focus chnges, the totl length of the string remins fixed. The line connecting the foci defines n xis of symmetry, the mjor xis, 3
for the ellipse. In this ppendix, the foci nd mjor xis will lwys be locted on the x xis. The perpendiculr line pssing through the mid-point between the foci is lso n xis of symmetry. This line defines the minor xis. The intersection of the two xes is the center of the ellipse. The coordinte systems used to define the ellipse will be locted either t the ellipse center or t one of the foci. The convention is tht the distnces from the origin to the foci re ±c nd re referred to s the hlf focl seprtion (Figures A2 nd A3). Figure A2. Illustrtion of the pencil positioned t one vertex to show the semimjor xis,, nd the focl point hlf seprtion, c. When the point P is locted on the x xis, P(±x,0), (Figure A2), the two string segments will lie on top of one nother (note tht in the digrm the segments re drwn curved so tht they cn be observed). This point of intersection of the ellipse with the x xis is clled the vertex. The distnce between the origin nd one vertex is hlf the length of the string. With the string length given s 2 the distnces long the x xis from the origin to the vertices re ±. The distnce from the center of the ellipse to the vertex is clled the semimjor xis of length. 4
Figure A3. Position of the pencil on the minor xis so tht S 1 S 2. The isosceles tringle forms two right tringles on the y xis so tht 2 b2 + c2. When the ellipse intersects the y xis, P(0,±y), the two segments of the string re equl, forming n isosceles tringle (Figure A3). This tringle is divided into two similr right tringles by the y xis. The distnce from the ellipse center to the ellipse is clled the semiminor xis of length b. (Note tht the hypotenuse of ech tringle is equl in length to the semimjor xis.) By the Pythgorin theorem, 2 b2 + c2. (A1) 1. Ellipse xes terminology The foci re lwys locted on the mjor xis nd, b, nd c re used to represent the semimjor xis, the semiminor xis, nd the hlf focl seprtion (or the distnce from the center of the ellipse to one foci) (Figure A2). The center of the ellipse is the point of intersection of its two xes of symmetry. 2. Eccentricity 5
One wy to specify the shpe of n ellipse is given by the eccentricity, e. The ellipse eccentricity, e, cn vry between 0 nd 1. An eccentricity of 0 mens the foci coincide, nd the shpe will be circle of rdius. At the other limit, e1, is line 2 in length pssing through the foci. The usul geodetic defintion is. e 2 1 b 2 2 (A2) Other forms of the eccentricity common for geodetic pplictions include e 2 2 b 2 2, e e 2 c 2 2. 2 b 2, (A3) Other common forms of this reltionship re: b 2 2 1 e 2 b e ( 1 e 2 ) c c 2 b 2 1 2 b 2 2 ( 1 e 2 ) b 1 e 2 c e (A4) The ngle α in Figure A3 is referred to s the ngulr eccentricity since sin α c e. (A5) (NOTE: in different texts ε ppers in one of three wys, either s the hlf focl seprtion, c, s the eccentricity, e, nd s the liner 6
eccentricity, ε 2 b 2.) 3. Ellipse fltness Ellipse shpe is lso expressed by the fltness, f. In geodesy the shpe of the ellipsiod (ellippsoid of rottion) tht represents erth models is usully specified by the fltness. The fltness is computed s: f 1 or b, b f. The reltionship between e nd f is e 2 2 f f 2, f 1 1 e 2. (A6) (A7) 4. Second eccentricity nd second fltness The eccentricity nd fltness, e nd f, re both defined by rtio with the semimjor xis. These re lso referred to the first eccentricity nd the first fltness. Anlogous quntities defined s the rtio to the semiminor xis re referred to s the second eccentricity, e, nd the second fltness, f, e ' 2 f ' 2 b 2, b b 1. (A8) Other forms for the second eccentricity re: 2 b 2 e' 2 1, b ( 1 + e' 2 ) 1 2 (A9) 7
