# Co-ordinates Sentence Examples

It is convenient to use these rather than Cartesian co-ordinates.

If the errors of the rectangular co-ordinates of these lines are known, the problem of determining the co-ordinates of any star-image on the plate becomes reduced to the comparatively simple one of interpolating the co-ordinates of the star relative to the sides of the 5 mm.

The microscope or viewing telescope is fitted with a spider-line micrometer having two screws at right angles to each other, by means of which readings can be made first on one reseau-line, then on the star, and finally on the opposite reseau-line in both co-ordinates.

This form of micrometer is of course capable of giving results of high precision, but the drawback is that the process involves a minimum of six pointings and the entering of six screw-head readings in order to measure the two co-ordinates of the star.

The problem of the curves is solved by their reduction to a problem of straight lines; and the locus of any point is determined by its distance from two given straight lines - the axes of co-ordinates.

At the head of the financial organization of France, and exercising a general jurisdiction, is the minister of finance, who co-ordinates in one general budget the separate budgets prepared by his colleagues and assigns to each ministerial department the sums necessary for its expenses.

Kepler's Problem, namely, that of finding the co-ordinates of a planet at a given time, which is equivalent - given the mean anomaly - to that of determining the true anomaly, was solved approximately by Kepler, and more completely by Wallis, Newton and others.

For instance, those of a ternary form involve two classes which may be geometrically interpreted as point and line co-ordinates in a plane; those of a quaternary form involve three classes which may be geometrically interpreted as point, line and plane coordinates in space.

Or, instead of looking upon a linear substitution as replacing a pencil of lines by a projectively corresponding pencil retaining the same axes of co-ordinates, we may look upon the substitution as changing the axes of co-ordinates retaining the same pencil.

As new axes of co-ordinates we may take any other pair of lines through the origin, and for the X, Y corresponding to x, y any new constant multiples of the sines of the angles which the line makes with the new axes.

AdvertisementIn the applications of the calculus the co-ordinates of a quaternion are usually assumed to be numerical; when they are complex, the quaternion is further distinguished by Hamilton as a biquaternion.

Taking co-ordinates in the plane of the screen with the centre of the wave as origin, let us represent M by, n, and P (where dS is situated) by x, y, z.

The form of (3) shows immediately that, if a and b be altered, the co-ordinates of any characteristic point in the pattern vary as a-'- and b-1.

The origin of co-ordinates 0 corresponds to v = 0; and the asymptotic points J, J', round which the curve revolves in an ever-closing spiral, correspond to v= =co .

The co-ordinates of J, J' being (- z, - z), I 2 is 2; and the phase is, period in arrear of that of the element at 0.

AdvertisementLet x, y, z be the co-ordinates of any particle of the medium in its natural state, and, 7 7, the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively.

In the limiting case in which the medium is regarded as absolutely incompressible S vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a 2 3 by a new function of the co-ordinates.

Thus if the plane is normal to Or, the resultant thrust R =f fpdxdy, (r) and the co-ordinates x, y of the C.P. are given by xR = f f xpdxdy, yR = f f ypdxdy.

Let us apply the above theorem to the case of a small parallelepipedon or rectangular prism having sides dx, dy, dz respectively, its centre having co-ordinates (x, y, z).

Its angular points have then co-ordinates (x t Zdx, y t Zdy, z * zdz).

AdvertisementThis is mathematically expressed by the statement that dE is an exact differential of a function of the co-ordinates defining the state of the body, which can be integrated between limits without reference to the relation representing the path along which the variations are taken.

Observing that F is a function of the co-ordinates expressing the state of the substance, we obtain for the variation of S with pressure at constant temperature, dS/dp (0 const) '=' 2 F/dedp =-0d 2 v/d0 2 (p const) (12) If the heat supplied to a substance which is expanding reversibly and doing external work, pdv, is equal to the external work done, the intrinsic energy, E, remains constant.

The final state of the substance, when equilibrium has been restored, may be deduced from this condition, if the energy can be expressed in terms of the co-ordinates.

If the system is supposed to obey the conservation of energy and to move solely under its own internal forces, the changes in the co-ordinates and momenta can be found from the Hamiltonian equations aE aE qr = 49 - 1 57., gr where q r denotes dg r ldt, &c., and E is the total energy expressed as a function of pi, qi,.

Since the values of the co-ordinates and momenta at any instant during the motion may be treated as " initial " values, it is clear that the " extension " of the range must remain constant throughout the whole motion.

AdvertisementThis result at once disposes of the possibility of all the systems acquiring any common characteristic in the course of their motion through a tendency for their co-ordinates or momenta to concentrate about any particular set, or series of sets, of values.

