Curvature of the Ellipsoid with Cartesian Coordinates

Home , Radius of curvature

Landscape Architecture and Regional Planning 2017; 2(2): 61-66 http://www.sciencepublishinggroup.com/j/larp doi: 10.11648/j.larp.20170202.13

Sebahattin Bektas

Geomatic Engineering, Faculty of Engineering, Ondokuz Mayis University, Samsun, Turkey

Email address: [email protected] To cite this article: Sebahattin Bektas. Curvature of the Ellipsoid with Cartesian Coordinates. Landscape Architecture and Regional Planning. Vol. 2, No. 2, 2017, pp. 61-66. doi: 10.11648/j.larp.20170202.13

Received: January 19, 2017; Accepted: February 4, 2017; Published: March 4, 2017

Abstract: This study aims to show how to obtain the curvature of the ellipsoid depending on azimuth angle. The curvature topic is quite popular at an interdisciplinary level. It can be to the friends of geometry, geodesy, satellite orbits in space, in studying all sorts of elliptical motions (e.g., planetary motions), curvature of surfaces and concerning eye-related radio-therapy treatment, for example the anterior surface of the cornea is often represented as ellipsoidal in form. On the calculation of the curvature, there is a famous Euler formula for rotating ellipsoid that everyone knows. Let θ be the angle, in the tangent plane, measured clockwise from the direction of minimum curvature κ1. Then the normal curvature κn(θ) in direction θ is given by 2 2 2 κn(θ) = κ1 cos θ + κ2 sin θ = κ1 + (κ2 - κ1) cos θ I wonder how can a formula for a triaxial ellipsoid? So we started to work. And we finally found the formula for the triaxial ellipsoid. Keywords: General Ellipsoid, Normal Section Curve, Principal Curvatures, Gaussian Curvature, Mean Curvature

with a new formula. And I think it has not been previously in 1. Introduction the related. On the other hand, we also notice the lack of The curvature issue is very important in geodesy and also numerical practical studies in the literature; and therefore, we in ophthalmology. To make geodetic computations on the have added a comprehensive numeric application to our ellipsoid (rotational or triaxial) first we need to know the work. normal section curve that combines observation points. The normal section curve is also available from the intersection of 2. General Surfaces the ellipsoid and a plane which contains normal of surface on the station point and passes destination point. Geodetic A point on surface can be represented by means of its computational formulas are contain the curvature parameters. position vector relative to the origin of an orthogonal This current study aims to pave the way for our further study coordinate system: on triaxial ellipsoid work. X=(x, y, z) Considerable several numbers of relevant studies were Suppose each of the coordinates is a function of two found in the literature. Some of them, [11, 15, 2, 6, 8]. parameters u, v Concerning the study of Harris in 2006, I think he was made x = x(u, v) a mistake. Curvature calculation was produced based on the y = y(u, v) Cartesian coordinates. However, the curvature calculation z = z(u, v) should have been based on the surface parameters (u, v 2.1. Ellipsoid parameters) not on the Cartesian coordinates. When we look at the literature, we see that the curvature An ellipsoid is a closed quadric surface that is analogue of calculation is usually given depending on the angle of an ellipse. Ellipsoid has three different axes (a > b > c) as parameters. However, in practice, the azimuth angle is used shown in Fig. 1. Mathematical literature often uses instead of the angle of parameters and azimuth angle can be “ellipsoid” in place of “Triaxial ellipsoid or general easily calculated from the Cartesian coordinates. At this point ellipsoid”. Scientific literature (particularly geodesy) often the importance of our study appears. We give the curvature uses “ellipsoid” in place of “biaxial ellipsoid, rotational calculation depending on the angle of the ellipsoid azimuth ellipsoid or ellipsoid revolution” (a = b > c). Older literature 62 Sebahattin Bektas: Curvature of the Ellipsoid with Cartesian Coordinates

