Note: The minimum curvature algorithm requires a work array that is M 2 in size, where M is the total number of input data points. If the SPHERE keyword is set, X and Y are given in degrees of longitude and latitude, respectively. If Z is specified as a double-precision value, the computation will be performed in double-precision arithmetic. For irregular grids, all three parameters must be present and have the same number of elements. For regularly gridded input data, X and Y are not used: the grid spacing is specified via the XGRID and YGRID (or XVALUES and YVALUES) keywords, and Z must be a two-dimensional array. Arguments Z, X, YĪrrays containing the Z, X, and Y coordinates of the data points on the surface. This function returns a two-dimensional floating-point array containing the interpolated surface, sampled at the grid points. Result = MIN_CURVE_SURF( Z ] |, XVALUES= array] |, YVALUES= array] ] ] ) Return Value
#Gridded sphere code#
Its source code can be found in the file min_curve_surf.pro in the lib subdirectory of the IDL distribution. This routine is written in the IDL language. The only difference is in the basis functions: TPS uses d 2*alog(d 2), and MCS uses d 2*alog(d), where d is the distance from point (x i,y i). The results obtained with the thin plate spline (TPS) and the minimum curvature surface (MCS) methods are very similar. x, y, and z are the 3D cartesian coordinates of the point (l,t) on the unit sphere.įor a sphere with the CONST keyword set, the value is: C(L i,l, T i,t) is the basis function above, with distance between the two points, (L i,T i), and (l, t), measured in radians of arc length. On the sphere, l and t are longitude and latitude. For any interpolation point, the interpolated value is: For n data points, a system of n+3 simultaneous equations are solved for the coefficients of the surface. Where d is the distance between ( x 0, y 0), ( x 1, y 1) and k = 1 for minimum curvature surface or k = 2 for Thin Plate Splines. TheoryĪ minimum curvature spline surface is fitted to the data points described by x, y, and z. Note: The accuracy of this function is limited by the single-precision floating-point accuracy of the machine. The MIN_CURVE_SURF function interpolates a regularly- or irregularly-gridded set of points, over either a plane or a sphere, with either a minimum curvature surface or a thin-plate- spline surface. Syntax | Return Value | Arguments | Keywords | Examples | Version History | See Also "Spherical Coordinates."įrom MathWorld-A Wolfram Web Resource.Routines (alphabetical) > Routines: M > MIN_CURVE_SURF MIN_CURVE_SURF Referenced on Wolfram|Alpha Spherical Coordinates Cite this as: Standard Mathematical Tables and Formulae. "Tensor Calculations on Computer: Appendix." Comm. Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Orlando, FL: Academic Press, pp. 102-111, "Spherical Polar Coordinates." §2.5 in Mathematical To Differential Equations and Probability. Apostol,Ģnd ed., Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications Spherical coordinates of vector (1, 2, 3) Extreme care is therefore needed when consulting the literature. The following table summarizes a number of conventions The symbol is sometimes also used in place of, instead of, and and instead of. Typically means (radial, azimuthal, polar) to a mathematician but (radial, polar,Īzimuthal) to a physicist. This is especially confusing since the identical Unfortunately, the convention in which the symbols and are reversed (both in meaning and in order listed) is alsoįrequently used, especially in physics. Used in the physics literature is retained (resulting, it is hoped, in a bit lessĬonfusion than a foolish rigorous consistency might engender). Is in spherical harmonics, where the convention The sole exception to this convention in this work Remaining the angle in the - plane and becoming the angle out of that Note that this definition provides a logicalĮxtension of the usual polar coordinates notation, In this work, following the mathematics convention, the symbols for the radial, azimuth, and zenith angleĬoordinates are taken as, , and, respectively. This is the convention commonly used in mathematics. To be distance ( radius) from a point to the origin. Is the latitude) from the positive z-axis Known as the zenith angle and colatitude, When referred to as the longitude), to be the polar angle (also Define to be the azimuthal angle in the - plane from the x-axis That are natural for describing positions on a sphere Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates
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