Ellipsoid

“ Excluding these degenerate forms, the surface is of the order, rank and class each =2, and it has no singularities. Distinguishing the forms according to reality, we have the ellipsoid, the hyperboloid of two sheets, the hyperboloid of one sheet, the elliptic paraboloid and the hyperbolic paraboloid (see GEOMETRY: § Analytical) . A particular case of the ellipsoid is/the sphere; in abstract geometry this is a quadric surface passing through a given quadric curve, the circle at infinity. ”

“ Ellipsoids and the Geoid as Reference Surfaces Since the surface of the geoid is irregular and the surface of an ellipsoid is regular, no ellipsoid can provide more than an approximation of part of the geoidal surface. Figure 203 illustrates an example. A variety of ellipsoids are necessary to cover the entire earth. ”

“ Writing for , (8) and (9) become
This value of is constant over the surface of the ellipsoid , for at all points of this surface; it also vanishes at infinity, and it satisfies (7) . It is therefore the value of required. To find the constant A we make have its proper value at the end of axis .
Now
at the end of the axis. ”

“ For a long time the assumption of an ellipsoid with three unequal axes has been held possible for the figure of the earth, in consequence of an important theorem due to K. G. Jacobi, who proved that for a homogeneous fluid in rotation a spheroid is not the only form of equilibrium; an ellipsoid rotating round its least axis may with certain proportions of the axes and a certain time of revolution be a form of equilibrium. ”

“ For many purposes the mathematical surface is assumed to be a plane; in other cases a sphere of radius 6371 kilometres (20,900,000 ft.) . In the case of extensive operations the surface must be considered as a compressed ellipsoid of rotation, whose minor axis coincides with the earth’s axis, and whose compression, flattening, or ellipticity is about 1/298. ”

“ If we are given a surface (e.g. an ellipsoid) in Euclidean three-dimensional space, then there exists for this surface a two-dimensional geometry, just as much as for a plane surface. Gauss undertook the task of treating this two-dimensional geometry from first principles, without making use of the fact that the surface belongs to a Euclidean continuum of three dimensions. ”

“ Thus a plane perpendicular to the optic axis touches the wave-surface along a circle. Similarly the normals to the circular sections of the reciprocal ellipsoid, or the axes of the tangent cylinders to the polarization-ellipsoid that have circular normal sections, are directions of single-ray velocity or ray-axes, and it may be shown as above that corresponding to a ray-axis there is a cone of wave-normals with circular sections parallel to the normal section of the corresponding tangent cylinder, and its plane of Contact with the ellipsoid. ”

“ In the international exhibition of 1851 at Hyde Park was exhibited a small model ellipsoid made in Berlin, on which the lines of curvature were traced after a method invented by Roberts. The lines of curvature and asymptotic lines on the surface, at any point of which the sum of the principal curvatures is zero, were also discussed in Liouville's ‘Journal de Mathématiques,’ 1850. ”

“ Since for the optic axis, it would be difficult to distinguish a case of this kind from an ordinary uniaxial crystal, unless the ellipsoid (24) should approach very closely to a sphere. [12]
It is only in the very limited case described in the last paragraph that a medium which is identical in its internal structure with its image by reflection can have the property of circular or elliptic polarization. To media which are unlike their images by reflection, and have the property of circular polarization, we may apply the following general principles. ”

“ All the above forms, the sphere, spheroid, ellipsoid, and ovaloid are geometrical forms which represent with more and more exactness the true form of the earth, but even this increasing exactness will not account for the discrepancies and irregularities which have been found at various places, and which cannot be explained on the ground of inaccurate work. ”

“ The period from Eratosthenes to Picard has been called the spherical era of geodesy; the period from Picard to the end of the 19th century has been called the ellipsoidal period. During the latter period the earth was conceived to be an ellipsoid, and the determination of its ellipticity, that is, the difference of equatorial radius and polar radius divided by the equatorial radius, became an important geodetic problem. A significant contribution to the solution of this problem was made by determinations of gravity by the pendulum. ”

“ Elements (Euclid's) , 36–39, 104, 114, 125, 127, 128, 133, 135, 136, 138. See Euclid.
Elimination, 250, 308, 310, 330, 331. See Equations.
Elizabeth, Princess, 188.
Ellipsoid (attraction of) , 215, 277, 280, 285, 298, 366, 378, 379; motion of, 383.
Elliptic co-ordinates, 379.
Elliptic functions, 241, 279, 280, 296, 329, 345, 346, 347–354, 363, 367, 370.
Elliptic geometry. See Non-Euclidean geometry.
Elliptic integrals, 247, 252, 328, 349, 350. ”

