SRM Tutorial, STC 2000, Snowbird, Utah Section IIB A Few More Facts and Diagrams

SRM Tutorial, STC 2000, Snowbird, Utah P(W,Z) h Z W ø ZeZe WeWe PePe P(X, Y, Z) or P ( Ø,  h) ø X Y Z PePe Where W 2 = X 2 + Y 2 Generic ERM & notationMeridian plane geometry Ellipsoids are standard in current geodesy practice. For SNE data modeling, spheres are often used to simplify dynamics equations. Geocentric coordinates (GCC) are defined by the point P(X, Y, Z). Geodetic coordinates (GDC) are defined by the point P(Ø,, h). Ellipsoidal Earth Reference Model (ERM) Geometry & Notation

SRM Tutorial, STC 2000, Snowbird, Utah For ellipsoids Latitude, longitude and geodetic height are defined as per this diagram. The line through P is perpendicular to the ellipsoid. Longitude is generally referenced to the Prime Meridian. For spheres Longitude is the same as for the ellipsoidal case,  is the geocentric latitude, and hos is height above the sphere. The line through P is per- pendicular to the sphere. In mapping, charting and geodesy, spherical ERMs are almost never used. P(X, Y, Z) or P( Ø,  h) ø X Y Z PePe h Y Z PsPs X P(X, Y, Z) or P(,  h) hos Latitude, Longitude and Height for Ellipsoids & Spheres

SRM Tutorial, STC 2000, Snowbird, Utah Ellipsoid Earth's Physical Surface Geoid h H The geoid is a gravity equipotential surface selected to match mean sea level as well as possible. For more on this see NIMA’s “Geodesy for the Layman” on Geoid Separation: + N Geoid Separation: - N h is the geodetic height H is the orthometric height N is the separation of the geoid Ellipsoid Geoid Cross-Section of the Geoid, Ellipsoid and Earth Surface

SRM Tutorial, STC 2000, Snowbird, Utah Ellipsoid Earth's Physical Surface Geoid Separation: + N Geoid Gravity potential results in a gravity field Gravity vector depends on: latitude, longitude, and H (or h) P H h Geoid Separation: - N h is the geodetic height H is the orthometric height N is the separation of the geoid Geoid Gravitational Field and the Geoid, Ellipsoid & Earth Surface

SRM Tutorial, STC 2000, Snowbird, Utah Universal Transverse Mercator (60) Lambert Conformal Conic Polar Stereographic GCS Geocentric (ECEF) Earth Referenced, Projection-based 2D and 3D SRFs Earth Referenced 3D (and 2D) SRFs Geodetic (3D and 2D) Geomagnetic Geocentric Equatorial Inertial Geocentric Solar Ecliptic Geocentric Solar Magnetospheric Solar Magnetic Transverse Mercator Oblique Mercator Local Tangent Plane (3D and 2D) SRM Refresher Local Space Rectangular 3D Equidistant Cylindrical Mercator Local Space Rectangular 2D Universal Polar Stereographic (2) With and without Augmentation

SRM Tutorial, STC 2000, Snowbird, Utah Map Projections Map projections were invented to support paper map development — a long time ago Section III

SRM Tutorial, STC 2000, Snowbird, Utah A cone or cylinder can be cut and laid out flat. Non-developable Surfaces Developable Surfaces The surface of an ellipsoid cannot be cut so it will lie flat without tearing or stretching. Development of Surfaces to Generate Maps

SRM Tutorial, STC 2000, Snowbird, Utah Since spheres and ellipsoids are not developable, distortions must occur. Note that the transformation is from three to two dimensions, and that there is no vertical axis in the plane. X Y Map Projections Associate points on the surface of an ERM with points on an X-Y plane – or more formally... A map projection is a mathematical transformation from a three dimensional ellipsoidal or spherical ERM surface onto a two dimensional plane.

SRM Tutorial, STC 2000, Snowbird, Utah Projection from the point N of all points on the circle onto a line. Note the stretching of the length of the arc s after the projection. The concept of a projection can be extended to projecting the points on the surface of an ERM onto a plane. N Note that the red points do not map! Projecting from 2D to 1D

SRM Tutorial, STC 2000, Snowbird, Utah Cylindrical Projections Tangent Secant

SRM Tutorial, STC 2000, Snowbird, Utah Planar Projections Tangent Secant

SRM Tutorial, STC 2000, Snowbird, Utah A Stereographic Projection Tangent (North Polar Aspect)

SRM Tutorial, STC 2000, Snowbird, Utah Conic Projections * From N. Bowditch, American Practical Navigator, U.S. Navy Hydrographic Office, 1966 Ed. Tangent Secant

SRM Tutorial, STC 2000, Snowbird, Utah A Mercator projection is a cylindrical projection. Mercator Projection * From N. Bowditch, American Practical Navigator, U.S. Navy Hydrographic Office, 1966 Ed.