X and Y are the distances measured with reference to two axes. X is the distance from the Y-Axis, measured parallel to the X-axis. The Y coordinate is similarly defined as the distance from the X-Axis, measured parallel to the Y-Axis. The axes are usually perpendicular with the X-Axis (the abscissa) horizontal and the Y-Axis (the ordinate) vertical, but oblique axis systems are possible. As conventionally defined, the value of X increases to the right and the value of Y increases upward. In plotting functions, X is generally but by no means universally treated as the independent variable and Y as the dependent variable. The point where the axes cross represents the coordinates (0,0) and is called the origin.
Polar coordinates are defined in terms of the distance (R) from the point being represented to an origin, and the azimuth (A) of the line from the origin to the point, measured from some reference direction. Conventionally, the reference line extends horizontally to the right of the origin and A is measured in a counterclockwise sense. Functions are usually represented in terms of A being the independent variable and R the dependent variable.
Compass directions are much like polar coordinates, but the reference direction usually is oriented vertically upward on the map and azimuth is measured clockwise. Thus compass azimuth C is related to polar coordinate azimuth A by:
lane polar (R,A) to plane Cartesian (X,Y) coordinates X = R * COS(A), Y = R * SIN(A)
Plane Cartesian (X,Y) to plane polar (R,A) coordinates R = SQR(X*X + Y*Y) A = ARCCOS (X/R) * SGN(Y) A = 2 * ATN(SQR((R-X)/(R+X))) * SGN (Y) The above formulas for A yield values in the range -180 to +180. To obtain values in the range 0-360, add 360 to negative values of A. The formula below gives values in the range 0-360 but cannot be used if X or Y are exactly zero. A = ATN(Y/X) + 45*(2 - SGN(Y) - SGN(X)*SGN(Y)) When X or Y are exactly zero use the following formula. A = 90*(1 - SGN(X) - SGN(Y) - SGN(X)*SGN(Y))
Coordinates are measured with reference to three intersecting axes, in exactly the same way as coordinates in the plane. For general purposes the axes are perpendicular but oblique axis systems are occasionally used.
Coordinates in two dimensions (R,A) are identical to plane polar coordinates. The third coordinate (Z) is simply the vertical distance from the origin plane.
The coordinates of a point are measured with reference to a line from the point to the origin. R is simply the length of the line, A is the azimuth of the vertical plane that contains the line and P is the angle the line makes with the vertical axis.
Geographic coordinates are simply a variation on spherical coordinates. The coordinate R is a constant, longitude W corresponds to A, and latitude L is measured with reference to the horizontal plane rather than the polar axis. Thus L = 90 - P. If A and W are to have the same directional sense, and the north pole is regarded as up, west longitude must be negative and east positive.
The position of a point may be described in terms of its distance from the origin (R) and the cosines of the angles which the line between the point and the origin makes with the coordinate axes (L,M,N). L, M, and N are related by L*L + M*M + N*N = 1. The Cartesian coordinates of the point are simply (R*L,R*M,R*N).