uses ‘spheroid’ in place of rotational ellipsoid. The standard The parameter lines (u, v) and geodetic (planetographic) equation of an ellipsoid centered at the origin of a Cartesian coordinates (Φ, λ) are orthogonal on rotational ellipsoid but coordinate system and aligned with the axes. General are not orthogonal on triaxial ellipsoid. ellipsoid equation as below in [3-5]. In this parametrization, the coefficients of the first fundamental form are E=[b2cos2 u + a2sin2 u] sin2 v (5) F=(b2-a2) cos u sin u cos v sin v (6) G=[a2 cos2u + b2sin2 u] cos2 v + c2 sin2 v (7)

I. fundamental form

I = E. du2 + 2. F . du . dv + G . dv2 (8)

and of the second fundamental form are

2 = a. b . c .sin v e (a . b .cos v )2+ c 2 ( b 2 cos 2 u + a 2 sin 2 u )sin 2 v (9) Figure 1. Triaxial Ellipsoid.

x2 y 2 z 2 + + −1 = 0 (1) f =0 (10) a2 b 2 c 2 a.. b c g = Ellipsoid equation (u,v) Gauss Parametric form (..cos)a b v2+ c 2 ( b 2 cos 2 u + a 2 sin 2 u )sin 2 v (11) x= a cos u sin v y= b sin u sin v (2) II. fundamental form z= c cos v II = edu.2 + 2. fdudv . . + gdv . 2 (12)

- π /2 ≤ u ≤ π/2, -π ≤ v ≤ π Also in this parametrization, the Gaussian curvature is

Parametric coordinates calculated from Cartesian  2 = a.. b c coordinates as below formula K  2 2 2 2 2 2 2  (..cos)a b v+ c ( b cos u + a sin u )sin v  (13) z  v = arccos   (3) c  and the Mean curvature is

a. y  u = arctan   (4) b. x 

abcab...[3(2+ 2 ) + 2 cabc 2 + ( 2 + 2 − 2 2 )cos2 vab − 2(2 − 2 )cos2.sin u2 v ] H = 8[(a22 b cos 2 v+ c 22 ( b cos 2 u + a 22 sin u )sin 23/2 v ] (14)

The Gaussian curvature and Mean curvature can be We will compute H and K in terms of the first and the calculated from Cartesian coordinates given below formulas second fundamental form. [14, 17] e. g− f 2 II K = = (17) − 2 = 1 EGF. I K 2 2 2 2 x y z   (15) − + + + = G. e 2 F . f E . g a.b.c.4 4 4   H 2 (18) a b c   2(EGF .− ) 2.2. Principal Curvatures, Gaussian Curvature, Mean x2+ y 2 + z 2 − a 2 − b 2 − c 2 H = (16) Curvature 2 2 2 3/2 2 x y z  2()a . b . c + +  We will now study how the normal curvature at a point a4 b 4 c 4   varies when a unit tangent vector varies. In general, we will Landscape Architecture and Regional Planning 2017; 2(2): 61-66 63

see that the normal curvature has a minimum value κ1 and a associated with that particular normal section. This κ maximum value κ2,. This was shown by Euler in 1760. The curvature n (α) is called the normal curvature of the quantity surface at point Po in the direction α. Then the normal curvature at point Po is given by K = κ1.κ2 called the Gaussian curvature (19) edu.2 + 2. fdudv . . + gdv . 2 and the quantity κ (,)du dv = (23) n Edu.2 + 2. Fdudv . . + Gdv . 2 H = (κ1 + κ2)/2 called the mean curvature, (20) where E, F, G, e, f, g are the fundamental coefficients of the play a very important role in the theory of surfaces. first and second order. Formula (39) above can be re-written in the following way =1 = 1 Maximum radii of curvature (21) R1 κ 2 2 1 HHK− − dv   dv  e+2. f .  + g .   κ dv  = du   du  n   2 (24) 1 1 du  = = +dv  +  dv  R2 Minimum radii of curvature (22) EFG2. .  .   κ 2 du   du  2 HHK+ − 2 The formula for the radius of curvature at arbitrary simply by dividing the numerator and denominator by du . In κ azimuth points up that the fact that the fundamental this form it is obvious that n is a function of the ratio dv/du. mathematical quantity is the inverse of these radii, which are If we let cot α = dv / du then (24) becomes simply called curvatures e+2. f .cotα + g .cot2 α κ() α = (25) 2.3. Normal Section of a Surface n EFG+2. .cotα + .cot 2 α