Let us imagine that the systems had the initial values of their co-ordinates and momenta so arranged that the number of systems for which the co-ordinates and momenta were within a given range was proportional simply to the extension of the range.

Ow are any momenta or functions of the co-ordinates and momenta or co-ordinates alone which are subject only to the condition that they do not enter into the coefficients a 1, a 2, &c.

This involves the use of Cartesian co-ordinates, and leads to important general formulae, such as Simpson's formula.

The most important formulae are those which correspond to the use of rectangular Cartesian co-ordinates.

In the case of the briquette the position of the foot of the ordinate u is expressed by co-ordinates x, y, referred to a pair of axes parallel to a pair of sides of the base of the briquette.

In analytical geometry, the equation to the sphere takes the forms x 2 +y 2 +z 2 =a 2, and r=a, the first applying to rectangular Cartesian co-ordinates, the second to polar, the origin being in both cases at the centre of the sphere.

R=wx (I+x2/4Y2) Let i be the angle between the tangent at any point having the co-ordinates x and y measured from the vertex, then 3..

But the nakshatras are twenty-eight, and are represented by as many " junction stars " (yogatara), carefully determined by their spherical co-ordinates.

The equations of motion are now, the co-ordinates x and y being measured in feet, 2 (26) - -rr- - C, dt2 dty - g' * These numbers are taken from a part omitted here of the abridged ballistic table.

The physical and the ethical are not distinguished, and in this respect the character of the system is thoroughly materialistic; for when Mani co-ordinates good with light, and evil with darkness, this is no mere figure of speech, but light is actually good and darkness evil.

The co-ordinates of its centre are - g/c, f/c; and its radius is (g 2 +f 2 - c) I.

In various systems of triangular co-ordinates the equations to circles specially related to the triangle of reference assume comparatively simple forms; consequently they provide elegant algebraical demonstrations of properties concerning a triangle and the circles intimately associated with its geometry.

The general equation to the circle in trilinear co-ordinates is readily deduced from the fact that the circle is the only curve which intersects the line infinity in the circular points.

The corresponding equations in areal co-ordinates are readily derived by substituting x/a, ylb, z/c for a, 1 3, y respectively in the trilinear equations.

Essentially, therefore, Descartes's process is that known later as the process of isoperimeters, and often attributed wholly to Schwab.2 In 16J5 appeared the Arithmetica Infinitorum of John Wallis, where numerous problems of quadrature are dealt with, the curves being now represented in Cartesian co-ordinates, and algebra playing an important part.

The phenomena of equilibrium can be represented on diagrams. Thus, if we take our co-ordinates to represent pressure and temperature, the state of the systems p with ice, water and vapour in equilibrium is represented by the point 0 where the pressure is that of the vapour of water at the freezing point and the temperature is the freezing point under that pressure.

The simplest equation to the parabola is that which is referred to its axis and the tangent at the vertex as the axes of co-ordinates, when it assumes the form y 2 = 4ax where as = semilatus rectum; this may be deduced directly from the definition.

An equation of similar form is obtained when the axes of co-ordinates are any diameter and the tangent at the vertex.

The various forms in areal co-ordinates may be derived from the above by substituting Xa for 1, µb for m and vc for n, or directly by expressing the condition for tangency of the line x+y+z = o to the conic expressed in areal coordinates.

The instrument is thus a theodolite, although, compared with its other dimensions, feeble as an apparatus for the measurement of absolute altitudes and azimuths, although capable of determining these co-ordinates with considerable precision.

This refers to the use of the x, y, z co-ordinates, - associated, of course, with i, j, k.

The physical properties of a heterogeneous body (provided they vary continuously from point to point) are known to depend, in the neighbourhood of any one point of the body, on a quadric function of the co-ordinates with reference to that point.

The problem of determining the possible configurations of equilibrium of a system of particles subject to extraneous forces which are known functions of the positions of the particles, and to internal forces which are known functions of the distances of the pairs of particles between which they act, is in general determinate For if n be the number of particles, the 3n conditions of equilibrium (three for each particle) are equal in number to the 351 Cartesian (or other) co-ordinates of the particles, which are to be found.

Since the four co-ordinates (Cartesian or other) of these two points are connected by the relation which expresses the invariability of the length AB, it is plain that virtually three inde pendent elements are re quired and suffice to specify the position of the lamina.

They may be called (in a generalized sense) the co-ordinates of the lamina.

If we equate these to zero we get the co-ordinates of the instantaneous centre.

