PDS_VERSION_ID = PDS3 OBJECT = DATA_SET_MAP_PROJECTION DATA_SET_ID = "MGN-V-RDRS-5-GVDR-V1.0" OBJECT = DATA_SET_MAP_PROJECTION_INFO MAP_PROJECTION_TYPE = SINUSOIDAL MAP_PROJECTION_DESC = " GVDR MAP PROJECTIONS ==================== GVDR data are present in three map projections: sinusoidal, Mercator, and polar stereographic, using the Venus body-fixed coordinate system approved by the IAU in 1985 [DAVIESETAL1989]. The polar stereographic projection is provided in two aspects, north polar and south polar. Although the GVDR data files are presented in tabular form, the underlying organization is an image composed of rectangular pixels. These pixels are centered on the integral values of a two-dimensional cartesian coordinate system. The relationship between cartesian coordinates (x,y) and geographic coordinates (latitude,longitude) is mathematically defined by the particular map projection. In all GVDR images, the x-coordinate increases from left to right and the y-coordinate increases from bottom to top. Longitude on Venus conventionally increases toward the east, and '330 E' is generally preferred to '-30 E' or '30 W' [LYONS1988]. In summary, discussions of the GVDR images refer to three coordinate systems: geographic (longitude and latitude), cartesian (map projection coordinates x and y) and image-based (pixel addresses line and sample). The geographic and cartesian coordinates are related by the map projection equations given below, which in general can be nonlinear and non-separable. The cartesian and image coordinates have the same scale and are related by a simple translation; the image is a rectangular window over some region of cartesian coordinate space. SINUSOIDAL MAP PROJECTION ========================= In the sinusoidal projection, parallels of latitude are straight lines, with constant distances between equal latitude intervals. Lines of constant longitude on either side of the projection meridian are curved since longitude intervals decrease with the cosine of latitude to account for their convergence toward the poles. The projection is pseudocylindrical, so named due to its similar appearance to true cylindrical projections. Areas on the map are proportional to the same areas on the planet. Distances are correct along all parallels and the central meridian, but shapes are increasingly distorted away from the central meridian and near the poles [ALPHA&SNYDER1982; SNYDER1987]. The sinusoidal equal-area projection is characterized by a projection longitude and a scale. The projection longitude defines the center meridian (X=0) of the projection; the planet's equator is coincident with the line Y=0. The scale is given in units of pixels/degree along the projection equator, which can be converted to km/pixel using the planetary radius. The GVDR projection longitude is the prime meridian, and the scale is 18.225 km/pixel at the equator. The object IMAGE_MAP_PROJECTION in the IMPSINU.LBL file provides the specific projection parameters for the Sinusoidal projection used with this data set. The most important parameters are A_AXIS_RADIUS Planetary radius CENTER_LATITUDE Center latitude (always 0 N) CENTER_LONGITUDE Center longitude (always 0 E) MAP_SCALE Pixel scale (18.225 km/pixel) at equator LINE_PROJECTION_OFFSET Image pixel address of Y=0 SAMPLE_PROJECTION_OFFSET Image pixel address of X=0 (See the PDS Data Dictionary [PDSDD1992] for a more detailed description of these and other parameters.) The transformation from latitude and longitude (LAT,LON) in radians to Sinusoidal cartesian coordinates (X,Y) is given by the following equations. SCALE = A_AXIS_RADIUS / MAP_SCALE X = SCALE * (LON - CENTER_LONGITUDE) * COS(LAT) = SCALE * LON * COS(LAT) Y = SCALE * LAT; The inverse relations are LAT = Y / SCALE LON = X / (SCALE * COS(LAT)) + CENTER_LONGITUDE = X / (SCALE * COS(LAT)) The transformation between cartesian coordinates (X,Y) and image pixel coordinates (SAMPLE,LINE) is LINE = 1 + LINE_PROJECTION_OFFSET - Y SAMPLE = 1 + SAMPLE_PROJECTION_OFFSET + X In the above definitions, integral values of LINE and SAMPLE correspond to the center of a GVDR pixel, and the top left image pixel has LINE=1 and SAMPLE=1. Note that while SAMPLE increases from left to right (the same as X), LINE increases from top to bottom (the reverse of Y)." ROTATIONAL_ELEMENT_DESC = "See [DAVIESETAL1989]." OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "DAVIESETAL1989" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "SNYDER1987" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "LYONS1988" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "PDSDD1992" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "ALPHA&SNYDER1982" END_OBJECT = DS_MAP_PROJECTION_REF_INFO END_OBJECT = DATA_SET_MAP_PROJECTION_INFO OBJECT = DATA_SET_MAP_PROJECTION_INFO MAP_PROJECTION_TYPE = STEREOGRAPHIC MAP_PROJECTION_DESC = " GVDR MAP PROJECTIONS ==================== GVDR data are present in three map projections: sinusoidal, Mercator, and polar stereographic, using the Venus body-fixed