1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
|
User documentation
==================
.. currentmodule:: stsci.sphere
The `stsci.sphere` library is a pure Python package for handling spherical
polygons that represent arbitrary regions of the sky.
Requirements
------------
- Python 2.7
- Numpy 1.4 or later
- astropy 0.3 or later
Coordinate representation
-------------------------
Coordinates in world space are traditionally represented by right
ascension and declination (*ra* and *dec*), or longitude and latitude.
While these representations are convenient, they have discontinuities
at the poles, making operations on them trickier at arbitrary
locations on the sky sphere. Therefore, all internal operations of
this library are done in 3D vector space, where coordinates are
represented as (*x*, *y*, *z*) vectors. The `stsci.sphere.vector` module
contains functions to convert between (*ra*, *dec*) and (*x*, *y*,
*z*) representations.
While any (*x*, *y*, *z*) triple represents a vector and therefore a
location on the sky sphere, a distinction must be made between
normalized coordinates that fall exactly on the unit sphere, and
unnormalized coordinates which do not. A normalized coordinate is
defined as a vector whose length is 1, i.e.:
.. math::
\sqrt{x^2 + y^2 + z^2} = 1
To prevent unnecessary recomputation, many methods in this library
assume that the vectors passed in are already normalized. If this is
not the case, `stsci.sphere.vector.normalize_vector` can be used to
normalize an array of vectors.
When not working in Cartesian vectors, the library allows the user to
work in either degrees or radians. All methods that require or return
an angular value have a `degrees` keyword argument. When `degrees` is
`True`, these measurements are in degrees, otherwise they are in
radians.
.. warning::
Due to constraints in the precision of intersection calculations,
points on the sphere that are closer than :math:`2^{-32}` along a
Cartesian axis are automatically merged into a single point. This
prevents intersections from being missed due to floating point
rounding error. There is currently no implemented solution to
deal with points that need to be closer together.
Spherical polygons
------------------
Spherical polygons are arbitrary areas on the sky sphere enclosed by
great circle arcs. They are represented by the
`~stsci.sphere.polygon.SphericalPolygon` class.
Representation
``````````````
The points defining the polygon are available from the
`~polygon.SphericalPolygon.points` property. It is a Nx3 array where
each row is an (*x*, *y*, *z*) vector, normalized. The polygon points
are explicitly closed, i.e., the first and last points are the same.
Where is the inside?
^^^^^^^^^^^^^^^^^^^^
The edges of a polygon serve to separate the “inside” from the
“outside” area. On a traditional 2D planar surface, the “inside” is
defined as the finite area and the “outside” is the infinite area.
However, since the surface of a sphere is cyclical, i.e., it wraps
around on itself, the a spherical polygon actually defines two finite
areas. To specify which should be considered the “inside” vs. the
“outside”, the definition of the polygon also has an “inside point”
which is just any point that should be considered inside of the
polygon.
In the following image, the inside point (marked with the red dot)
declares that the area of the polygon is the green region, and not the
white region.
.. image:: inside.png
The inside point of the the polygon can be obtained from the
`~polygon.SphericalPolygon.inside` property.
Cut lines
^^^^^^^^^
If the polygon represents two disjoint areas or the polygon has holes,
those areas will be connected by cut lines. The following image shows
a polygon made from the union of a number of cone areas which has both
a hole and a disjoint region connected by cut lines.
.. image:: cutlines.png
Creating spherical polygons
```````````````````````````
.. currentmodule:: stsci.sphere.polygon
`SphericalPolygon` objects have 4 different constructors:
- `SphericalPolygon`: Takes an array of (*x*, *y*, *z*)
points and an inside point.
- `SphericalPolygon.from_radec`: Takes an array of (*ra*, *dec*)
points and an inside point.
- `SphericalPolygon.from_cone`: Creates a polygon from a cone on the
sky shere. Takes (*ra*, *dec*, *radius*).
- `SphericalPolygon.from_wcs`: Creates a polygon from the footprint
of a FITS image using its WCS header keywords. Takes a FITS
filename or a `astropy.io.fits.Header` object.
Operations on Spherical Polygons
````````````````````````````````
Once one has a `SphericalPolygon` object, there are a number of
operations available:
- `~SphericalPolygon.contains_point`: Determines if the given point is inside the polygon.
- `~SphericalPolygon.intersects_poly`: Determines if one polygon intersects with another.
- `~SphericalPolygon.area`: Determine the area of a polygon.
- `~SphericalPolygon.union` and `~SphericalPolygon.multi_union`:
Return a new polygon that is the union of two or more polygons.
- `~SphericalPolygon.intersection` and
`~SphericalPolygon.multi_intersection`: Return a new polygon that
is the intersection of two or more polygons.
- `~SphericalPolygon.overlap`: Determine how much a given polygon
overlaps another.
- `~SphericalPolygon.to_radec`: Convert (*x*, *y*, *z*) points in the
polygon to (*ra*, *dec*) points.
- `~SphericalPolygon.same_points_as`: Determines if one polygon has the
same points as another. When only sorted unique points are considered
(default behavior), polygons with same points might not be the same
polygons because the order of the points matter.
- `~SphericalPolygon.draw`: Plots the polygon using matplotlib’s
Basemap toolkit. This feature is rather bare and intended
primarily for debugging purposes.
Great circle arcs
-----------------
.. currentmodule:: stsci.sphere.great_circle_arc
As seen above, great circle arcs are used to define the edges of the
polygon. The `stsci.sphere.great_circle_arc` module contains a number of
functions that are useful for dealing with them.
- `length`: Returns the angular distance between two points on the sphere.
- `intersection`: Returns the intersection point between two great
circle arcs.
- `intersects`: Determines if two great circle arcs intersect.
- `intersects_point`: Determines if a point is along the great circle
arc.
- `angle`: Calculate the angle between two great circle arcs.
- `midpoint`: Calculate the midpoint along a great circle arc.
|