CONVERSION TO CARTESIAN THREE-DIMENSIONAL COORDINATES From Cylindrical Coordinates (R0,A0,Z0) X = R0*COS(A0), Y = R0*SIN(A0), Z = Z0 From Spherical Coordinates (R0,A0,P0) X = R0*SIN(P0)*COS(A0) Y = R0*SIN(P0)*SIN(A0) Z = R0*COS(P0) From Latitude and Longitude (R0,L0,W0) X = R0*COS(L0)*COS(W0) Y = R0*COS(L0)*SIN(W0) Z = R0*SIN(L0) From Direction Cosines (R0,L0,M0,N0) X = R0*L0, Y = R0*M0, Z = R0*N0Conversion to Cylindrical Three-dimensional Coordinates From Cartesian Coordinates (X0,Y0,Z0) R = SQR(X0*X0 + Y0*Y0) A = 2 * ATN(SQR((R-X)/(R+X))) * SGN (Y) Refer to conversion from Cartesian to polar plane coordinates for other formulas. Z = Z0 From Spherical Coordinates (R0,A0,P0) R = R0 * SIN(P0) A = A0 Z = R0 * COS(P0) From Latitude and Longitude (R0,L0,W0) R = R0 * COS(L0) A = -W0 Z = R0 * SIN(L0) From Direction Cosines (R0,L0,M0,N0) R = R0 * SQR(L0*L0 + M0*M0) A = 2 * ATN(SQR((1-L0)/(1+L0))) * SGN (M0) Z = R0 * N0
5-1f CONVERSION TO SPHERICAL THREE-DIMENSIONAL COORDINATES (R,A,P) From Cartesian Coordinates (X0,Y0,Z0) R = SQR(X0*X0 + Y0*Y0 + Z0*Z0) A = 2 * ATN(SQR((R-X0)/(R+X0))) * SGN (Y0) P = ATN(SQR(X0*X0+Y0*Y0)/Z0) From Cylindrical Coordinates (R0,A0,Z0) R = SQR(R0*R0 + Z0*Z0) A = A0 P = ATN(R0/Z0) From Latitude and Longitude (R0,L0,W0) R = R0 A = -W0 P = 1.5707963 - L0 (in radians) From Direction Cosines (R0,L0,M0,N0) R = R0 A = 2 * ATN(SQR((1-L0)/(1+L0))) * SGN (M0) P = ATN(SQR(L0*L0+M0*M0)/n0)
5-1g CONVERSION TO LATITUDE AND LONGITUDE COORDINATES (R, L, W) From Cartesian Coordinates (X0,Y0,Z0) R = SQR(X0*X0 + Y0*Y0 + Z0*Z0) L = ATN(Z0/SQR(X0*X0+Y0*Y0)) W = 2 * ATN(SQR((R-X0)/(R+X0))) * SGN (-Y0) From Cylindrical Coordinates (R0,A0,Z0) R = SQR(R0*R0 + Z0*Z0) L = ATN(Z0/R0) W = -A0 From Spherical Coordinates (R0,A0,P0) R = R0 L = 1.5707963 - P0 (in radians) W = -A0 From Direction Cosines (R0,L0,M0,N0) R = R0 L = ATN(N0/SQR(L0*L0+M0*M0)) W = 2 * ATN(SQR((1-L0)/(1+L0))) * SGN(-M0)
5-1h CONVERSION TO DIRECTION NUMBERS OR COSINES (R,L,M,N) From Cartesian Coordinates (X0,Y0,Z0) R = SQR(X0*X0 + Y0*Y0 + Z0*Z0) L = X0/R M = Y0/R N = Z0/R From Cylindrical Coordinates (R0,A0,Z0) R = SQR(R0*R0 + Z0*Z0) L = R0*COS(A0)/R M = R0*SIN(A0)/R N = Z0/R From Spherical Coordinates (R0,A0,P0) R = R0 L = COS(A0)*SIN(P0) M = SIN(A0)*SIN(P0) N = COS(P0) From Latitude and Longitude (R0,L0,W0) R = R0 L = COS(L0)*COS(W0) M = -SIN(L0)*COS(W0) N = SIN(L0)
5-2 ROTATION AND TRANSLATION OF COORDINATES
5-2A TRANSLATION IN THE PLANE
5-2B ROTATION IN THE PLANE
5-2C TRANSLATION IN THREE DIMENSIONS
5-2D ROTATION IN THREE DIMENSIONS
5-2E COMBINED ROTATION AND TRANSLATION
5-3 PLANE PROJECTIONS
The projections we will consider here all involve relativelysimple projection of three-dimensional data directly onto a flatplane, as opposed to a curved surface like a cylinder or conewhich is later unrolled into a plane. In all the definitions ofcoordinates we will assume we are looking down the Z-axis towardthe negative direction and that X- and Y- axes are oriented inthe conventional sense of plane Catresian coordinates. All theprojections in this section are azimuthal; the azimuth of a pointas measured around the projection direction is the same as theazimuth of the projected point in the projection plane.
