SKEW-NORMAL CORRECTION TO GEODETIC DIRECTIONS ON AN ELLIPSOID

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Aron Simmons
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1 Geoptil Science SKEW-NORMAL CORRECTION TO GEODETIC DIRECTIONS ON AN ELLIPSOID In Figure, n re two point t height n h bove n ellipoi of P h emi-mjor xi, n flttening f. The norml n PH (piercing the PH ellipoi t n ) re kewe with repect to ech other. An oberver t P, Q Q whoe theoolite i et up o tht it xi of revolution i coincient with the norml t, ight to trget P ; the verticl plne of the theoolite contining, P n H i norml ection plne n will interect the ellipoi long the norml ection curve QQ hving n zimuth α but thi i not the correct norml ection curve. N P h Q α α δα meriin of norml P h Q' Q norml rottionl xi of ellipoi centre of ellipoi A O H ν e in ν e in H B equtor of ellipoi Figure. Sectionl view of point t height n t height h bove n ellipoi of h P revolution. C:\Project\Geoptil\Geoey\Skew-Norml\SKEW NORMAL CORRECTION.oc

2 Geoptil Science The correct norml ection curve i the plne curve hving n zimuth α tht QQ i crete by the interection of the norml ection plne contining, n H, Q n the ellipoi urfce. The ngulr ifference between thee two curve i δα n i pplie correction to ny oberve irection to trget bove the ellipoi. Thi correction, known the kew-norml correction, i ue entirely to the height of the trget ttion. It i ometime known the "height of trget" correction n formul for thi correction i erive in the following mnner. From Figure = ν e in νe in ν ν = e in in () i the itnce between n H where the norml interect the xi of H revolution of the ellipoi, i the ltitue of P, ν = QH = e in i the riu of curvture of the prime verticl ection of the ellipoi t Q n e = f ( f) i the qure of the eccentricity of the ellipoi. In eqution () the term ν will be very cloe to unity n we my write where ε ν re mll poitive quntitie, n ν = + ε n = + ε = e {( + ε ) in ( + ε ) in } = e {( in in ) + ( εin εin ) } The n term in the brce will be quite mll, ince n ε will be pproximtely ε 0.00 for mi-ltitue vlue n my be neglecte in n pproximtion, giving e ( in in ) () A+ B A B Uing the trigonometric ition formul: in A in B co = in e + co in = C:\Project\Geoptil\Geoey\Skew-Norml\SKEW NORMAL CORRECTION.oc

3 Geoptil Science + Letting the men ltitue m = n the ltitue ifference δ = e com in δ δ Now if δ i mll then in n δ = give δ e co m = e ( ) co (3) m On the ellipoi, the elementl itnce long the meriin i ( e ) ( e in ) ρ 3 where ρ = i the riu of curvture in the meriin plne n from Figure A ρ α λ λ + λ B + ν co ρ = co α λ Figure. Elementl rectngle on the ellipoi ρ + ρ Letting = n rerrnging give we my write for mll rectngle on the ellipoi co α ρ δ δ m co α = (4) Subtituting eqution (4) in eqution (3) give co m (5) e co α C:\Project\Geoptil\Geoey\Skew-Norml\SKEW NORMAL CORRECTION.oc 3

4 Geoptil Science P γ Figure 3 i extrcte from Figure n A h Q ellipoi B Q' y norml 90 x rottionl xi of ellipoi O H H how chemtic view of the meriin ection through P. The meriin rc itnce QQ = y γ i the ngle between the norml the line PH PH x i the perpeniculr itnce from the norml to H n x = co (6) n Figure 3. Meriin ection through P Replcing with in eqution (5) ince it will not introuce ny pprecible error give m co co e α Multiplying both ie by co give co e co co = α n the left-hn-ie equl x of eqution (6) hence co x = e α co (7) Referring to Figure 3, the itnce PH cn be pproximte by the emi-mjor xi x length, n the ngle γ t P pproximte by γ ; hence iviing both ie of eqution (7) by give n pproximtion for the ngle γ C:\Project\Geoptil\Geoey\Skew-Norml\SKEW NORMAL CORRECTION.oc 4