Hence in trilinear co-ordinates, with ABC as fundamental triangle, its equation is Pa+Q/1+R7=o.

The six independent quantities, or co-ordinates, which serve to specify the position of a rigid body in space may of course be chosen in an endless variety of ways.

We may, for instance, employ the three Cartesian co-ordinates of a particular point 0 of the body, and three angular co-ordinates which express the orientation of the body with respect to 0.

We first suppose that one point 0 of the body is fixed, and take this as the origin of a right-handed system of rectangular co-ordinates; i.e.

Let (x1, yi, Zi) be the co-ordinates of a point Pi on the line of action of one of the forces, whose components are (say) X1, Yi, Zi.

If any other point 0, whose co-ordinates are x, y, z, be chosen in place of 0, as the point to which the forces are transferred, we have to write x1x, yiy, ZiZ for xl, Yi, z1,and so on, in the preceding process.

In the analytical investigations of line geometry, these six quantities, supposed subject to the relation (4), are used to specify a line, and are called the six co-ordinates of the line; they are of course equivalent to only four independent quantities.

The omission of the additive arbitrary constants of integration in (8) is equivalent to a special choice of the origin 0 of co-ordinates; viz.

In astronomical and other investigations relating to central forces it is often convenient to use polar co-ordinates with the centre of force as pole.

Proceeding to the general motion of a rigid body in two dimensions we may take as the three co-ordinates of the body the rectangular Cartesian co-ordinates x, y of the mass-centre G and the angle C through which the body has turned from some standard position.

Again, if x, y, z be the co-ordinates of P, the component velocities of m are qzry, rxpz, pyqx, (6)

The equation of the latter, referred to its principal axes, being as in II (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or X, u, v.

The component velocities of any point whose co-ordinates relative to 0 are x, y, z are then u+qzry, v+rxpz, w+Pyqx (12)

Let a be the radius of the rolling sphere, c that of the spherical surface which is the locus of its centre, and let x, y, I be the co-ordinates of this centre relative to axes through 0, the centre of the fixed sphere.

These co-ordinates may be chosen in an endless variety of ways, but their number is determinate, and expresses the number of degrees of freedom of the system.

We denote these co-ordinates by qi,qi,..

It is implied in the above description of the system that the Cartesian co-ordinates x, y, z of any particle of the system are known functions of the qs, varying in form (of course) from particle to particle.

As a first application of Lagranges formula (II) we may form the equations of motion of a particle in spherical polar co-ordinates.

To apply the equations (11) to the case of the top we start with the expression (15) of 22 for the kinetic energy, the simplified form (i) of 20 being for the present purpose inadmissible, since it is essential that the generalized co-ordinates employed should be competent to specify the position of every particle.

A necessary and sufficient condition of equilibrium is therefore that the value of the potential energy should be stationary for infinitesimal variations of the co-ordinates.

In discussing the small oscillations of a system about a configuration of stable equilibrium it is convenient so to choose the generalized co-ordinates qi, q2,..

By a suitable choice of the generalized co-ordinates it is possible to reduce T and V simultaneously to sums of squares.

These variables represent the whole assemblage of generalized co-ordinates qr; they are continuous functions of the independent variables x, y, 1 whose range of variation corresponds to that of the index r, and of 1.

Parallel Projections of Figures.If any figure be referred to a system of co-ordinates, rectangular or oblique, and if a second figure be constructed by means of a second system of co-ordinates, rectangular or oblique, and either agreeing with or differing from the first system in rectangularity or obliquity, but so related to the co-ordin.

For example, all ellipsoids referred to co-ordinates parallel to any three conjugate diameters are parallel projections of each other and of a sphere referred to rectangular co-ordinates.

Each of the four co-ordinates, n', x', y are functions of %, rj, x, y; and if it be assumed that the field of view and the aperture be infinitely small, then, n, x, y are of the same order of infinitesimals; consequently by expanding ', ii', x', y in ascending powers of E, rt, x, y, series are obtained in which it is only necessary to consider the lowest powers.

It may be assumed that the planes I' and II' are drawn where the images of the planes I and II are formed by rays near the axis by the ordinary Gaussian rules; and by an extension of these rules, not, however, corresponding to reality, the Gauss image point 0', with co-ordinates 'o, of the point 0 at some distance from the axis could be constructed.

Let the surface of separation be originally in the plane of the orifice, and let the co-ordinates x and y be measured from one corner parallel to the sides a and b respectively, and let z be measured upwards.

In this work he introduced the use of linear functions in place of the ordinary co-ordinates; he also made the fullest use of the principles of collineation and reciprocity.

