Estimate Hypersurface Location

**HYPERSPHERE_SURFACE**
is a MATLAB library which
investigates a procedure for estimating the location of points on a
hypersurface implicitly defined by a characteristic function or by
a signed function.

In M dimensions, a characteristic function F for a region of points S has the property that F(X) is 1 for points inside S, and 0 otherwise. A characteristic function is thus discontinous. The set of discontinuity points for F is the boundary of S.

In M dimensions, suppose that the function F, presumably continuous, takes on both positive and negative values. Then we may be interested in the set of points X such that F(X) = 0, which we regard as the boundary between the sets where F is negative or positive. Assuming that F is continuous, or better yet, differentiable, then this boundary may be substantially easier to locate than in the characteristic function case.

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

**HYPERSPHERE_SURFACE** is available in
a MATLAB version.

EDGE, a MATLAB library which defines some test functions in 1D, 2D and 3D for the detection of edges.

HYPERSPHERE_PROPERTIES, a MATLAB library which carries out various operations for an M-dimensional hypersphere, including converting between Cartesian and spherical coordinates, stereographic projection, sampling the surface of the sphere, and computing the surface area and volume.

LEVELS, a MATLAB library which makes a contour plot, choosing the contour levels using random sampling.

SHORELINE, a MATLAB program which tries to identify and triangulate the 2D domain over which some function f(x,y) is nonnegative.

SHORELINE2, a MATLAB program which tries to identify and triangulate the 2D domain over which some function f(x,y) is roughly zero.

- bisect_characteristic.m, uses bisection to outline the transition surface for a given characteristic function.
- cartesian_to_hypersphere.m, converts from a cartesian to a hypersphere coordinate system.
- circle_characteristic.m, a characteristic function for a circle.
- cube_characteristic.m, a characteristic function for a cube (any dimension).
- exterior_point_characteristic.m, searches for a point that is outside a surface defined by a characteristic, and for which a base point inside the surface is given, as well as the desired hypersphere angles of the exterior point.
- hypersphere_to_cartesian.m, converts from a hypersphere to a cartesian coordinate system.
- interior_point_characteristic.m searches a hyperrectangle for a point that is inside a surface defined by a characteristic.
- r8vec_print.m prints an R8VEC.
- sphere_characteristic.m, a characteristic function for a sphere (any dimension).
- timestamp.m prints the YMDHMS date as a timestamp.
- triangle_characteristic.m, a characteristic function for a triangle.

- hypersphere_surface_test.m, calls all the tests.
- hypersphere_surface_test_output.txt, the output file.
- exterior_point_characteristic_test.m, tests exterior_point_characteristic().
- interior_point_characteristic_test.m, tests interior_point_characteristic().

The **circle** example works with the characteristic function of
a circle.

- circle_plots.m, creates plots of the surface and R(Theta) function for a circle, using a centered and an offcentered base point.
- circle_centered_plot.png, a plot of the radial distance R as a function of angle THETA for a circle defined by a characteristic function, using a centered base point.
- circle_centered_surface.png, a plot of the transition surface for a circle defined by a characteristic function, using a centered base point.
- circle_offcentered_plot.png, a plot of the radial distance R as a function of angle THETA for a circle defined by a characteristic function, using an offcentered base point.
- circle_offcentered_surface.png, a plot of the transition surface for a circle defined by a characteristic function, using an offcentered base point.

The **cube** example works with the characteristic function of
a cube.

- cube_plots.m, creates plots of the surface and R(Theta1,Theta2) function for a cube, using a centered and an offcentered base point.
- cube_centered_plot.png, a plot of the radial distance R as a function of the angles for a cube defined by a characteristic function, using a centered base point.
- cube_centered_surface.png, a plot of the transition surface for a cube defined by a characteristic function, using a centered base point.
- cube_offcentered_plot.png, a plot of the radial distance R as a function of the angles for a cube defined by a characteristic function, using an offcentered base point.
- cube_offcentered_surface.png, a plot of the transition surface for a cube defined by a characteristic function, using an offcentered base point.

The **sphere** example works with the characteristic function of
a sphere.

- sphere_plots.m, creates plots of the surface and R(Theta1,Theta2) function for a sphere, using a centered and an offcentered base point.
- sphere_centered_plot.png, a plot of the radial distance R as a function of the angles for a sphere defined by a characteristic function, using a centered base point.
- sphere_centered_surface.png, a plot of the transition surface for a sphere defined by a characteristic function, using a centered base point.
- sphere_offcentered_plot.png, a plot of the radial distance R as a function of the angles for a sphere defined by a characteristic function, using an offcentered base point.
- sphere_offcentered_surface.png, a plot of the transition surface for a sphere defined by a characteristic function, using an offcentered base point.

The **triangle** example works with the characteristic function of
a triangle.

- triangle_plots.m, displays plots of the surface and R(Theta) function for a triangle, using a centered and an offcentered base point.
- triangle_centered_plot.png, a plot of the radial distance R as a function of angle THETA for a triangle defined by a characteristic function, using a centered base point.
- triangle_centered_surface.png, a plot of the transition surface for a triangle defined by a characteristic function, using a centered base point.
- triangle_offcentered_plot.png, a plot of the radial distance R as a function of angle THETA for a triangle defined by a characteristic function, using an offcentered base point.
- triangle_offcentered_surface.png, a plot of the transition surface for a triangle defined by a characteristic function, using an offcentered base point.

You can go up one level to the MATLAB source codes.