# HYPERSPHERE_SURFACE 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.

### Languages:

HYPERSPHERE_SURFACE is available in a MATLAB version.

### Related Data and Programs:

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.

### Examples and Tests:

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.

Last revised on 05 May 2013.