Let us construct a normal to a surface at a point Po. Then where the curve that is described on the surface by any plane passing through the normal (i.e. containing the normal) is EF+ tanθ cotα = (26) called a normal section of the surface (Fig. 1). In other words W tanθ a normal section is a plane section formed by a plane containing a normal to the surface [15, 12, 1]. A surface may be curved in many ways and consequently one might think that the dependence of the curvature κ on the 2.4. Curvature of a Surface at a Point angle α might be arbitrary. In fact this is not so. The following theorem is due to Euler. Let us construct a unit normal and a tangent plane at given point Po on surface and consider the curves that are 2.5. Euler’s Theorem formed on the surface by planes passing through P o θ containing the normal i.e. the various normal sections Let be the angle, in the tangent plane, measured clockwise from the direction of minimum curvature κ1. Then passing through point Po. Each normal section passing the normal curvature κ (θ) in direction θ is given by through Po possesses a particular curvature at point Po. We n can specify a particular normal section by use of a polar κ (θ) = κ cos2θ + κ sin2θ = κ + (κ - κ ) cos2θ (27) coordinate system constructed on the tangent plane, origin n 1 2 1 2 1 at point P , polar axis as some arbitrarily chosen initial ray o κn(θ) curvature at azimuth θ in the tangent plane, and an angle α measured clockwise For spheroid (rotational ellipsoid): from the polar axis to the plane of the normal section (Fig. 2). The curvature at point Po in direction α is thus given as the function κn (α). For each value of α there is a curvature 1 N = R = Radius of Curvature in Prime Vertical (Max. radii of curvature) (28) 1 κ 1

1 M = R = Radius of Curvature in Meridian (Min. radii of curvature) (29) 2 κ 2

2 2 N, M can be easily calculated the latitude of point Po as NM.cosθ+ .sin θ κ() θ = (30) below n NM. N= c / V, M = c / V3, c = a2/ c, V2=1+ e 2 cos2Φ p p p x o same as Eq. 26 64 Sebahattin Bektas: Curvature of the Ellipsoid with Cartesian Coordinates

θ is between azimuth angle of P and P points. For this we α= E θ o 1 cot cot (31) need the P geodetic coordinates (Φ, λ). We may obtain the G o Po geodetic coordinates (Φ, λ) from its Cartesian coordinates 2.6. Computing the Principal Directions and Curvatures at (xo, yo, zo). Formulas related the geodetic and Cartesian a Point Po coordinates conversion on a triaxial ellipsoid were expressed on in [9, 15, 3]. For detailed information on this subject Given a point Po on a surface S, the directions at which the please refer to [3]. The following link can be used for the normal curvature at Po attains its minimum and maximum conversion of Cartesian coordinates to geodetic coordinates values can be computed as follows. Let the normal curvature on triaxial ellipsoid [18]. at Po be given as ∆x = x1-xO ∆y = y1-yO ∆z = z1-zO e+2. f .λ + g . λ 2 κ() λ = (32) n EFG+2. .λ + . λ 2 The azimuth angle of (Po-P1) from Cartesian coordinates [16]. where λ = dv/du. We wish to find those values of λ at which θ = (Po-P1) = arc tan the function κn (λ) has its minimum and maximum values. We are thus faced with a problem of finding the maxima and  −∆xsinλ + ∆ y cos λ    (37) minima of a function. A necessary condition for the function  −∆xsin Φ cosλ − ∆ y sin λ sin Φ + ∆ z cos Φ  κn (λ) to have a maxima or minima at a point is that at that Φ λ point d κn (λ) /dλ = 0. Using the usual formula for computing Po geodetic coordinates ( , ) calculated from its (xo,yo,zo) the derivative of a quotient we obtain Cartesian coordinates [18]. The directions corresponding to the minimum and In order to calculation for the curvature, we need to add maximum values of curvature are called the principal the reduction of the direction of rmin to the angle θ directions of the surface. The values κ1 and κ2 are called the Let’s assume that r is the reduction of the direction of principal curvatures of the surface [12, 1]. minimum curvature rmin