5. Specifying n ellipse The shpe nd size of n ellipse cn be specified by ny pir combintion of or b with c, e, e, f or f. Different pplictions use different sets. The common combintions re: 6. The directrix semimjor nd semiminor xes (, b), semimjor nd eccentricity (, e), semimjor nd fltness (, f). The directrix is stright line perpendiculr to the mjor xis. The unique property of the directrix is tht the horizontl distnce from point P on the ellipse to the directrix is proportionl to the distnce from the closest focus to tht point (see the left hnd side of figure A4). The constnt of proportionlity is e. Since the ellipse hs two foci, the ellipse hs two directrices nd they re locted ±(/e) from the ellipse center (Figure A4). Figure A4. Digrm of n ellipse illustrting the distnces of the focus, vertex nd directrix from the ellipse center. Note tht the horizontl 8
distnce to the directrix from the ellipse is proportionl by e to the distnce from tht point to the closest focus. III. The mthemticl ellipse A mthemticl definition of the ellipse is the locus of points P(x,y) whose sum of distnces from two fixed points, the foci, is constnt. When the foci coincide, the ellipse is circle, nd s e is incresed, the distnce between the foci increses nd the shpe becomes more elongted, or squshed, until in the limit it is stright line. In specifying n ellipse mthemticlly, it is importnt to know the loction chosen for the origin. There re two common conventions, the origin t either the center or t one focus of the ellipse. This section presents the equtions for both the origin t the ellipse center nd the origin t one focus. For consistency, ll focus centered equtions hve the right hnd focus (F 2 of figure A4) s the origin. A. The ellipse eqution 1. Crtesin coordintes. Centered origin The eqution for the ellipse in Crtesin coordintes with the origin t the ellipse center is: x 2 2 + y 2 b 2 1. (A10) The constnts nd b re the semimjor nd semiminor xes. b. Focus origin When using the origin t the focus the Crtesin form of the eqution is ( x c ) 2 2 + y 2 b 2 1 (A11) 9
The offset c is the distnce from the origin to focus. Other formuls cn be derived using the reltionships between, b, nd c (Eqution A1). 2. Ellipse in polr coordintes When using polr coordintes, the ellipse cn be specified with the origin t either the ellipse center or the origin t one focus. The reder should be wre tht r nd θ re regulrly used to define point on the ellipse for both coordinte systems. When mesured from the ellipse center, θ is the centrl ngle nd r is the distnce from the ellipse center. When θ is mesured t one focus it is clled the true nomly nd r is the distnce from the focus to the point on P. To distinguish the two, upper cse symbols Θ nd R will be used in equtions centered on the focus. Mny pplictions fil to mke cler the distinction between the true nomly nd centrl ngle nd the different distnces represented by r.. Ellipse centered origin The ellipse cn be drwn s distnce r from the ellipse center where the length of r depends on the centrl ngle θ (Figure A5). 10
Figure A5. The ellipse defined by the centrl ngle θ nd the rdius r. The ngle θ is mesured counter clockwise from the semimjor xis. Using the centrl ngle, θ, the length of r is determined by ny of the following: r 2 r 2 r 2 r 2 b 2 1 e 2 cos 2 θ, 2 1 e 2 sin 2 θ, 2 ( 1 e 2 ) 1 e 2 cos 2 θ, 2 b 2 2 sin 2 θ + b 2 cos 2 θ. (A12) One cn check equtions like A12 by evluting r for θ equls 0 nd 90. For θ equls 0, r is on the x xis nd r2 equls 2. For θ equls 90, r is on the y xis nd r2 equls b2. The conversion between Crtesin nd polr coordintes is obtined from x r cosθ, y r sinθ. (A13) b. Ellipse centered t one focus Figure A6 presents the definitions used to locte point on the ellipse when mesured from focus. The origin is set t one focus nd the vertex closest to the origin is clled the point of perigee for erth stellites. The vertex frthest to the origin is then clled the pogee. The ngle Θ, the true nomly, is mesured t the focus, moving counter clockwise from perigee. In terms of these vribles the rdius from the focus is given by: R ( 1 e 2 ) ( 1 + e cos Θ ). (A14) 11
Figure A6. Ellipse defined by the true nomly Θ, mesured from the focus, nd the rdil distnce R. i. The semiltus rectum When the true nomly, Θ, is 90, the rdius, R, is clled the semiltus rectum, p. The semiltus rectum is the line prllel to the minor xis from the focus to the ellipse (Figure A7). p ( 1 e 2 ), p b 2. (A15) Equtions relting p to, b, nd c re: 12
