Coordinate system

“ We choose to impose the additional condition that the resulting metric space be a Euclidean geometry with a rectangular coordinate system. [Pg 67]
Even this always does not yield uniqueness, but we will show the additional restriction that will guarantee uniqueness after the necessary language is developed. Since all metrizations of a given metrizable topology are isomorphic, in the quotient class the orthogonal Euclidean geometry serves the purpose of being a convenient representative of the unique element resulting from a given metrizable topology. ”

“ As external observers we have a prior knowledge of this coordinate [Pg 59] system; however, the NPO is given only the vectors X1 and X2 in the i0 ⨉ i1 and i0 ⨉ i2 planes. The NPO can reconstruct the entire geometry but the actual output Ξ obviously is constrained to lie in the plane of the input vector X. The following formulas are typical of the relations present. ”

“ It is readily seen that the position of a heavenly body is completely defined when these co-ordinates are given.
One of the systems of co-ordinates is familiar to every one, and may be used as a general illustration of the method. It is our system of defining the position of a point on the earth’s surface by its latitude and longitude. Regarding O (fig. 1) as the centre of the earth, and P as a point on the earth’s surface, a city for example, it will be seen that OZ being the earth’s axis, the circle MN will be the equator. ”

“ The treatment of the problems that are now coming into play is most simple, when instead of the co-ordinate system used above, we introduce another one which is rigidly connected with ponderable matter and therefore shares its displacement.
While the coordinates of a point with respect to the fixed system were called x, y, z, let those, which refer to the moving system and which I call the relative coordinates, denoted by (x) , (y) , (z) for the time being. ”

“ If we disregard the existence of the co-ordinate system, nothing has been defined, but if we consider the co-ordinate system as given, then every point of the plane can be considered as defined unambiguously. ”

“ Latitude and longitude, which lie at the basis of all descriptive geography, are determined by observation. No map deserves the name, on which the position of important points has not been astronomically determined. Some even of our most important political and administrative arrangements depend upon the coöperation of this science. Among these I may mention the land system of the United States, and the determination of the boundaries of the country. ”

“ The resulting equation in (α, β) represents the locus of the centre of curvature. This is the evolute of the curve.
22. Polar Coordinates.—In plane geometry the distance of any point P from a fixed origin (or pole) O, and the inclination xOP of OP to a fixed line Ox, determine the point: r, the numerical measure of OP, the radius vector, and θ, the circular measure of xOP, the inclination, are called polar coordinates of P. The formulae x = r cos θ, y = r sin θ connect Cartesian and polar coordinates, and make transition from either system to the other easy. ”

“ It would be, for example, evidently more difficult to figure to one's self clearly the changes of place of a point which is determined by the intersection of two right lines, which each turn around a fixed point, making a greater or smaller angle with a certain axis, as in the system of co-ordinates previously noticed. ”

“ Let us imagine that each point has noted upon it the three numbers that give its position, then we have something comparable with a measure with numbered subdivisions; only we now have to do, one might say, with a good many imaginary measures in three common perpendicular directions. In this "system of co-ordinates" the numbers that fix the position of one or the other of the bodies may now be read off at any moment. ”

“ And now the feeblest among us can repeat the words of power, and take part in dynamical discussions which a few years ago we should have left to our betters."
A very remarkable feature in this discussion is the use made of the idea of "ignoration of coordinates." The variables made use of in the Lagrangian equations must be such as to enable the positions of the parts of the system which determine the motion to be expressed for any instant of time. These parts, by their displacements, control those of the other parts, through the connections of the system. ”

“ If the ratio of to is zero or unity, the system of surfaces becomes one system of meridian planes, one system of right cones having a common axis, and a system of concentric spherical surfaces of which the radius is inversely proportional to . ”

“ The same necessity which has caused the metrical system to be adopted for the measurement of terrestrial spaces, and which is now endeavouring to establish a common meridian, renders it equally indispensable that a common era should be sought, so as to establish a concordance for the events of various nations. ”

“ This proposal was not favorably received, and the Academy, at the request of government, appointed as a commission Borda, Lagrange, Laplace, Monge, and Condorcet, to decide whether the seconds pendulum, the quarter of the equator, or the quarter of a meridian, should be used as the natural standard for the new system of measures. [10] They settled on the last as best for the purpose, and resolved that the ten millionth of the meridian quadrant, or distance from equator to pole, measured at sea level, be taken for basis of the new system, and be called a mètre. ”

“ The system of circles described above is by far the most extensively used, and positions determined by its coordinates are comparatively constant, but there is still a third system of circles which was used and handed down to us by the ancients. ”

“ Hence a couple of moment G, whose axis has the direction (l, m, n) relative to a right-handed system of rectangular axes, is equivalent to three couples lG, mG, nG in the co-ordinate planes. The analytical reduction of a three-dimensional system can now be conducted as follows. Let (x1, y1, z1) be the co-ordinates of a point P1 on the line of action of one of the forces, whose components are (say) X1, Y1, Z1. Draw P1H normal to the plane zOx, and HK perpendicular to Oz. In KH introduce two equal and opposite forces ±X1. ”

“ A system coaxal with the two given circles is readily constructed by describing circles through the common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis—considered as a circle of infinite radius. ”

“ By “a system of correlations” we mean the adequate result of the combination of all the intervals from every single term to every other within the limits assumed, in every direction. Such a result will therefore represent either a volume, or a surface, or a line, according as the terms considered within the given limits are differently disposed in space. Thus a spherical place results from the mutual relations intervening between all the terms of its geometric surface ”