Instead of confining himself, as before, to the fruitless integration of three differential equations of the second degree, which are furnished by mathematical principles, he reduced them to the three co-ordinates which determine the place of the moon; and he divided into classes all the inequalities of that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit.

We find on the left-hand scale of yield that the height of the ordinate drawn to the 50-inch mean rainfall curve from 200,000 on the capacity scale, is 1457 gallons per day per acre; and the straight radial line, which cuts the point of intersection of the curved line and the co-ordinates, tells us that this reservoir will equalize the flow of the two driest consecutive years.

Similarly, if we wish to equalize the flow of the three driest consecutive years we change the co-ordinates to the radial line figured 3, and thus find that the available capacity of the reservoir must be 276,000 gallons per acre, and that in consideration of the additional expense of such a reservoir we shall increase the daily yield to 1612 gallons per acre.

The idea was to represent any curve whatever by means of a relation between the co-ordinates (x, y) of a point of the curve, or say to represent the curve by means of its equation.

It is to be noticed here that the axes of co-ordinates may be any two lines at right angles to each other whatever; and that the equation of a curve will be different according to the selection of the axes of co-ordinates; but the order is independent of the axes, and has a determinate value for any given curve.

Trilinear and Tangential Co-ordinates.---The Geometrie descriptive, by Gaspard Monge, was written in the year 1794 or 1 795 (7th edition, Paris, 1847), and in it we have stated, in piano with regard to the circle, and in three dimensions with regard to a surface of the second order, the fundamental theorem of reciprocal polars, viz.

Plucker, moreover, imagined a system of line-co-ordinates (tangential co-ordinates).

It is possible, and (not so much for any application thereof as in order to more fully establish the analogy between the two kinds of co-ordinates) important, to give independent quantitative definitions of the two kinds of co-ordinates; but we may also derive the notion of line-co-ordinates from that of point-coordinates; viz.

Similarly, if we have a curve U= o derived from the curve u = o in a manner independent of the particular axes of co-ordinates, then from the transformed equation u' = o deriving in like manner the curve U' = o, the two equations U= o, U' = o must each of them imply the other; and when this is so, U will be a covariant of u.

In particular if D =o, that is, if the given curve be unicursal, the transformed curve is a line, 4 is a mere linear function of 0, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of 0; it is easy to see that for a given curve of the order m, these functions of 0 must be of the same order m.

And so if D =2, then the transformed curve is a nodal quartic; 4 can be expressed as the square root of a sextic function of 0 and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of 0, and of the square root of a sextic function of 0.

It is a form of the theorem for the case D = r, that the coordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be r -0 2 .r - k 2 0 2, and writing 8 =snu, the radical becomes = cnu, dnu; and we have expressions of the form in question.

The analytical theory by Cartesian co-ordinates was first considered by Alexis Claude Clairaut, Recherches sur les courbes et double courbure (Paris, 1731).

A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a conic became the intersections of the tangents from the circular points at infinity; the centre of a conic the pole of the line at infinity, &c. In analytical geometry the line at infinity plays an important part in trilinear co-ordinates.

The numerical quantities by which the distance and direction, and therefore the position, are defined, are termed co-ordinates of the point.

Co-ordinates referred to.a point of observation as the origin are termed " apparent," those referred to the centre of the earth are " geocentric," those referred to the centre of the sun, " heliocentric."

A system of equatorial co-ordinates may also be used when the origin is on the earth's surface.

This angle is called the Longitude, Right Ascension or Azimuth of the body, in the various systems of co-ordinates.

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.

The conception of the co-ordinates we have defined is facilitated by introducing that of the celestial sphere.

The relation of geocentric to apparent co-ordinates depends upon the latitude of the observer.

The changes which the aspect of the heaven undergoes, as we travel North and South, are so well known that they need not be described in detail here; but a general statement of them will give a luminous idea of the geometrical co-ordinates we have described.

We have next to point out the relation of the co-ordinates we have described to the annual motion of the earth around the sun.

The vernal equinox is taken as the initial point on the sphere from which co-ordinates are measured in the equatorial and ecliptic systems. Referring to fig.

The vertical line at any place being the fundamental axis of the apparent system of co-ordinates, this system rotates with the earth, and so seems to us as fixed.

Differential equations which express the changes of the co-ordinates are then constructed.

In this case the two bodies really revolve round their common centre of gravity; but a very slight modification of the equations of motion reduces them to the relative motion of the planet round the sun, regarding the moving centre of the latter as the origin of co-ordinates.

