Cartesian coordinate system

“ The analytical theory by Cartesian co-ordinates was first considered by Alexis Claude Clairaut, Recherches sur les courbes à double courbure (Paris, 1731) . Such a curve may be considered as described by a point, moving in a line which at the same time rotates about the point in a plane which at the same time rotates about the line; the point is a point, the line a tangent, and the plane an osculating plane, of the curve; moreover the line is a generating line, and the plane a tangent plane, of a developable surface or torse, having the curve for its edge of regression. ”

“ In an Euclidean for orthogonal Cartesian coordinates as well as , covariants and contravariants coincide due to the properties of the orthogonal matrix of determinant . In addition, for instance, the homogeneous coordinates of the plane of are comparable to the covariants of first order as long as the transformation of the homogeneous to the homogeneous is projective (i.e. linear) , and the point coordinates of are comparable to the contravariants of first order. ”

“ It should be noted that inasmuch as space-time appears as an absolute continuum, existing independently of our measurements, there is no reason to limit ourselves to Cartesian mesh-systems when we wish to explore it. The mesh-system is, as we know, a mere mathematical device facilitating our investigations; and from a purely mathematical point of view we may always with equal justification split up a given continuum with one type of mesh-system or another. ”

“ 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. ”

“ Just as it was there impossible to construct a Cartesian co-ordinate system from equal rods, so here it is impossible to build up a system (reference-body) from rigid bodies and clocks, which shall be of such a nature that measuring-rods and clocks, arranged rigidly with respect to one another, shall indicate position and time directly. ”

“ So far as the essential characteristics of space-time itself are concerned, these Gaussian mesh-systems are just as legitimate as the Cartesian ones. As a matter of fact, when space-time becomes curved, owing to the presence of gravitation, Cartesian co-ordinates cannot be realised, being incompatible with the curvature of the continuum. ”

“ When the previous formulas are applied to the special case of the passage from Cartesian orthogonal coordinates of an Euclidean to orthogonal curvilinear coordinates, the known Jacobian integral transformation follows. ”

“ Here, the line integral is orbiting around the normal in the negative sense, thus clock-wise if the coordinate system is a right-system; because by § 1 the directions , , are following each other like the coordinate axes, where is the normal (which is directed outwards of the area framed on ) of the framing-. Ordinarily, one prefers a positive sense of circulation and therefore the generalized rotation:
etc. ”

“ Straight lines and conics are represented in trilinear and in areal, because in Cartesian, coordinates by equations of the first and second degrees respectively, and these degrees are preserved when the equations are made homogeneous. What must be said about points infinitely far off in order to make universal the statement, to which there is no exception as long as finite distances alone are considered, that every homogeneous equation of the first degree represents a straight line? ”

“ These equations of motion are therefore valid in every other system of Cartesian co-ordinates in the space of reference. In the case where the forces are conservative we can easily recognize the vector character of () . For a potential energy, , exists, which depends only upon the mutual distances of the particles, and is therefore an invariant. The vector character of the force, = , is then a consequence of our general theorem about the derivative of a tensor of rank 0. ”

“ They therefore set to work to reconstruct geometry on a basis which did not assume the universal possibility of numerical measurement—a reconstruction which, as may be seen in Euclid, they effected with extraordinary skill and with great logical acumen. The moderns, under the influence of Cartesian geometry, have reasserted the universal possibility of numerical measurement, extending arithmetic, partly for that purpose, so as to include what are called “irrational” numbers, which give the ratios of incommensurable lengths. ”

“ In order to exhibit the correspondence between the short and methods of vector analysis and the more familiar formulae of Cartesian geometry, we take a right-handed system of three mutually perpendicular axes Ox, Oy, Oz, and adopt three fundamental unit vectors i, j, /1, having the positive directions of these axes respectively. ”

“ If we treat as covariant, then this is evidently somewhat arbitrary in terms of Cartesian orthogonal coordinates, since covariants and contravariantes coincide here. If we would require (following Abraham) that the transform like the products of the corresponding components of the radius vector, then we would have to postulate the as contravariants. By passage to the reciprocal system we can, however, represent them as convariants. For the sake of being easier related to the integral forms, we stick to the covariant description. ”

“ In two dimensions there is no necessity for a special line geometry, inasmuch as the straight line and the point are interchangeable by the principle of duality; but in three dimensions the straight line is its own reciprocal, and for the better discussion of systems of lines we require some new apparatus, e.g., a system of coordinates applicable to straight lines rather than to points. ”

“ From two observers moving uniformly but at different velocities on parallel paths, each of them can claim with the same justification, that they are at rest in respect to each other. Expressed in terms of geometry, this means that we can transform any point of the plane into a state of rest by relating it to a new coordinate system. From the related point, let fall the perpendicular upon the X-axis and use this perpendicular as the new ordinate axis. If we take, for example, the perpendicular through M, then XY is the coordinate system S, in which observer M (as he thinks) is at rest. ”

