A NOTE OF DIFFERENTIAL GEOMETRY

Transcription

1 A NOTE OF DIFFERENTIAL GEOMETRY - APPLICATION OF THE METHOD OF THE REPÈRE MOBILE TO THE ELLIPSOID OF REFERENCE IN GEODESY - by ABDELMAJID BEN HADJ SALEM INGÉNIEUR GÉNÉRAL RETIRED FROM THE Offce de la Topographe et du Cadastre (Tunsa) (abenhadjsalem@gmal.com) JANUARY 2017 VERSION 1.

2 Contents 1 THE REPÈRE MOBILE INTRODUCTION THE DIFFERENTIAL EXPRESSIONS OF da AND de 4 3 RELATIONS SATISFIED BY DIFFERENTIAL FORMS ω AND ω j CASE OF AN ORTHONORMAL REPÈRE MOBILE APPLICATION TO THE ELLIPSOID OF REFERENCE DETERMINATION OF THE ω DETERMINATION OF THE ω j VERIFICATION OF THE FORMULAS dω AND dω j CASE WHERE h = 0 8 2

3 A NOTE OF DIFFERENTIAL GEOMETRY - APPLICATION OF THE METHOD OF THE REPÈRE MOBILE TO THE ELLIPSOID OF REVOLUTION ABDELMAJID BEN HADJ SALEM 1 1 THE REPÈRE MOBILE 1.1 INTRODUCTION Let E the geodetc ellpsod of reference defned by the parameters a, e respectvely the sem-major axs and the frst eccentrcty. Let R(O, X, Y, Z) the geocentrc frame assocated to ths ellpsod. A pont A est defned by ts trdmensonal Cartesan coordnates (X, Y, Z) : X = (N + h)cosϕcosλ (1) Y = (N + h)cosϕsnλ (2) Z = (N(1 e 2 ) + h)snϕ (3) wth N = a 1 e 2 sn 2 ϕ (4) At A(ϕ, λ, h), we consder the local frame defned by the orthonormal frame (e λ, e ϕ, e n ) gven n the normed bass (, j, k) of R as: e λ = snλ cosλ 0 To facltate the notatons, let: ; e ϕ = snϕcosλ snϕsnλ cosϕ ; e n = cosϕcosλ cosϕsnλ snϕ (5) e 1 = e λ (6) e 2 = e ϕ (7) e 3 = e n (8) When the geodetc coordnates (ϕ, λ, h) of A change, the local frame at A moves that t called moble frame or repère moble of the pont A. 1 6, rue du Nl, Cté Solman Er-Radh, 8020 Solman, TUNISIA. 3

4 Fgure 1: The Moble Frame 2 THE DIFFERENTIAL EXPRESSIONS OF da AND de The poston of A s gven n R by: OA = X + Y j + Zk (9) wth (, j, k) the canoncal bass of R. As (e 1, e 2, e 3 ) s also a orthonormal bass of R, the lason of (, j, k) to (e 1, e 2, e 3 ) (A. Ben Hadj Salem, 2010) s gven by: The dfferental of (9) gves : snλ snϕcosλ cosϕcosλ e 1 j = cosλ snϕsnλ cosϕsnλ e 2 (10) k 0 cosϕ snϕ e 3 da = dx + jdy + kdz Now, we replace, j et k n functon of e 1, e 2 and e 3, we obtan the dfferental expressons: da = =3 =1 ω e (11) Smlarly, the dfferental of the vector e s expressed n (, j, k), and wll be transformed n the bass (e ) usng the nverse of the matrx gven by (10), then we obtan: de = j=3 j=1 ω je j (12) 4

5 The two formulas (11) and (12) defne the nfntesmal shftng of the repère moble at A when A moves. The coeffcents ω and ω respectvely n the prevous two formulas are dfferental forms of degree 1 n terms of dfferental forms dλ, dϕ and dh (A. Ben Hadj salem, 2012). 3 RELATIONS SATISFIED BY DIFFERENTIAL FORMS ω AND ω j Returnng to past formulas (11) and (12), they represent dfferentals of vector functons, as a result, then: The formula (13) gves: d(da) = 0 (13) d(de ) = 0 = 1, 3 (14) d(da) = d( ω e ) = d(ω e ) = (dω e ω de ) = dω e ω de = 0 Let: dω e = ω de Now replace de by ts expresson (12), we get: dω e = ω de = ω ω k e k = k k Gvng by changng the ndex and k for the rght term: dω e = ω k ω k e k ω ω k e k (15) So that: Now back to the formula (14): But: dω = k=3 ω k ω k (16) j=3 j=3 d(de ) = 0 = d( ω j e j ) = d(ω j e j ) (17) j=1 j=1 d(ω j e j ) = j j dω j e j j ω j d(e j ) = j dω j e j j k=3 ω j ( ω jk e k ) = 0 5

6 Then: k=3 dω j e j = ω j ω jk e k (18) j j=3 j=1 The (e ) form a bass of R, e j coeffcents must be equal to 0, gvng after handlng: dω j = k=3 ω k ω kj 1 3, 1 j 3 (19) Dfferental forms gven by (16) and (19) consttute the Ele Cartan formulas (H. Cartan, 1979). 3.1 CASE OF AN ORTHONORMAL REPÈRE MOBILE In the case studed n ths note, the base (e ) s an orthonormal bass that s to say: { = 1 s = j = 1 e.e j = δ j (20) = 0 s j Dfferencng (20), we get : de.e j + e.de j = 0 (21) Usng the formula gvng de, that s to say (12), the above expresson becomes:: k=3 ( As e.e j = δ j, we obtan: k=3 ω k e k ).e j + e.( ω jk e k ) = 0 (22) ω j + ω j = 0, j = 1, 2, 3 (23) When = j, we get: ω 11 = ω 22 = ω 33 = 0 (24) 4 APPLICATION TO THE ELLIPSOID OF REFERENCE 4.1 DETERMINATION OF THE ω Takng the dfferental of the formulas (1-3) n the bass (, j, k), then : da = dx + jdy + kdz Replacng, j and k by ts expressons n functon of e 1, e 2 and e 3, we obtan: da = cosϕ(n + h)dλe 1 + (ρ + h)dϕe 2 + dhe 3 (25) Let: ω 1 = cosϕ(n + h)dλ; ω 2 = (ρ + h)dϕ; ω 3 = dh (26) 6