coordinate system approved by the IAU in 1985 [DAVIESETAL1989]. The polar stereographic projection is provided in two aspects, north polar and south polar. Although the GVDR data files are presented in tabular form, the underlying organization is an image composed of rectangular pixels. These pixels are centered on the integral values of a two-dimensional cartesian coordinate system. The relationship between cartesian coordinates (x,y) and geographic coordinates (latitude,longitude) is mathematically defined by the particular map projection. In all GVDR images, the x-coordinate increases from left to right and the y-coordinate increases from bottom to top. Longitude on Venus conventionally increases toward the east, and '330 E' is generally preferred to '-30 E' or '30 W' [LYONS1988]. Discussions of the GVDR images refer to three coordinate systems: geographic (longitude and latitude), cartesian (map projection coordinates x and y) and image-based (pixel addresses line and sample). The geographic and cartesian coordinates are related by the map projection equations given below, which in general can be nonlinear and non-separable. The cartesian and image coordinates have the same scale and are related by a simple translation; the image is a rectangular window over some region of cartesian coordinate space. POLAR STEREOGRAPHIC MAP PROJECTIONS =================================== In the polar stereographic projection, parallels of latitude are circles centered on the pole, and meridians of longitude are straight lines radiating from the pole. The projection is azimuthal, formed by projecting onto a tangent plane making contact at the center of the projection; the point of projection is antipodal to the point of tangency. Directions are true only from the center point of projection. Scale increases away from the center point. Any straight line through the center point is a great circle. The distortion of areas and large shapes increases away from the center point. The projection is conformal (meaning that small shapes are preserved) and perspective, but not equal area or equidistant [ALPHA&SNYDER1982; SNYDER1987]. GVDR images in this projection are centered on either the North or South pole of Venus, with the central longitude (specified by the value of CENTER_LONGITUDE in the IMAGE_MAP_PROJECTION label) running vertically from the pole to the bottom of the image for the North Polar projection, and to the top of the image for the South Polar projection. South polar GVDR images extend to 60S latitude and North polar GVDR images extend to 60N. The IMAGE_MAP_PROJECTION objects in the IMPNORTH.LBL and IMPSOUTH.LBL files specify the specific projection parameters for the stereographic projections used with this data set. The most important parameters are A_AXIS_RADIUS Planetary radius CENTER_LATITUDE Center latitude (either 90N or 90S) CENTER_LONGITUDE Center longitude (always 0E) MAP_SCALE Pixel scale (18.225 km/pixel) at the pole LINE_PROJECTION_OFFSET Image pixel address of Y=0 SAMPLE_PROJECTION_OFFSET Image pixel address of X=0 (See the PDS Data Dictionary [PDSDD1992] for a more detailed description of these and other parameters.) The transformation from latitude and longitude (LAT,LON) in radians to North Polar Stereographic cartesian coordinates (X,Y) is given by the following equations. SCALE = A_AXIS_RADIUS / MAP_SCALE T = TAN(PI/4 - LAT/2) X = 2 * SCALE * T * SIN(LON - CENTER_LONGITUDE) = 2 * SCALE * T * SIN(LON) Y = -2 * SCALE * T * COS(LON - CENTER_LONGITUDE) = -2 * SCALE * T * COS(LON) and for South Polar Stereographic by X = 2 * SCALE * T * SIN(LON - CENTER_LONGITUDE) = 2 * SCALE * T * SIN(LON) Y = 2 * SCALE * T * COS(LON - CENTER_LONGITUDE) = 2 * SCALE * T * COS(LON) The inverse relations for the northern aspect are RHO = SQRT(X*X + Y*Y) C = 2 * ARCTAN2(RHO,2*SCALE) LAT = ARCSIN(COS(C)) = PI/2 - C LON = ARCTAN2(X,-Y) + CENTER_LONGITUDE = ARCTAN2(X,-Y) where ARCTAN2(a,b) is the four-quadrant arctangent function similar to ARCTAN(a/b). For the southern aspect, the values of RHO and C are the same and LAT = ARCSIN(-COS(C)) = C - PI/2 LON = ARCTAN2(X,Y) + CENTER_LONGITUDE = ARCTAN2(X,Y) The transformation between cartesian coordinates (X,Y) and image pixel coordinates (SAMPLE,LINE) is LINE = 1 + LINE_PROJECTION_OFFSET - Y SAMPLE = 1 + SAMPLE_PROJECTION_OFFSET + X In the above definitions, integral values of LINE and SAMPLE correspond to the center of a GVDR pixel, and the top left image pixel has LINE=1 and SAMPLE=1. Note that while SAMPLE increases from left to right (the same as X), LINE increases from top to bottom (the reverse of Y)." ROTATIONAL_ELEMENT_DESC = "See [DAVIESETAL1989]." OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "DAVIESETAL1989" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "SNYDER1987" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "LYONS1988" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "PDSDD1992" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "ALPHA&SNYDER1982" END_OBJECT = DS_MAP_PROJECTION_REF_INFO