“ It is evident that in the Euclidean geometry only such, and all such, quantities have an objective significance, independent of the particular choice of the Cartesian co-ordinates, as can be expressed by an invariant with respect to linear orthogonal transformations. This is the reason that the theory of invariants, which has to do with the laws that govern the form of invariants, is so important for analytical geometry. ”

“ If we turn our attention to experimental physics we see that there the coordinate system is invariably represented by a "practically rigid" body. Furthermore it is assumed that such rigid bodies can be positioned in rest relative to one another in common with the bodies of Euclidian geometry. Insofar as we may think of the rigid measuring body as existing as an object which can be experienced, the "system of coordinates" concept as well as the concept of the motion of matter relative thereto can be accepted in the sense of the "stipulation of meaning". ”

“ As many different constructions as can be imagined for determining the position of a point, whether on a plane or in space, so many distinct systems of co-ordinates may be conceived; they are consequently susceptible of being multiplied to infinity. But, whatever may be the system adopted, we shall always have reduced the ideas of situation to simple ideas of magnitude, so that we will consider the change in the position of a point as produced by mere numerical variations in the values of its co-ordinates. ”

“ If we imagine constructions to be made with rigid rods in the surface (similar to that above with the marble slab) , we should find that different laws hold for these from those resulting on the basis of Euclidean plane geometry. The surface is not a Euclidean continuum with respect to the rods, and we cannot define Cartesian co-ordinates in the surface. Gauss indicated the principles according to which we can treat the geometrical relationships in the surface, and thus pointed out the way to the method of Riemann of treating multi-dimensional, non-Euclidean continua. ”

“ In the first place, we have to select the space in which the problem is to be set. The most general space that we feel capable of tackling is a metrizable topology. On the other hand, anything less general would be unnecessarily restrictive. Thus, we choose a metrizable topological space. [Pg 66]
As soon as we have made this choice, we regret it. In order to improve the situation somewhat, we show that there is no (additional) loss of generality in using an orthogonal Euclidean space times [9] a denumerable random cartesian product of irreducible (wrt direct product) denumerable groups. ”

“ We may, for instance, employ the three Cartesian co-ordinates of a particular point O of the body, and three angular co-ordinates which express the orientation of the body with respect to O. Thus in fig. 36, if OA, OB, OC be three mutually perpendicular lines in the solid, we may denote by θ the angle which OC makes with a fixed direction OZ, by ψ the azimuth of the plane ZOC measured from some fixed plane through OZ, and by φ the inclination of the plane COA to the plane ZOC. In fig. ”

“ In examining the motion of the Earth nobody will decide to use a coordinate system of the last kind, and neither will anyone decide from considering this example that the coordinate system, whose origin is co-moving with the center of mass of the considered mechanical system, is in principle privileged over other coordinate systems. It is the same in the example you mentioned. Nobody will use a coordinate system that is at rest relative to the planet Earth, because that would be impractical. However as a matter of principle such a theory of relativity is equally valid as any other. ”

“ In the article Geometry: Analytical, it is shown that the general equation to a circle in rectangular Cartesian co-ordinates is x2 + y2 + 2gx + 2fy + c = 0, i.e. in the general equation of the second degree the co-efficients of x2 and y2 are Cartesian co-ordinates. equal, and of xy zero. The co-ordinates of its centre are -g/c, -f/c; and its radius is (g2 + f2 - c) 1⁄2. The equations to the chord, tangent and normal are readily derived by the ordinary methods. ”

“ 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. ”

“ These formulas can be derived in the shortest way as follows: The static case, i.e. a straight world line parallel to the -axis, is only a specialization of the coordinate system for the potentials of a pointlike charge in , as one can see e.g. from the solution of Herglotz, because
etc.
thus
where means the “rest”-density, and the assumption of the static case is equivalent with a rotation of the coordinate system in the plane (Lorentz transformation) , so that
.
However, since it is indeed
,
the solution is
. ”

“ Imagine a curve, real or imaginary, represented by an equation (involving, it may be, imaginary coefficients) between the Cartesian co-ordinates u, u′; then, writing u = x + iy, u′ = x′ + iy′, the equation determines real values of (x, y) , and of (x′, y′) , corresponding to any given real values of (x′, y′) and (x, y) respectively; that is, it establishes a real correspondence (not of course a rational one) between the points (x, y) and (x′, y′) ”

“ 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. ”

“ But, if we demand, in particular, that each point of the manifold may be fixed by means of rectangular Cartesian co-ordinates, whereby particular assumptions are made concerning the possible ways of placing the measuring rods, then the line-element, [Pg 26] expressed in these special variables, assumes the form Hitherto this expression has always been introduced for the length of the line-element in all physical laws. ”

“ Such tables are usually of single entry, i.e. to a certain value of one variable there corresponds one, and only one, value of the other variable. To construct the graph, as it is called, of such a table, Cartesian co-ordinates are usually employed. Two lines or axes at right angles to each other are chosen, intersecting at a point called the origin; the horizontal axis is the axis of abscissae, the vertical one the axis of ordinates. Along one, say the axis of abscissae, distances are taken from the origin corresponding to the values of one of the variables ”