import numpy from shapely.geometry import LineString, Point from mpas_tools.vector import Vector def subdivide_great_circle(x, y, z, maxRes, earthRadius): """ Subdivide each segment of the transect so the horizontal resolution approximately matches the requested resolution Uses a formula for interpolating unit vectors on the sphere from https://en.wikipedia.org/wiki/Slerp Parameters ---------- x : numpy.ndarray The Cartesian x coordinate of a transect, where the number of segments is ``len(x) - 1``. ``x``, ``y`` and ``z`` are of the same length. y : numpy.ndarray The Cartesian y coordinate of the transect z : numpy.ndarray The Cartesian z coordinate of the transect maxRes : float The maximum allowed spacing in m after subdivision earthRadius : float The radius of the Earth in m Returns ------- xOut : numpy.ndarray The Cartesian x values of the transect subdivided into segments with segment length at most ``maxRes``. All the points in ``x``, ``y`` and ``z`` are guaranteed to be included. yOut : numpy.ndarray The Cartesian y values of the subdivided transect zOut : numpy.ndarray The Cartesian y values of the subdivided transect dIn : numpy.ndarray The distance along the transect before subdivision dOut : numpy.ndarray The distance along the transect after subdivision """ angularDistance = angular_distance(x=x, y=y, z=z) dx = angularDistance * earthRadius nSegments = numpy.maximum( (dx / maxRes + 0.5).astype(int), 1) dIn = numpy.zeros(x.shape) dIn[1:] = numpy.cumsum(dx) frac = [] outIndices = [] delta = [] for index in range(len(dIn) - 1): n = nSegments[index] frac.extend(numpy.arange(0, n)/n) outIndices.extend(index*numpy.ones(n, int)) delta.extend(angularDistance[index]*numpy.ones(n)) frac.append(1.) outIndices.append(len(dIn) - 2) delta.append(angularDistance[-1]) frac = numpy.array(frac) delta = numpy.array(delta) outIndices = numpy.array(outIndices) a = numpy.ones(delta.shape) b = numpy.zeros(delta.shape) mask = delta > 0. denom = 1./numpy.sin(delta[mask]) a[mask] = denom*numpy.sin((1.-frac[mask])*delta[mask]) b[mask] = denom*numpy.sin(frac[mask]*delta[mask]) xOut = a*x[outIndices] + b*x[outIndices+1] yOut = a*y[outIndices] + b*y[outIndices+1] zOut = a*z[outIndices] + b*z[outIndices+1] dOut = (-frac + 1.)*dIn[outIndices] + frac*dIn[outIndices+1] return xOut, yOut, zOut, dIn, dOut def cartesian_to_great_circle_distance(x, y, z, earth_radius): """ Cartesian transect points to great-circle distance Parameters ---------- x : numpy.ndarray The Cartesian x coordinate of a transect y : numpy.ndarray The Cartesian y coordinate of the transect z : numpy.ndarray The Cartesian z coordinate of the transect earth_radius : float The radius of the earth Returns ------- distance : numpy.ndarray The distance along the transect """ distance = numpy.zeros(x.shape) for segIndex in range(len(x)-1): transectv0 = Vector(x[segIndex], y[segIndex], z[segIndex]) transectv1 = Vector(x[segIndex+1], y[segIndex+1], z[segIndex+1]) distance[segIndex+1] = distance[segIndex] + \ earth_radius*transectv0.angular_distance(transectv1) return distance def subdivide_planar(x, y, maxRes): """ Subdivide each segment of the transect so the horizontal resolution approximately matches the requested resolution Uses a formula for interpolating unit vectors on the sphere from https://en.wikipedia.org/wiki/Slerp Parameters ---------- x : numpy.ndarray The planar x coordinate of a transect, where the number of segments is ``len(x) - 1`` y : numpy.ndarray The planar y coordinates of the transect, the same length as ``x`` maxRes : float The maximum allowed spacing in m after subdivision Returns ------- xOut : numpy.ndarray The x coordinate of the transect, subdivided into segments with length at most ``maxRes``. All the points in ``x`` are guaranteed to be included. yOut : numpy.ndarray The y coordinate of the transect. All the points in ``y`` are guaranteed to be included. dIn : numpy.ndarray The distance along the transect before subdivision dOut : numpy.ndarray The distance along the transect after subdivision """ dx = numpy.zeros(len(x)-1) for index in range(len(x)-1): start = Point(x[index], y[index]) end = Point(x[index+1], y[index+1]) segment = LineString([start, end]) dx[index] = segment.length nSegments = numpy.maximum( (dx / maxRes + 0.5).astype(int), 1) dIn = numpy.zeros(x.shape) dIn[1:] = numpy.cumsum(dx) frac = [] outIndices = [] for index in range(len(dIn) - 1): n = nSegments[index] frac.extend(numpy.arange(0, n)/n) outIndices.extend(index*numpy.ones(n, int)) frac.append(1.) outIndices.append(len(dIn) - 2) frac = numpy.array(frac) outIndices = numpy.array(outIndices) xOut = (-frac + 1.)*x[outIndices] + frac*x[outIndices+1] yOut = (-frac + 1.)*y[outIndices] + frac*y[outIndices+1] dOut = (-frac + 1.)*dIn[outIndices] + frac*dIn[outIndices+1] return xOut, yOut, dIn, dOut def lon_lat_to_cartesian(lon, lat, earth_radius, degrees): """Convert from lon/lat to Cartesian x, y, z""" if degrees: lon = numpy.deg2rad(lon) lat = numpy.deg2rad(lat) x = earth_radius * numpy.cos(lat) * numpy.cos(lon) y = earth_radius * numpy.cos(lat) * numpy.sin(lon) z = earth_radius * numpy.sin(lat) return x, y, z def cartesian_to_lon_lat(x, y, z, earth_radius, degrees): """Convert from Cartesian x, y, z to lon/lat""" lon = numpy.arctan2(y, x) lat = numpy.arcsin(z/earth_radius) if degrees: lon = numpy.rad2deg(lon) lat = numpy.rad2deg(lat) return lon, lat def angular_distance(x, y, z): """ Compute angular distance between points on the sphere, following: https://en.wikipedia.org/wiki/Great-circle_distance Parameters ---------- x : numpy.ndarray The Cartesian x coordinate of a transect, where the number of segments is ``len(x) - 1``. ``x``, ``y`` and ``z`` are of the same length. y : numpy.ndarray The Cartesian y coordinate of the transect z : numpy.ndarray The Cartesian z coordinate of the transect Returns ------- distance : numpy.ndarray The angular distance (in radians) between segments of the transect. """ first = Vector(x[0:-1], y[0:-1], z[0:-1]) second = Vector(x[1:], y[1:], z[1:]) distance = first.angular_distance(second) return distance