“ But however we may choose a system of co-ordinates, the laws of configuration of rigid rods do not agree with those of Euclidean geometry; in other words, we cannot choose any system of co-ordinates so that the co-ordinate differences, , , , corresponding to the ends of a unit measuring rod, oriented in any way, shall always satisfy the relation . ”

“ The creators of the metrical system were persuaded that the measurements of the meridian effected in their day could never be surpassed in precision; and on the other hand, by borrowing from nature a definite basis, they thought to take from the definition of the unit some of its arbitrary character, and to ensure the means of again finding the same unit if by any accident the standard became altered. ”

“ The coordinates of a point taken at random in the plane will, as a rule, not satisfy the equation, but infinitely many points, and in most cases infinitely many real ones, have coordinates which do satisfy it, and these points are exactly those which lie upon some locus of one dimension, a straight line or more frequently a curve, which is said to be represented by the equation. ”

“ As we examined our mesh-system at regions farther and farther removed from the equator, we should notice that the meridians had a tendency to run closer and closer together, so that the quadrilaterals bounded by the meridians and parallels would depart more and more in shape from those of Euclidean squares. ”

“ If this is the case, then they differ only by their position against the coordinate axes. By appropriate use of the arbitrary constants and the functions U, V, W we can obtain vivid special cases. By coordinate transformation we are lead to a (at least formally) general case, in which the shift of the surface is not parallel to the A-axis parallel, but directed in an arbitrary way. ”

“ The co-ordinates of the principal stations of a trigonometrical survey are usually the spherical co-ordinates of latitude and longitude; those of a traverse survey are always rectangular, plane {or a small area but spherical for a large one. ”

“ It is necessary, in order to conveniently define the position of heavenly bodies to mark this celestial sphere with imaginary circles to serve as coordinates, as we mark the earth with meridians of longitude and parallels of latitude. Before going into the explanation of these coordinates, it may be well to consider a few facts concerning circles. A great circle is of course understood to be one whose plane passes through the center of a sphere, dividing it into two equal parts. ”

“ The centre of a circle is a point in itself indivisible, it fills no space; but in order that it be of any use as centre, we must be able to refer all the radii to it, and this is impossible unless it occupy a determinate position equidistant from all points of the circumference. As a general rule, geometry acts upon dimensions, and these dimensions require points in which they commence, points through which they pass, and points in which they end, and by which distances, inclinations, and all that relates to the position of lines and planes, are measured. ”

“ Measurement of Distance.—The shortest distance between two places on the surface of a globe is represented by the arc of a great circle. If the two places are upon the same meridian or upon the equator the exact distance separating them is to be found by reference to a table giving the lengths of arcs of a meridian and of the equator. In all other cases recourse must be had to a map, a globe or mathematical formula. Measurements made on a topographical map yield the most satisfactory results. Even a general map may be trusted, as long as we keep within ten degrees of its centre. ”

“ These measures of arcs of the meridian, as measurements of a known and selected portion of the surface of the spheroidal globe of the earth, have, more or less, fixed mathematical relations with the axis of the earth; as the circumference of a sphere has an exact mathematical ratio to its diameter. ”

“ Here the radius, a straight line, is clearly the determining factor.
We now move to the other pole of the primary polarity, that is to the plane, and let the sphere arise by imagining the plane approaching an infinitely distant point evenly from all sides. We view the process realistically only by imagining ourselves in the plane, so that we surround the point from all sides, with the distance between us and
the point diminishing gradually. ”

“ In the case of the spheroid this equation becomes
which integrated gives ydx − xdy = Cds. This again may be put in the form r sin a = C, where a is the azimuth of the geodetic at any point—the angle between its direction and that of the meridian—and r the distance of the point from the axis of revolution. ”

“ In order that directions on the map and on the ground shall correspond, it is necessary for the map to be oriented, that is, the true meridian of the map must lie in the same direction as the true meridian through the observer's position on the ground, which is only another way of saying that the lines that run north and south on the map must run in the same direction as the lines north and south on the ground. ”

“ After this geometrical representation of the relations of the horizontal magnetic force, we proceed to develope the mode of submitting them to calculation. On the surface of the earth becomes a simple function of two variable magnitudes, for which we will take the geographical longitude reckoned eastward from an arbitrary first meridian,—and the distance from the north pole of the earth ”

“ With the master and secondary stations in known geographic positions, the only unknowns are the two geographic coordinates of the observer.
Each hyperbolic line of position in Figure 1204 represents the locus of points for which (D) is held constant. ”

“ Thus a plane is the simplest and most symmetrical boundary by which a solid can be divided; and a straight line is the simplest and most symmetrical boundary by which a plane can be separated. These conceptions are obtained by considering the boundaries of an interminable space, capable of imaginary division in every direction. And as a limited space may be separated into two parts by a plane, and a plane again separated into two parts by a straight line, so a line is divided into two portions by a point, which is the common boundary of the two portions ”

“ He who solves scientifically a geometrical, physical, or technical problem, easily remarks that his procedure is a methodical mental quest, rendered possible by the economical order of the province—a simplified purposeful quest as contrasted with unmethodical, unscientific guess-work. The geometer, for example, who has to construct a circle touching two given straight lines, casts his eye over the relations of symmetry of the desired construction, and seeks the centre of his circle solely in the line of symmetry of the two straight lines. ”