In the actual problems of celestial mechanics three co-ordinates necessarily enter, leading to three differential equations and six equations of solution.

By taking this plane, which is that of the orbit in which the planet performs its revolution, as the plane of xy, we have only two co-ordinates to consider.

This is expressed in three equations, one for each of the three rectangular co-ordinates.

The result of the integration is that the co-ordinates x and y and their derivatives as to the time, which express the position, direction of motion and speed of the planet at any moment, are found as functions of the four constants and of the time.

This may be done because, since the elements and co-ordinates completely determine each other, we may concentrate our attention on either, ignoring the other.

The results which are required to compare with observations are not merely the elements, but the co-ordinates.

The position and velocity being given in all three co-ordinates, a certain osculating plane is determined for each instant in which the planet is moving at that instant.

The longitudes, latitudes and radii vectores of a planet, being algebraically expressed as the sum of an infinite periodic series of the kind we have been describing, it follows that the problem of finding their co-ordinates at any moment is solved by computing these expressions.

This is facilitated by the construction of tables by means of which the co-ordinates can be computed at any time.

The fundamental problem of practical astronomy is that of determining by measurement the co-ordinates of the heavenly bodies as already defined.

Of the three co-ordinates,the radius vector does not admit of direct measurement, and must be inferred by a combination of indirect measurements and physical theories.

In the first system of co-ordinates already described the fundamental axis is the vertical line or direction of gravity at the point of observation.

In the measurement of equatorial co-ordinates, the polar distance is determined in an analogous way.

To measure the difference between the longitudinal co-ordinates of two objects by means of a graduated circle the instruments must turn on an axis parallel to the principal axis of the system of coordinates, and the plane of the graduated circle must be at right angles to that axis, and, therefore, parallel to the principal co-ordinate plane.

The motion of the graduated circle in passing from one pointing to the other is the measure of the difference between the longitudinal co-ordinates of the two objects.

By the second general method the moon's co-ordinates are obtained in terms of the time by the direct integration of the differential equations of motion, retaining as algebraic symbols the values of the various elements.

We have just mentioned the four small quantities e, e', y and m, in terms of the powers and products of which the moon's co-ordinates have to be expressed.

Metrical relations between the axes, eccentricity, distance between the foci, and between these quantities and the co-ordinates of points on the curve (referred to the axes and the centre), and focal distances are readily obtained by the methods of geometrical conics or analytically.

He diverged from Ptolemy when he placed the asterisms Coma Berenices and Antinous upon the level of formal constellations, Ptolemy having 1 The historical development of star-catalogues in general, regarded as statistics of the co-ordinates, &c., of stars, is given in the historical section of the article 'ASTRONOMY.

This ambiguity can be resolved by transforming the measured color co-ordinates to compensate for the thickness of the papillary dermis.

Eastings and Northings - This searches the Image Library by spatial location, using eastings and Northings - This searches the Image Library by spatial location, using easting and northing co-ordinates to define an area for searching.

Ecliptic and equatorial co-ordinates are referred to the mean equinox of a given epoch.

The computer simply works out whose co-ordinates are closest and, hey presto, some jammy git wins the car.

The County Council co-ordinates planned work on the highway through regular liaison with all parties involved.

When the user activates the mouse in this region the co-ordinates are downloaded wrapped as a chemical/x-pdb mime type.

Eastings and northings - This searches the Image Library by spatial location, using easting and northing co-ordinates to define an area for searching.

If (u, v, w, t) be the co-ordinates of any point, then the relation u+v-Fw-fit=R, where R is a constant, invariably holds.

Besides many papers communicated to the academy of sciences, of which he became a member in 1714, he published Memoires pour servir d l'histoire et au progres de l'astronomie (St Petersburg, 1738), in which he gave the first method for determining the heliocentric co-ordinates of sun-spots; Memoire sur les nouvelles decouvertes au nord de la mer du sud (Paris, 1752), &c.

The equation to a parabola in triangular co-ordinates is generally derived by expressing the condition that the line at infinity is a tangent in the equation to the general conic. For example, in trilinear co-ordinates, the equation to the general conic circumscribing the triangle of reference is 113y+mya+naf3=o; for this to be a parabola the line as + b/ + cy = o must be a tangent.

A plane frame of n joints which is just rigid (as regards deformation in its own plane) has 2n3 bars, for if one bar be held fixed the 2(n2) co-ordinates of the remaining fl2 joints must just be determined by the lengths of the remaining bars.

This is now covered by co-ordinates for where she adopted her children Maddox, Pax and Zahara as well as the birth places of Shiloh, Vivienne, Knox and Brad Pitt.