5-3A ORTHOGRAPHIC Projection point: Infinity Useful or Distinctive Properties: Provides a fairly realistic view of objects. Distances parallel to the projection plane are true. Circles on a sphere project as ellipses. Straight lines project as straight lines. Distortions and Limitations: Exactly half of the sphere can be portrayed. Objects at a high angle to the projection plane are very much foreshortened. Plotting Coordinates: X- and Y- coordinates of points are plotted directly, Z- coordinates ignored.
5-3B PERSPECTIVE Projection point: Observer's location a finite distance from objects being viewed Useful or Distinctive Properties: Inclined circles project as ellipses. Parallel lines project as converging lines. Distortions and Limitations: Less than half of the sphere can be portrayed. Objects at a high angle to the projection plane are very much foreshortened. Scale varies with distance from projection point. Plotting Coordinates:
5-3C GNOMONIC Projection point: Center of a sphere Useful or Distinctive Properties: Great circles on the sphere project as straight lines on the projection plane. Small circles on the sphere project as ellipses, parabolas or hyperbolas. Distortions and Limitations: Less than one hemisphere can be portrayed; in practice, scale considerations usually permit only about a third of the sphere to be mapped. Radial distortion is severe at large angles to the projection axis. Plotting Coordinates:
5-3D STEREOGRAPHIC Projection point: Edge of a sphere Useful or Distinctive Properties: All circles on the sphere plot as circles on the plane. This remarkable property makes this projection one of the most useful in the earth sciences. The stereographic projection is also conformal or angle-preserving. In theory the entire sphere can be plotted except for the projection point; in practice it is rare to plot more than a complete hemisphere. The conformal nature of this projection makes it useful for portraying small areas of the earth and also for star maps. in fact, medieval astrolabes used this projection. Distortions and Limitations: Moderate radial distortion within the near hemisphere, becoming more severe at large distances. Plotting Coordinates:
5-3E AZIMUTHAL EQUIDISTANT Projection point: A point on the circumference of a sphere. This is not a projection in the sense of the preceding cases, where it would actually be possible to simulate the projection by putting a light bulb at the appropriate point on a transparent sphere. Instead, the arc length from the projection point to a point on the sphere becomes the radius of the plotted point on the projection. Useful or Distinctive Properties: Useful for portraying distances from a central point. Distortions and limitations: No radial distance distortion, but severe circumferential distortion at large distances from the projection point. Circles on the sphere have no simple representation on the plane. Plotting Coordinates:
5-3F AZIMUTHAL EQUAL-AREA Projection point: A point on the circumference of a sphere. Like the azimuthal equidistant case, this is not a projection in the optical sense. Instead, the chord length from the projection point to a point on the sphere becomes the radius of the plotted point on the plane. Useful or Distinctive Properties: Area preserving, hence useful for portaying the statistical distribution of spatial data. Equal areas on the sphere plot as equal areas on the plane. Circles on the sphere plot as complex curves in the plane. It is possible to portay the entire sphere; the point opposite the projection pole is distorted into the circumference of a circle. For geological applications it is rare to plot more than a hemisphere, but whole-earth maps employing this projection (usually centered on a pole) are fairly common. Distortions and Limitations: Radial distances are compressed and circumferential distances are stretched in this projection. Circles on the sphere lack simple representations in the plane. Plotting Coordinates:
5-3E AZIMUTHAL EQUIDISTANT Projection point: A point on the circumference of a sphere. This is not a projection in the sense of the preceding cases, where it would actually be possible to simulate the projection by putting a light bulb at the appropriate point on a transparent sphere. Instead, the arc length from the projection point to a point on the sphere becomes the radius of the plotted point on the projection. Useful or Distinctive Properties: Useful for portraying distances from a central point. Distortions and limitations: No radial distance distortion, but severe circumferential distortion at large distances from the projection point. Circles on the sphere have no simple representation on the plane. Plotting Coordinates: R = P (if P is stated in radians)