2 p b 2 4 e 2 p 2 ( 1 e 2 ) 2, 1 e 2 2 e 4 e 2 p 2 ( 1 e 2 ),, (A16) p c ( 1 e 2 ) 2 e 2 e 2 p ( 1 e 2 ) 2. 1 2 b, The rdil distnce from the foci, R, is given by: R e( 2 p + R cos Θ ), R 2 ep 1 cos Θ. (A17) Figure A7. The semiltus rectum, p, is the line norml to the semimjor xis, Θ 90, from the focus to the intersection with the ellipse. B. Ellipse from circles nd rditing lines 13
1. Ellipse from the intersection of two concentric circles with rditing lines An importnt wy to construct n ellipse is illustrted in Figure A8. Two concentric circles of rdii, nd b, define the ellipse. The rdius of the inscribed circle, b, defines the minor xis nd the rdius of the circumscribed circle,, defines the mjor xis. Next, rdil line is drwn from the center t ngle E. The ngle E is clled the eccentric nomly in stellite work nd the reduced ltitude in geodesy. It is lso clled the prmetric ngle. Often the development of the ellipse is given only showing the circumscribing circle. Figure A8. Formtion of n ellipse from the intersection of rdil line with two concentric circles. The ngle E is the reduced or prmetric ngle. The points tht the rdil line intersect with the two circles give the x nd y coordintes. The y coordinte is tken from the intersection with the smller inscribed circle of rdius b. The x coordinte is tken from the intersection with the lrger circumscribing circle of rdius. The x nd y coordintes of the ellipse re given by: 14
x cos E, y b sin E, 1 y ( 1 e 2 2 ) sin E (A18) 2. One-wy reduction of circle An lternte wy to define n ellipse from circumscribed circle nd rdil lines is one-wy reduction of circle, kind of foreshortening (Figure A9). Rdil lines of ngle E re drwn to the circumscribed circle. The point of intersection is (x i, y i ). The x vlue is found s in the previous exmple. The y i vlue of this intersection is scled by b/ to give the y vlue of the ellipse. x i cos E x x i y i sin E y b y i y b sin E (A18) 15
Figure A9. Ellipse from one wy reduction of lines norml to the semimjor xis. The orthogonl lines from the intersection of the rdil t ngle E on the circle, dshed line, re reduced by the constnt b/, hevy line. C. Conic section An ellipse is lso formed by the intersection of right circulr cone nd plne inclined less steeply thn the side of the cone (Figure A10). When the plne does not pss through the bse the shpe of the intersection is n ellipse. The eccentricity is determined by the steepness of the cone nd the ngle of intersection between the cone nd the intersecting plne. When the intersecting plne is prllel to the cone bse, the intersecting line is circle. 16
Figure A10. The ellipse formed by the intersection of plne, light grey, with right circulr cone. This exploded view shows the ellipse of intersection in drk grey. The circle, ellipse, prbol, nd hyperbol re cll conic sections becuse they cn be generted in this mnner. D. Ellipse from stright edge A stright edge of length +b cn be used to construct 1/4 of n ellipse. 17
Let the ends be ttched to, but ble to slide long the xes (Figure A11). The point P is distnce from the end of the stright edge on the y xis. As the one end of the stright edge slides down the y xis from y +b to y 0, the loction of point P will mp out the curve of the ellipse. The ngle E between the stright edge nd the x xis is used to define the x nd y coordintes. This ngle hs the sme mgnitude s the eccentric nomly. x cos E, y b sin E. (A19) The full ellipse is creted by repeting the exercise in ll four qudrnts. Figure A11. The formtion of n ellipse by sliding stright edge long pir of norml lines (xes). As the end of the stright edge moves down the y xis nd the other end moves out the x xis ny point, P, on the stright edge mps out qurter ellipse. E. Tngent to the ellipse nd rdius of curvture 18
