stripy provides a python interfact to STRIPACK and SSRFPACK (Renka 1997a,b) as a triangulation class that would typically be used as follows:
import stripy as stripy
spherical_triangulation = stripy.sTriangulation(lons=vertices_lon_as_radians, lats=vertices_lat_as_radians)
s_areas = spherical_triangulation.areas()
The methods of the sTriangulation class include interpolation, smoothing and gradients (from SSRFPACK), triangle areas, point location by simplex and nearest vertex, refinement operations by edge or centroid, and neighbourhood search / distance computations through a k-d tree algorithm suited to points on the surface of a unit sphere. stripy also includes template triangulated meshes with refinement operations.
In this notebook we introduce the sTriangulation class itself.
Renka, R. J. (1997), Algorithm 772: STRIPACK: Delaunay triangulation and Voronoi diagram on the surface of a sphere, ACM Transactions on Mathematical Software (TOMS).
Renka, R. J. (1997), Algorithm 773: SSRFPACK: interpolation of scattered data on the surface of a sphere with a surface under tension, ACM Transactions on Mathematical Software (TOMS), 23(3), 435–442, doi:10.1145/275323.275330.
Renka, R. J. (1996), Algorithm 751; TRIPACK: a constrained two-dimensional Delaunay triangulation package, ACM Transactions on Mathematical Software, 22(1), 1–8, doi:10.1145/225545.225546.
Renka, R. J. (1996), Algorithm 752; SRFPACK: software for scattered data fitting with a constrained surface under tension, ACM Transactions on Mathematical Software, 22(1), 9–17, doi:10.1145/225545.225547.
The next example is Ex2-SphericalGrids
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import stripy as stripy
import numpy as np
# Vertices of an icosahedron as Lat / Lon in degrees
vertices_LatLonDeg = np.array(
[[ 90, 0.0 ],
[ 26.57, 0.0 ],
[-26.57, 36.0 ],
[ 26.57, 72.0 ],
[-26.57, 108.0 ],
[ 26.57, 144.0 ],
[-26.57, 180.0 ],
[ 26.57, 360.0-72.0 ],
[-26.57, 360.0-36.0 ],
[ 26.57, 360.0-144.0 ],
[-26.57, 360.0-108.0 ],
[-90, 0.0 ]])
vertices_lat = np.radians(vertices_LatLonDeg.T[0])
vertices_lon = np.radians(vertices_LatLonDeg.T[1])
spherical_triangulation = stripy.sTriangulation(lons=vertices_lon, lats=vertices_lat)
This creates a triangulation object constructed using the wrapped fortran code of Renka (1997). The triangulation object has a number of useful methods and attached data which can be listed with
help(spherical_triangulation)
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print(spherical_triangulation.areas())
print(spherical_triangulation.npoints)
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refined_spherical_triangulation = stripy.sTriangulation(lons=vertices_lon, lats=vertices_lat, refinement_levels=2)
print(refined_spherical_triangulation.npoints)
We can make a plot of the two grids and the most straightforward way to display the information is through a standard map projection of the sphere to the plane.
(Here we superimpose the points on a global map of coastlines using the cartopy map library and use the Mollweide projection.
Other projections to try include Robinson, Orthographic, PlateCarree)
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%matplotlib inline
import gdal
import cartopy
import cartopy.crs as ccrs
import matplotlib.pyplot as plt
fig = plt.figure(figsize=(20, 10), facecolor="none")
ax = plt.subplot(121, projection=ccrs.Mollweide(central_longitude=0.0, globe=None))
ax.coastlines(color="#777777")
ax.set_global()
ax2 = plt.subplot(122, projection=ccrs.Mollweide(central_longitude=0.0, globe=None))
ax2.coastlines(color="#777777")
ax2.set_global()
## Plot the vertices and the edges for the original isocahedron
lons = np.degrees(spherical_triangulation.lons)
lats = np.degrees(spherical_triangulation.lats)
ax.scatter(lons, lats, color="Red",
marker="o", s=150.0, transform=ccrs.Geodetic())
segs = spherical_triangulation.identify_segments()
for s1, s2 in segs:
ax.plot( [lons[s1], lons[s2]],
[lats[s1], lats[s2]],
linewidth=0.5, color="black", transform=ccrs.Geodetic())
## Plot the vertices and the edges for the refined isocahedron
lons = np.degrees(refined_spherical_triangulation.lons)
lats = np.degrees(refined_spherical_triangulation.lats)
ax2.scatter(lons, lats, color="Red", alpha=0.5,
marker="o", s=50.0, transform=ccrs.Geodetic())
segs = refined_spherical_triangulation.identify_segments()
for s1, s2 in segs:
ax2.plot( [lons[s1], lons[s2]],
[lats[s1], lats[s2]],
linewidth=0.5, color="black", transform=ccrs.Geodetic())
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import lavavu
lv = lavavu.Viewer(border=False, resolution=[666,666], background="#FFFFFF")
lv["axis"]=False
lv['specular'] = 0.5
ghost = lv.triangles("ghost", wireframe=False, colour="#777777", opacity=0.1)
ghost.vertices(refined_spherical_triangulation.points*0.99)
ghost.indices(refined_spherical_triangulation.simplices)
tris = lv.triangles("coarse", wireframe=False, colour="#4444FF", opacity=0.8)
tris.vertices(spherical_triangulation.points)
tris.indices(spherical_triangulation.simplices)
trisw = lv.triangles("coarsew", wireframe=True, colour="#000000", opacity=0.8, linewidth=3.0)
trisw.vertices(spherical_triangulation.points)
trisw.indices(spherical_triangulation.simplices)
nodes = lv.points("coarse_nodes", pointsize=10.0, pointtype="shiny", colour="#4444FF", opacity=0.95)
nodes.vertices(spherical_triangulation.points)
tris2 = lv.triangles("fine", wireframe=False, colour="#007f00", opacity=0.25)
tris2.vertices(refined_spherical_triangulation.points)
tris2.indices(refined_spherical_triangulation.simplices)
tris2w = lv.triangles("finew", wireframe=True, colour="#444444", opacity=0.25, linewidth=0.5)
tris2w.vertices(refined_spherical_triangulation.points)
tris2w.indices(refined_spherical_triangulation.simplices)
nodes2 = lv.points("fine_nodes", pointsize=3.0, pointtype="shiny", colour="FF0000", opacity=0.95)
nodes2.vertices(refined_spherical_triangulation.points)
lv.hide("fine")
lv.hide("finew")
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lv.control.Panel()
lv.control.Button(command="hide triangles; hide points; show ghost; show fine_nodes; show coarse; show coarsew; show coarse_nodes; redraw", label="Coarse")
lv.control.Button(command="hide triangles; hide points; show ghost; show coarse_nodes; show fine; show finew; show fine_nodes; redraw", label="Fine")
lv.control.show()
One common use of stripy is in meshing the sphere and, to this end, we provide pre-defined meshes for icosahedral and octahedral triangulations, each of which can have mid-face centroid points included. A triangulation of the six cube-vertices is also provided as well as a 'buckyball' mesh. The icosahedral meshes defined above can be created directly using:
spherical_triangulation = stripy.spherical_meshes.icosahedral_mesh(refinement_levels=0)
refined_spherical_triangulation = stripy.spherical_meshes.icosahedral_mesh(refinement_levels=3)
This capability is shown in a companion notebook Ex2-SphericalGrids