INTERSECTION OF THREE QUADRIC IN THE SPECIAL POSITION Milada Kočandrlová

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1 INTERECTION OF THREE QUADRIC IN THE PECIAL POITION Milada Kočandrlová Mathematical statement of the problem On the assumptions that we know coordinates of satellites and, the length of the signal from to, the velocity vector of the satellite and the parameters of reference ellipsoid we have three quadrics of the followng properties: Oblate ellipsoid of revolution Q, Earth reference ellipsoid. Prolate ellipsoid of revolution Q, the set of points P, that reflect a signal of the constant length d + d a, which is transmitted from the satellite to the satellite. The satellites and lie in the foci. traight circular cone Q with the vertex in, the set of all points P in which a signal transmitted from the satellite is reflected to the satellite so that the satellite velocity vector u and reflected ray vector make the angle θ. We investigate the intersection of the quadrics Q with the quadric Q and a common points of this intersection and the quadric Q. Then the found common point is the desired reflected point P. Intersection of the prolate ellipsoid of revolution and the straight circular cone We prove that the intersecting curve of the quadrics Q and Q consists of two ellipses in two planes ρ and ρ. If these planes exist they must be from a beam of quadrics determined by the quadrics Q and Q. We describe the quadrics Q and Q by quadric equations. Thus the equation of a beam quadric is a linear combination of the quadric equations describing the quadrics Q and Q. There are singular quadrics of the beam. The singular quadrics are planes hence the quadric equation of each of them is decomposed into two linear equations. We find these planes and choose one of them to search for all three quadrics intersection points. Let ρ be this plane. We solve the intersection of the quadrics Q and Q in the Cartesian coordinate system with the origin O in the center of the quadric Q and unit vectors e ( O), e3 ( e u), e e3 e. O e u Then the equation of the quadric Q is x x x () a b b Any point X lies on the circular cone Q if relation ( X ) u cosθ X () holds. The angle of the circular cone elements and the axis of rotation is θ hence cos θ >. Let u i ue i, i,,3, we use relation () to derive the equation of the quadric Q. It is (( x e) u + xu ) cos θ( x e) + x + x3 ). (3) The quadrics provide a beam of quadrics. Any quadric of the beam has the equation which is the linear combination of the equations () and (3). And vice versa any linear combination of the equations () and (3) sets a quadric of the beam. Accordingly the equation

2 (( x ) + ) θ ( ) + e u xu cos x e b x a derived from the substitution () and (3) and its modification e cos θ x (( x e) u + x u ) a determine a quadric of the beam. The last-mentioned equation is factorized into product e e ( x e) u + xu + cos θ x ( x e) u + xu cos θ x. a a Derived product is a singular quadric consists of two planes ρ and ρ. Their equations are e e ρ : ( x e) u + xu + cosθ x, ρ': ( x e) u + xu cosθ x. a a b 3 Intersection of the plane ρ with the quadrics Q and Q The cone Q and the ellipsoid Q intersect at an intersecting ellipse Q lzing in the plane ρ. A common point of the ellipse Q and the reference ellipsoid Q is the desired reflected point P. Therefore the point P is a common point of the ellipse Q and an ellipse Q which is the intersection of the ellipsoid Q with the plane ρ. To find a part of the cone cut bz the plane ρ we divided the cone Q into two cones. The first one is formed by points X such as ( X ) u > and the second one is of points X satisfying inequality ( X ) u <. The e inequality x a holds for all points of the ellipsoid Q therefore x a. Accordingly < the first cone intersects the plane ρ. Next we find the center O and the axes of the intersecting ellipse Q Q ρ Q ρ. The vertices A r, B r of the ellipse Q are in the coordinate plane x x. The center O is the center of the line segment A r B r. The center O and vectors of the major and minor axes define a local coordinate system in the plane ρ. denotes the coordinate system with respect to the global coordinate system with the origin in the center of the ellipsoid Q. In the coordinate szstem we find an equations f (x) and g (x) respectively of the quadric Q respectively where Q Q ρ. These equations are quadratic equations and their common solution is the reflected point P. This problem is transformed from the quadratic one into linear one again bz a beam of conic sections in the plane ρ. a 4 Beam of conic sections in the plane ρ The equation of any conic section of the beam (except Q ) is f ( x) + λ g ( x). (4) The common points of the ellipses Q and Q are the desired reflected points P. ingular conic sections beam defined bz the ellipses Q and Q in the plane ρ go through the points P. The determinant of the matrix of the form (4) is equal to zero for all singular beam conic

3 sections. The determinant is the cubic polynomial with variable λ. There is at least one real root λ of the polynomial. The singular conic section of the beam (4) has equation ( x) + g ( x). f λ (5) 5 Number of common points of the ellipses There are following options for the quadratic form (5): ) The quadratic form (5) is zero form. Then the conic sections Q are identical. ) The matrix of the quadratic form (5) is of the rank one. Then the conic section (5) is one line p. Any point of the line p is the singular point of the conic section (5). Afterwards one of the following possibilities must arise: a) p Q { A, B} - A, B are common points of the conic sections Q and the conic sections have the common tangent line at these points, Fig.b. b) p Q { A} - A is the only common point of the conic sections Q with the tangent line, Fig. f. c) The set p Q is a pair of imaginary complex associated points. The conic sections Q have no real common point. 3) The matrix of the quadratic form (5) is of the rank two. Then the conic section (5) is a pair of lines p, p. The lines are either real or imaginary with a real point of intersection which is the singular point T. Accordingly there are the following possibilities: a) T Q - hence T Q and the conic sections have the common tangent line at the point T, Fif.c. Q have no I. The lines p, p are imaginary. In this case the conic sections Q and next common point. II. The lines p, p are real: A. One line of the lines p, p is the tangent line of Q at the point q. Let p be this tangent line. Then the line p intersects the conic section Q at the next point A. The conic sections intersect but do not touch at the point A, Fig.d..

4 b) T Q : B. None of the lines p, p is the tangent line of Q. In this case each of them intersects Q at the next point. The set Q Q is of three points and the conic sections Q have the common tangent line at the point T. Fif.e. I. The lines p, p are imaginary. In this case Q Q Φ. II. The lines p, p are real and non-identical. Then there are the following three possibilities for each of them, Fig.a: A. Either it intersects the conic section Q at two points, B. or it is tangent to Q at one point and have no next point of intersection, C. or it has no common point with Q. a) b) c) d) e) f) The results are for these input data: All the Cartesian coordinates and lengths are in milions of meters. The angles are in degrees. Coordinates of satellites: [.74788, , ] [ , 7.5, ] velocity vector: u ( ,.7353,.837) angle θ 69.33

5 length of signal reflected by the earth surface: a semi-axis of the reference ellipsoid: a r, b r Results from the first approach: P P [ , , ] [ , , ] checking data: P P P + P angle (, u) P position error on the reference ellipsoid: P P P + P angle (, u) P position error on the reference ellipsoid: x + y a z + b Literatura [] Kostelecký J., Klokočník J.: Geometry and accuracy of reflecting points in bistatic satellite altimetry, J.Geod., in review, 4