5-3F AZIMUTHAL EQUAL-AREA Projection point: A point on the circumference of a sphere. Like the azimuthal equidistant case, this is not a projection in the optical sense. Instead, the chord length from the projection point to a point on the sphere becomes the radius of the plotted point on the plane. Useful or Distinctive Properties: Area preserving, hence useful for portaying the statistical distribution of spatial data. Equal areas on the sphere plot as equal areas on the plane. Circles on the sphere plot as complex curves in the plane. It is possible to portay the entire sphere; the point opposite the projection pole is distorted into the circumference of a circle. For geological applications it is rare to plot more than a hemisphere, but whole-earth maps employing this projection (usually centered on a pole) are fairly common. Distortions and Limitations: Radial distances are compressed and circumferential distances are stretched in this projection. Circles on the sphere lack simple representations in the plane. Plotting Coordinates:
5-4 CYLINDRICAL PROJECTIONS The projections we will consider here all are based onprojection of three-dimensional data from a sphere onto acircumscribing cylinder. In all the definitions of coordinates wewill assume that the Z-axis is the axis of the projectioncylinder in space. The coordinates of projected points will berepresented by (X,Y) with Y parallel to the cylinder axis. Inevery case, therefore, we will have X proportional to the azimuthmeasured around the projection axis.
5-3A ORTHOGRAPHIC CYLINDRICAL Projection: Perpendicular to the cylinder axis. Useful or Distinctive Properties: Provides a fairly realistic view of objects. Distances parallel to the projection plane are true. Circles on a sphere project as ellipses. Straight lines project as straight lines. Distortions and Limitations: Exactly half of the sphere can be portrayed. Objects at a high angle to the projection plane are very much foreshortened. Plotting Coordinates: X- and Y- coordinates of points are plotted directly, Z- coordinates ignored.
5-3b MERCATOR Projection point: Center of the sphere Useful or Distinctive Properties: Great circles on the sphere project as straight lines on the projection plane. Small circles on the sphere project as ellipses, parabolas or hyperbolas. Distortions and Limitations: Less than one hemisphere can be portrayed; in practice, scale considerations usually permit only about a third of the sphere to be mapped. Radial distortion is severe at large angles to the projection axis. Plotting Coordinates:
5-3c CYLINDRICAL EQUIDISTANT Projection: Distances measured from the equator along meridians are the same on both the sphere and the projection. Useful or Distinctive Properties: Useful for portraying distances from a central point. Distortions and limitations: No radial distance distortion, but severe circumferential distortion at large distances from the projection point. Circles on the sphere have no simple representation on the plane. Plotting Coordinates:
5-3d CYLINDRICAL EQUAL-AREA Projection point: A point on the circumference of a sphere. Like the azimuthal equidistant case, this is not a projection in the optical sense. Instead, the chord length from the projection point to a point on the sphere becomes the radius of the plotted point on the plane. Useful or Distinctive Properties: Area preserving, hence useful for portaying the statistical distribution of spatial data. Equal areas on the sphere plot as equal areas on the plane. Circles on the sphere plot as complex curves in the plane. It is possible to portay the entire sphere; the point opposite the projection pole is distorted into the circumference of a circle. For geological applications it is rare to plot more than a hemisphere, but whole-earth maps employing this projection (usually centered on a pole) are fairly common. Distortions and Limitations: Radial distances are compressed and circumferential distances are stretched in this projection. Circles on the sphere lack simple representations in the plane. Plotting Coordinates:
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Created 18 September 1998, Last Update 18 September 1998