mirror of https://github.com/CGAL/cgal
30 lines
1.6 KiB
TeX
30 lines
1.6 KiB
TeX
\ccHeading{Restricted Spherical Geometry}
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We introduce geometric objects that are part of the spherical surface
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$S_2$ and operations on them. We define types \ccc{Sphere_point},
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\ccc{Sphere_circle}, \ccc{Sphere_segment}, and \ccc{Sphere_direction}.
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\ccc{Sphere_point}s are points on $S_2$, \ccc{Sphere_circle}s are
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oriented great circles of $S_2$, \ccc{Sphere_segment}s are oriented
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parts of \ccc{Sphere_circles} bounded by a pair of
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\ccc{Sphere_point}s, and \ccc{Sphere_direction}s are directions that
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are part of great circles. (a direction is usually defined to be a
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vector without length, that floats around in its underlying space and
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can be used to specify a movement at any point of the underlying
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space; in our case we use directions only at points that are part of
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the great circle that underlies also the direction.)
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Note that we have to consider special geometric properties of the
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objects. For example two points that are part of a great circle define
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two \ccc{Sphere_segment}s, and two arbitrary \ccc{Sphere_segment}s can
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intersect in two points.
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If we restrict our geometric objects to a so-called perfect hemisphere
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of $S_2$\footnote{A perfect hemisphere of $S_2$ is an open half-sphere
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plus an open half-circle in the boundary of the open half-sphere
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plus one endpoint of the half-circle.} then the restricted objects
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behave like in classical geometry, e.g., two points define exactly one
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segment, two segments intersect in at most one interior point
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(non-degenerately), or three non-cocircular sphere points can be
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qualified as being positively or negatively oriented.
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