Two importnt spects of the ellipse needed for geodesy re the tngent to the ellipse nd the rdius of curvture. For the erth the tngent will be (pproximtely) the locl horizontl plne. The rdius of curvture of the ellipse is one of the effective rdii of the erth needed to convert ngulr differences to liner distnces. (Both effective rdii re described in the ellipsoid ppendix.) Figure A12 illustrtes the tngent line to the ellipse t point P. The line perpendiculr to the point of tngency is lso drwn nd is lbeled PQ. The distnce from P to Q, the point of intersection with the y xis, defines the rdius of curvture, R N. Note tht R N does not intersect the y xis t the origin nd it forms n ngle φ with the x xis. In geodesy pplictions φ is clled the geodetic ltitude nd is the ltitude found on mps. (In geodesy pplictions θ, the geocentric ngle, is signified by φ nd clled the geocentric ltitude.) The centered Crtesin coordintes in terms of φ re given by x y cos φ ( 1 e 2 sin 2 φ ) ( 1 e 2 ) sin φ ( 1 e 2 sin 2 φ ) 1 2 1 2,. (A20) And the rdius of curvture, R N, is obtined from R N ( 1 e 2 sin 2 φ ) 1 2. (A21) The Crtesin coordintes from R N re obtined by x R N cos φ, y R N ( 1 e 2 ) sin φ (A22) 19
Figure A12. The tngent line to the ellipse nd the norml, R N, to the tngent line. The length of R N from the y xis to the ellipse defines the rdius of curvture. IV. Coordinte conversions It is often necessry to convert the point on the ellipse to either different coordinte centered system or to convert between the different polr ngles. The conversion equtions re presented in this section. A. Crtesin conversion between centered origin nd focus origin The trnsformtion of the Crtesin coordinte systems with the origins t the center nd focl point re ccomplished by moving the origin of the x xis. When using the focus centered origin, only the x xis is n xis of symmetry. Using the following symbols for centered origin coordintes (x, y) focl point origin coordintes (x, y ) 20
x ' x c, x ' x e, y ' y (A23) B. Summry of ngles A very busy digrm with most of the lines nd ngles discussed in this ppendix is shown in Figure A13. Subsets of this figure re shown in Figures A14 nd A15. Figure A13 summrizes the four polr ngles, Ε, φ, θ, Θ, nd the two rdii, R, r, plus the rdius of curvture R N. A summry of the different equtions for converting between the four polr mesurements, nd crtesin coordintes, concludes this ppendix. Figure A13. Summry digrm showing the differences between r, R, R N nd E θ, Θ, nd φ. Figure A14 illustrtes the three center origin polr mesurements. The vrious conversions for the crtesin loction of the ellipse to the different mesurement ngles re provided in the following tbles. 21
Figure A14. Summry digrm illustrting the differences between r, R N nd E, φ, nd φ. Tble A-2. Conversion of crtesin coordintes to the different polr coordintes. Crtesin geocentric (centric) φ ( θ) eccentric E Geodetic φ x r cos φ ' cos E R N cos φ cos φ 1 2 sin 2 φ 2 cos φ 2 cos 2 φ + b 2 sin 2 φ 22
y r sin φ ' b sin E R N ( 1 e 2 ) sin φ b b sin φ 1 e 2 sin 2 φ b 2 sin φ 2 cos 2 φ + b 2 sin 2 φ y x tn φ ' b tn E b 2 2 tn φ Tble A-3. Conversion of the rdil distnce from the ellipse center for the three center origin ngles. r 2 2 1 e 2 sin 2 φ ' 2 ( 1 e 2 sin 2 E ) R N [ cos 2 φ + b 4 4 sin 2 φ b 2 1 e 2 cos 2 φ ' 2 cos 2 E + b 2 sin 2 E 4 cos 2 φ + b 4 sin 2 φ 2 cos 2 φ + b 2 sin 2 φ 2 b 2 2 sin 2 φ ' + b 2 cos 2 φ ' Center origin ngle conversions Tble A-4. cos E cos φ 1 e 2 sin 2 φ R N cos φ sin E b sin φ 1 e 2 sin 2 φ R N b 2 sin φ 23
cos φ ' r cos φ 1 e 2 sin 2 φ R N r cos φ sin φ ' b 2 r sin φ 1 e 2 sin 2 φ R N b 2 r 2 sin φ R N r ( 1 e 2 ) sin φ cos φ ' r cos E sin φ ' b r sin E 4. Conversion between the focus origin nd the eccentric nomly Unlike the center origin ngles, E nd θ, which re lwys in the sme qudrnt, Θ cn be in different qudrnt (s drwn in Figure A15). When the point on P lies between the center y xis nd the ltus rectum, Θ will be in different qudrnt thn the center ngles. The usul ngle conversion procedure is to find cos Θ nd sin Θ from E (Tble A-4) nd then use four qudrnt rc tngent to find Θ. Figure A15. The reltionship between the prmetric ngle, E, nd the true nomly, Θ. 24
Figure A15 illustrtes the mesurement loction for the eccentric nomly, E, nd the true nomly, Θ. Below re given the conversion equtions for trnsforming from one ngle to nother. Tble A-5. Conversion between eccentric nomly nd true nomly. x ' R cos Θ ( cos E e) y ' R sin Θ b sin E cos Θ cos E e 1 e cos E cos E e ( 1 e cos E) sin Θ 1 e 2 1 e cos E sin E b sin E 1 e cos E R ( 1 e 2 ) 1 e cos Θ ( 1 e cos E) b 2 1 1 e cos Θ 25