2 2 (E + 2Fλ + Gλ ) (f + g λ) - (e + 2f λ + g λ ) (F + Gλ) = 0 (33) Wtan( r )  r = arctan min  (38) E+ Ftan( r ) Upon expansion f=0 and rearrangement (33) becomes  min 

2 (F g) λ + (E g – G e) λ - F e = 0 (34) For spheroid r min and r becomes zero And we give a new formula for the curvature calculation One can then solve (34) obtaining the two principal depending on the θ angle of the azimuth on the triaxial ellipsoid directions λ1 and λ2. One can then substitute the two values λ1 κ() θ= κ + κ − κ2 θ − and λ2 into (32) to obtain the principal curvatures κ1 and κ2. n 1( 2 1 ) cos (r ) New Formula (39) The principal directions Curvature calculation can also be made as follows: First a λ rmax = arc tan( 1) (35) plane’s equation is determined which contains Po surface of normal and passes P point. And then we find elliptical r = arc tan(λ ) (36) 1 min 2 equation the intersection of the plane and the ellipsoid [10, 2.7. The Curvature of the Ellipsoid with Cartesian 13, 7, 19]. The curvature can be calculated at point Po on the Coordinates elliptical equation. 3. Numerical Example

Find the curvature of normal section curve at Po point which contains Po surface of normal and passes P1 point on a triaxial ellipsoid θ Angle is a azimuth angle Po-P1 direction and Cartesian coordinates of Po and P1 point are given below

x2 y 2 z 2 + + −1 = 0 (Ellipsoid equation) 25 16 9

(a= 5, b= 4, c= 3) semi-axis Φ Figure 2. (X,Y,Z) Cartesian and ( , λ, h) Geodetic coordinates on Triaxial Cartesian coordinates (x,y,z) ellipsoid. Po (3.000 2.500 1.4981) P1 (2.6189 2.4125 1.8047) First, we need to find the azimuth angle between the two Geodetic coordinates [18] Φ o λ o points known as the Cartesian coordinates. Let’s assume that o = 40.194814370 o = 52.47573738 ho = 0 Landscape Architecture and Regional Planning 2017; 2(2): 61-66 65