“ Thus, if a cone be cut by a plane at right angles to its axis we get a circle. If, instead of being perfectly at right angles, the plane subtends with the axis an angle of 89° 59′, we have an ellipse which no human eye, even when aided by an accurate pair of compasses can distinguish from a circle. Decreasing the angle minute by minute the ellipse becomes first perceptibly eccentric, then manifestly so, and by and by acquires so immensely elongated a form, as to bear no recognizable resemblance to a circle. By continuing this process the ellipse passes insensibly into a parabola ”

“ In isometric perspective everything that is round in reality is drawn in the shape of an ellipse, that is a closed curve that is longer than it is wide as shown at A in Fig. 28; there are different shaped ellipses but [33] there is only one used for isometric drawing and this is always in the ratio of 11⁄4 to 2; that is if an ellipse is 2 inches long it will be 11⁄4 inches wide; an ellipse 4 inches long will be 21⁄2 inches wide and so on.
An easy, though rough way to draw an isometric ellipse is to make a line as long as the diameter of the disk or wheel you intend to represent ”

“ If one axis of our ellipse vanish, while the other remain of finite length, then the ellipse is reduced to a straight line, and its roulette will appear as a succession of semicircles touching one another upon the axis (C) ; the solid of revolution will be a series of equal spheres. If as before one axis of the ellipse vanish, but the other be infinitely long, then the curve described by the rotation {219} of this latter will be a circle of infinite radius, i.e. a straight line infinitely distant from the axis; and the surface of rotation is now a plane. ”

“ Let us return to the subject of Analytical Geometry.
The tangent to an ellipse is an imaginary straight line, representing the path which would be followed by the body tracing the ellipse, if its connection with the attracting focus were suddenly to cease. In its essence, the tangent is a sublime effort of the scientific imagination; it pictures the result of a sudden cessation of the action of gravity. In practice the tangent is a convenient line for indicating the curvature at any given point. ”

“ If perturbations are ignored, a planet can be regarded as moving in an ellipse with the sun in one focus. The size and shape of the ellipse can be defined by the length of its axis and by the eccentricity; the plane in which the ellipse is situated is determined by the position of the line, called the line of nodes, in which it cuts a fixed plane, usually taken to be the ecliptic, and by the inclination of the two planes. ”

“ If we hold it parallel to the face, with the whole breadth opposite to the eye, we see it as an entire circle. If we turn it a little sideways, it appears oval, or as an ellipse; and, as we continue to turn it more and more round, the ellipse grows narrower and narrower, until, when the edge is presented to the eye, we see nothing but a line. Now imagine the ring to be near a perpendicular wall, and the eye to be removed at such a distance from it, as not to distinguish any interval between the ring and the wall ”

“ I shall now speak of elliptic motion. I must first state what an ellipse is. There are different ways of describing an ellipse. I dare say there are many mechanical persons near me who are acquainted with a carpenter's trammel. An ellipse, or an oval may be described in this way, in Figure 29. Suppose AB, DE, to represent the longer and shorter diameters of the ellipse, at right angles to each other; then, if we have a bar GFH, with two pins FG fixed in it, so arranged that the pin F shall always move along the long diameter AB, and the pin G shall always move along the short diameter DE ”

“ Theorem. Every section of a cone made by a plane passing through its vertex is a triangle.
At this time, or in speaking of the preliminary definitions, reference should be made to the conic sections. Of these there are three great types: (1) the ellipse, where the cutting plane intersects all the elements on one side of the vertex; a circle is a special form of the ellipse; (2) the parabola, where the plane is parallel to an element; (3) the hyperbola, where the plane cuts some of the elements on one side of the vertex, and the rest on the other side ”

“ Suppose then that a class, while becoming skilful in working problems, seems to have forgotten that the axes are no part of the curve itself. The teacher may wake it up by saying, "You don't imagine, surely, that axes form any necessary portion of a curve. An ellipse, for instance, is the path of a body moving round a focus of attraction. Suppose a planet were endowed with the power of leaving a track in the sky, the track would be an ellipse only, unencumbered with axes. ”

“ The equation of an ellipse expresses the universal conceivable relation between its co-ordinates, of which only the real values have geometrical significance. Similarly, the equations between the elements of phenomena express a universal, mathematically conceivable relation. Here, however, for many values only certain directions of change are physically admissible. As in the ellipse only certain values satisfying the equation are realised, so in the physical world only certain changes of value occur. Bodies are always accelerated towards the earth. ”

“ Let a p = o q, and when the motion parallel to a c has vanished, the particle will be found at p; hence p must be the extremity of the major axis of the ellipse. In the next revolution, the extremity of the axis will advance a little more, and thus the ellipse moves round gradually. ”