7 4.2 DETERMINATION OF THE ω j The expresson of de n functon of the vectors e of the repère moble at A s obtaned usng (5): de 1 = (cosλ + jsnλ)dλ de 2 = ( cosϕcosλdϕ + snϕsnλdλ) + ( cosϕsnλdϕ snϕcosλdλ)j kcosϕdϕ de 3 = ( snϕcosλdϕ cosϕsnλdλ) + ( snϕsnλdϕ + cosϕcosλdλ)j + kcosϕdϕ (27) Startng from (10), we get: also n matrx form: de 1 de 2 = de 3 de 1 = snϕdλe 2 cosϕdλe 3 (28) de 2 = snϕdλe 1 dϕe 3 (29) de 3 = cosϕdλe 1 + dϕe 2 (30) 0 snϕdλ cosϕdλ snϕdλ 0 dϕ cosϕdλ dϕ 0 e 1 e 2 e 3 = Ω e 1 e 2 e 3 (31) wth: Ω = 0 snϕdλ cosϕdλ snϕdλ 0 dϕ = cosϕdλ dϕ 0 0 ω 12 ω 13 ω 21 0 ω 23 (32) ω 31 ω 32 0 Then the elements ω j : ω 11 = ω 22 = ω 33 = 0 (33) ω 12 = ω 21 = snϕdλ (34) ω 13 = ω 31 = cosϕdλ (35) ω 23 = ω 32 = dϕ (36) 4.3 VERIFICATION OF THE FORMULAS dω AND dω j Let us check the formula (16) that s: k=3 dω = ω k ω k For example, we calculate dω 1 : dω 1 = d((n + h)cosϕdλ) = d(ncosϕdλ) + d(hcosϕdλ) (37) 7

8 but d(ncosϕ) = ρsnϕdϕ, we obtan: dω 1 = ρsnϕdϕ dλ hsnϕdϕ dλ + cosϕdh dλ (38) or : dω 1 = (ρ + h)snϕdϕ dλ + cosϕdh dλ (39) By the formula (16): k=3 dω 1 = ω k ω k1 = ω 1 ω 11 + ω 2 ω 21 + ω 3 ω 31 = 0 + ω 2 ω 21 + ω 3 ω 31 = (ρ + h)dϕ ( snϕdλ) + dh cosϕdλ = (ρ + h)snϕdϕ dλ + cosϕdh dλ (40) Whch s dentcal to the equaton (39) above. Now we check the formulas of the dω j : k=3 dω j = ω k ω kj and we calculate for example dω 12 : dω 12 = d(snϕdλ) = cosϕdϕ dλ (41) On the other hand, accordng to the formula (19), we get: k=3 dω 12 = ω 1k ω k2 = ω 11 ω 12 + ω 21 ω 22 + ω 13 ω 32 but: Fnally, we obtan: ω 11 = ω 22 = 0 k=3 dω 12 = ω 1k ω k2 = ω 13 ω 32 = cosϕdλ dϕ = cosϕdϕ dλ (42) It s the result of (41). 5 CASE WHERE h = 0 If h = 0, then the pont A s on the tangent plane on the ellpsod, n ths case, we obtan: ω 1 = Ncosϕdλ (43) ω 2 = ρdϕ (44) ω 12 = snϕdλ (45) 8

9 However, accordng to the fundamental theorem of the local Remannan geometry, there exsts a unque dfferental form ω 12 defned n the tangent plane whch satsfes the equatons (S.S Chern, 1985): dω 1 = ω 12 ω 2 (46) dω 2 = ω 1 ω 12 (47) et dω 12 = Kω 1 ω 2 (48) wth K s Gauss curvature or the total curvature at the pont A. then: but : Let us check the equatons (46-48). From (43-45), we obtan: dω 1 = ρsnϕdϕ dλ (49) dω 2 = ρ dϕ dϕ = 0 (50) dω 12 = cosϕdϕ dλ (51) dω 1 = ω 12 ω 2 = snϕdλ ρdϕ = ρsnϕdλ dϕ (52) dω 2 = ω 1 ω 12 = Ncosϕdλ snϕdλ = 0 (53) et dω 12 = cosϕdϕ dλ (54) ω 1 ω 2 = Ncosϕdλ ρdϕ = ρncosϕdϕ dλ (55) Comparng (51) and (55), we fnd that : wth: dω 12 = 1 ρn ω 1 ω 2 = Kω 1 ω 2 (56) K = 1 = the total curvature or Gauss curvature ρn It s the nverse of the product of two rad of curvature of the ellpsod. (57) Q.E.D References H. Cartan Cours de Calcul Dfférentel. Nouvelle édton refondue et corrgée. Hermann Pars. Collecton Méthodes. 365p. S.S Chern Movng Frames. Socété Mathématque de France. Astérsque, numéro hors sére "Ele Cartan et les mathématques d aujourd hu". Lyon, jun p A. Ben Hadj Salem Note de Géométre Dfférentelle - Le Repère 9

10 Local -. v1. 7p. A. Ben Hadj Salem Selected Papers. v pages. 10