END_OBJECT = DATA_SET_MAP_PROJECTION_INFO OBJECT = DATA_SET_MAP_PROJECTION_INFO MAP_PROJECTION_TYPE = MERCATOR MAP_PROJECTION_DESC = " GVDR MAP PROJECTIONS ==================== GVDR data are present in three map projections: sinusoidal, Mercator, and polar stereographic, using the Venus body-fixed coordinate system approved by the IAU in 1985 [DAVIESETAL1989]. The polar stereographic projection is provided in two aspects, north polar and south polar. Although the GVDR data files are presented in tabular form, the underlying organization is an image composed of rectangular pixels. These pixels are centered on the integral values of a two-dimensional cartesian coordinate system. The relationship between cartesian coordinates (x,y) and geographic coordinates (latitude,longitude) is mathematically defined by the particular map projection. In all GVDR images, the x-coordinate increases from left to right and the y-coordinate increases from bottom to top. Longitude on Venus conventionally increases toward the east, and '330 E' is generally preferred to '-30 E' or '30 W' [LYONS1988]. Discussions of the GVDR images refer to three coordinate systems: geographic (longitude and latitude), cartesian (map projection coordinates x and y) and image-based (pixel addresses line and sample). The geographic and cartesian coordinates are related by the map projection equations given below, which in general can be nonlinear and non-separable. The cartesian and image coordinates have the same scale and are related by a simple translation; the image is a rectangular window over some region of cartesian coordinate space. MERCATOR MAP PROJECTION ======================= In the Mercator projection, parallels of latitude and meridians of longitude are straight lines that intersect one another at right angles. The projection is cylindrical, formed by projecting onto a cylinder tangent to the equator. The projection's most famous feature is that any straight line on the map is a rhumb line (a line of constant direction) and that distances along a rhumb line are true between any two points. While this is useful for seafaring navigators, it provides no advantage to those studying the planet Venus, and the map projection has been chosen simply due to its familiarity. Distances are true only along the equator, but a special scale can be used to measure distances along other parallels. Although the map is conformal (small shapes are preserved), the shapes and surface areas of large regions are distorted. Distortion increases away from the equator and is extreme in polar regions. The projection is not perspective, equal area or equidistant [ALPHA&SNYDER1982; SNYDER1987]. Images in this projection are centered on the equator. GVDR images extend from 0 E to 360 E in longitude, and from 65 N to 65 S. The parameters required to specify the projection are a scale along the equator and a longitude origin. The scale is given by MAP_SCALE in the IMAGE_MAP_PROJECTION object in the IMPMERC.LBL file, the longitude origin by CENTER_LONGITUDE. The important parameters in this object are: A_AXIS_RADIUS Planetary radius CENTER_LONGITUDE Origin of projection (always 0E) MAP_SCALE Pixel scale (18.225 km/pixel) on equator LINE_PROJECTION_OFFSET Image pixel address of Y=0 SAMPLE_PROJECTION_OFFSET Image pixel address of X=0 (See the PDS Data Dictionary [PDSDD1992] for a more detailed description of these and other parameters.) The transformation from latitude and longitude (LAT,LON) in radians to Mercator cartesian coordinates (X,Y) is given by the following equations. SCALE = A_AXIS_RADIUS / MAP_SCALE X = SCALE * (LON - CENTER_LONGITUDE) = SCALE * LON Y = SCALE * LOG(TAN(PI/4 + LAT/2)) = SCALE * ARCTANH(SIN(LAT)) where LOG is the natural logarithm function and ARCTANH is the inverse hyperbolic tangent function. The inverse relations are LAT = PI/2 - 2 * ARCTAN(EXP(-Y/SCALE)) LON = X/SCALE + CENTER_LONGITUDE = X/SCALE where EXP is the natural exponential, E=2.71828... raised to the power of its argument. The transformation between cartesian coordinates (X,Y) and image pixel coordinates (SAMPLE,LINE) is LINE = 1 + LINE_PROJECTION_OFFSET - Y SAMPLE = 1 + SAMPLE_PROJECTION_OFFSET + X In the above definitions, integral values of LINE and SAMPLE correspond to the center of a GVDR pixel, and the top left image pixel has LINE=1 and SAMPLE=1. Note that while SAMPLE increases from left to right (the same as X), LINE increases from top to bottom (the reverse of Y)." ROTATIONAL_ELEMENT_DESC = "See [DAVIESETAL1989]." OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "DAVIESETAL1989" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "SNYDER1987" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "LYONS1988" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "PDSDD1992" END_OBJECT = DS_MAP_PROJECTION_REF_INFO OBJECT = DS_MAP_PROJECTION_REF_INFO REFERENCE_KEY_ID = "ALPHA&SNYDER1982" END_OBJECT = DS_MAP_PROJECTION_REF_INFO END_OBJECT = DATA_SET_MAP_PROJECTION_INFO END_OBJECT = DATA_SET_MAP_PROJECTION END