* only overload Manifold::project_to_manifold() for simple
* situations. This is often useful when describing manifolds that are
* embedded in higher dimensional space, e.g., the surface of a
- * sphere. In those cases, the desired new point maybe computed
- * simply by the (weighted) average of the provided point, projected
+ * sphere. In those cases, the desired new point may be computed
+ * simply by the (weighted) average of the provided points, projected
* back out onto the sphere.
*
*
* In its default implementation it uses a pair-wise reduction of
* the points in the quadrature formula by calling the function
* get_new_point() that takes three arguments on the first two
- * points, then on the resulting point and the next, untill all
+ * points, then on the resulting point and the next, until all
* points in the quadrature have been taken into account. User
* classes can get away by simply implementing the get_new_point()
* function that takes three arguments.
* Manifold description for a spherical space coordinate system.
*
* You can use this Manifold object to describe any sphere, circle,
- * hypersphere or hyperdisc in two or three dimensions, both as a co-dimension
- * one manifold descriptor or as co-dimension zero manifold descriptor.
+ * hypersphere or hyperdisc in two or three dimensions. This manifold
+ * can be used as a co-dimension one manifold descriptor of a
+ * spherical surface embedded in a higher dimensional space, or as a
+ * co-dimension zero manifold descriptor for a body with positive
+ * volume, provided that the center of the spherical space is excluded
+ * from the domain.
*
* The two template arguments match the meaning of the two template arguments
* in Triangulation<dim, spacedim>, however this Manifold can be used to
* it may not be suitable for domains that contain either the north or
* south poles. Consider for instance the pair of points
* \f$x_1=(1,\pi/3,0)\f$ and \f$x_2=(1,\pi/3,\pi)\f$ in polar
- * coordinates (laying on the surface of a sphere with radius one, on
- * a parallel at at height $\pi/3$. In this case connecting the points
+ * coordinates (lying on the surface of a sphere with radius one, on
+ * a parallel at at height $\pi/3$). In this case connecting the points
* with a straight line in polar coordinates would take the long road
* around the globe, without passing through the north pole.
*
* where $\kappa = \frac{x_1 \times \x_2}{\Vert x_1 \times \x_2 \Vert}$
* and $\alpha(t) = t * \arccos(x_1 * x_2) $ for $t\in[0,1]$.
* Indeed, this is a geodesic, and it is the natural choice when
- * connecting points on the surface of the sphere.
- *
- * If the codimension of the Manifold is one, than this Manifold
- * connects points using geodesics. In all other cases it is a
- * continuus extension of the codimension one case.
+ * connecting points on the surface of the sphere. In the examples above,
+ * the PolarManifold class implements the first way of connecting two
+ * points on the surface of a sphere, while SphericalManifold implements
+ * the second way, i.e., if the codimension of the Manifold is one,
+ * than this Manifold connects points using geodesics. In all other cases
+ * it is a continuus extension of the codimension one case.
*
* In particular, this class implements a Manifold that joins any two
* points in space by first projecting them onto the surface of a