5 Geoptil Science γ e ρ m co α co (8) Now the itnce QQ = y i pproximtely hγ n from eqution (8) y he co α co (9) Referring to Figure, the pheroil tringle the ngle δα i very mll QQQ cn be coniere plne ince 80 (δα +( 360 α )) Q α α δα Q Q' Q' Q α y y δα Plne Figure In the plne figure QQQ the itnce QQ = y, the ngle t Q cn be pproximte α 80 ince δα i very mll n the ine rule give y = in( α 80) in δα Since δα i mll in δα δα n α 80 α we my write δα y in α Subtituting eqution (9) into thi eqution give the kew-norml correction h δα e in α co α co ρ m Uing the trigonometric ouble ngle formul in A = in AcoA lterntive expreion h δα e in α co ρ m The correction i pplie in the following mnner give n (0) () correct norml ection = oberve norml ection + δα C:\Project\Geoptil\Geoey\Skew-Norml\SKEW NORMAL CORRECTION.oc 5

6 Geoptil Science To tet the vliity of formul for the kew-norml correction, eqution (0) or (), erie of tet line (geoeic) of 0, 0, 50, 00 n 00 km length with geoeic zimuth of 45 were compute. Thee line of vrying length rite from ( = 38, λ = 45 ) to point P, P3, P4, P5 n P6. All the point re relte to the GRS80 ellipoi ( = m, f = ). Note tht n zimuth of 45 will give the mximum vlue of in α co α the compute ltitue n longitue of the terminl point. in eqution (0). Tble how Point Geoeic Azimuth A Geoeic itnce Ltitue Longitue P , P , P , P , P , Tble Tet Line -P, -P3,, -P6 on the GRS80 ellipoi Auming ellipoil height h = 0 the "correct" norml ection zimuth cn be compute from Crtein coorinte ifference. Tble how the "correct" norml ection zimuth of the tet line -P, -P3,, -P6. α Point Norml ection zimuth α Riu of curvture of prime verticl ection (nu) Crtein coorinte X Y Z P P P P P Tble Norml ection zimuth α of Tet Line on the GRS80 ellipoi C:\Project\Geoptil\Geoey\Skew-Norml\SKEW NORMAL CORRECTION.oc 6

7 Geoptil Science Chnging the ellipoil height of ttion P, P3, P4, P5 n P6 to h = 000 m, recomputing the Crtein coorinte n then computing nother et of "oberve" norml ection zimuth give the vlue α for h = 000 m. Thee cn then be compre with the "correct" zimuth α to obtin n exct vlue of the kewnorml correction δα. Thi vlue cn then be compre with the vlue compute from eqution (0) to guge the ccurcy of the formul Norml ection true δ Point zimuth α α for h = 000m compute δ iff δα = α α eq (0) P P P P P Tble 3 Norml ection zimuth correction of Tet Line on the GRS80 ellipoi In Tble 3, column how the "correct" norml ection zimuth n column 3 how the "oberve" norml ection zimuth for trget 000 m bove the ellipoi. Column 4 how the true correction n column 5 how the correction compute from eqution (0). The ifference between the two correction re hown in column 5 n from thee we cn infer tht the correction compute from eqution (0) i ccurte to t let 0.00 for line up to 00 km in length for trget 000 m bove the ellipoi. The vlue in Tble 3 re "mximum" vlue for trget 000 m bove the ellipoi in ltitue outh. Inpection of eqution (0) how tht the correction i proportionl to the height of the trget, o mximum vlue for trget 500 m bove the ellipoi in the me ltitue woul be pproximtely Thi correction i very mll n i often ignore unle the terrin i mountinou. C:\Project\Geoptil\Geoey\Skew-Norml\SKEW NORMAL CORRECTION.oc 7

8 Geoptil Science REFERENCES Krkiwky, E.J. n Thomon, D.B., 974. Geoetic Poition Computtion, 990 re-print, Deprtment of Surveying Engineering, Univerity of Clgry, Clgry, Albert, Cn. C:\Project\Geoptil\Geoey\Skew-Norml\SKEW NORMAL CORRECTION.oc 8

A B C. Solution: a The cart moves down the incline with acceleration a

A B C. Solution: a The cart moves down the incline with acceleration a PHYS 154 Prctice Tet Solution Spring 18 Q1: [4] A cr trvel long the pth hown in the picture with uniforl increing pee. In which point i the ccelertion iu in gnitue? ) A b) B c) C ) Nowhere. The ccelertion