'' u, v surface parameters on Po point Eq.(4) 1 y k = = o = 0.2770411 u = 46.1691393 v = 60.0413669 R(1+ y '2 ) 3/ 2 E, F, G, e, f, g are the Fundamental Coefficients of the o First and Second Order evaluated at P point. Eqs. (5-12) E= 15.52562500 F= -1.94530956 G= 11.82202800 4. Conclusion e=2.91030887 f= 0 g= 3.87718085 This study aims to show how to obtain the curvature of the Gaussian Curvature, Mean Curvature ellipsoid depending on azimuth angle. We have developed an From fundamental coefficients of the first and second order. algorithm to obtain the curvature of the ellipsoid depending K- Gauss= 0.06277139 on azimuth angle. The efficiency of the new approaches is H-Mean Curve= 0.26313232 (Eqs. 17-18) demonstrated through a numerical example. The presented Main radii of curvatures algorithm can be applied easily for spheroid, sphere and also other quadratic surface, such as paraboloid and hyperboloid. =1 = 1 R1 =5.4730575 κ 2 Today, backward and forward problem between the two 1 HHK− − points on the triaxial ellipsoid with geodetic coordinates maximum radii of curvature (Eq. 21) could not be a clear solution. Our future work will be on this unsolvable problem. I hope, the result of this study will =1 = 1 = 2.9107724 minimum radii of R2 κ 2 contribute to the solution of the above problem. 2 HHK+ − curvature Eq.(22) Principal Curvatures κ 1 = 0.182713 minimum curvature References κ 2 = 0.343551 maximum curvature [1] Aleksandrov A. D., Kolmogorov A. N., Lavrent'ev M. A., Principal directions Eq.(34) Mathematics: Its Content, Methods and Meaning, Dover -7.54231697 λ2 + 25.7899 λ + 5.66145 = 0 Publications, ISBN: 9780486409163, 1999. λ1 = 3.62635252 [2] Barbero, S. “The concept of geodesic curvature applied to λ2 = -0.20699173 λ o optical surfaces”, Ophthalmic Physiol Opt 2015. doi: rmax =arc tan( 1) = 74.583317 10.1111/opo.12216 [1]. o rmin =arc tan(λ2)= -11.694599 Azimuth angle (Po-P1) from Eq.(37) [3] Bektas, S, “Orthogonal Distance From An Ellipsoid”, Boletim θ = (P -P ) = 30.136633 o de Ciencias Geodesicas, Vol. 20, No. 4 ISSN 1982-2170, o 1 http://dx.doi.org/10.1590/S1982-217020140004000400053, 2014.  Wtan( r )  r = arctan  min  o + Ï r = –9.88360428  E Ftan( rmin )  [4] Bektas, S, “Least squares fitting of ellipsoid using orthogonal distances”, Boletim de Ciencias Geodesicas, Vol. 21, No. 2 θ - r =30.136633o – (-9.88360428o) = 40.02023728o ISSN 1982-2170, http://dx.doi.org/10.1590/S1982- and curvature 21702015000200019, 2015a. [5] Bektas, S, Geodetic Computations on Triaxial Ellipsoid, κ( θ) = κ +( κ − κ )cos2 ( θ −r ) = 0.277041 n 1 2 1 International Journal of Mining Science (IJMS) Volume 1, Issue 1, June 2015b, PP 25-34 www.arcjournals.org ©ARC Control Page | 25. Equation of plane which contains Po surface of normal and passes P point [6] Benett, A. G., Aspherical and continuous curve contact lenses, 1 Part Three, Optom. Today 28, 238-242, 1988 0.3125 x -0.50134 y + 0.245316 z -0.051655= 0 [7] C. C. Ferguson, “Intersections of Ellipsoids and Planes of Arbitrary Orientation and Position,” Mathematical Intersection ellipse’s equation [19] Geology, Vol. 11, No. 3, pp. 329-336. doi: 10.1007/BF01034997, 1979. x2 y 2 x2 + = 1 ==> y =ξ 1 − η2 ξ 2 η 2 [8] Douthwaite, W. A., Pardhan, S., Surface tilt measured with the EyeSys Videokeratoscope: influence on corneal asymmetry, Invest. Ophthalmol., 1998, Vis. Sci 39, 1727-1737. η = 4.65998 ξ = 3.11078 Transformed coordinates of Po point in intersection’s plane [9] Feltens J., Vector method to compute the Cartesian (X, Y, Z) to ϕ xo = -3.7331 yo = -1.8619 geodetic ( , λ, h) transformation on a triaxial ellipsoid. J Geod '' 83: 129–137, 2009. yo ' = 0.89347 yo = -0.66809 and curvature [10] Gray, A. "The Ellipsoid" and "The Stereographic Ellipsoid." k §13.2 and 13.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 301-303, 1997. 66 Sebahattin Bektas: Curvature of the Ellipsoid with Cartesian Coordinates

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