Problem Definition ::

### Shape

Our discussion of the shape from probing problem is limited to a special class of compact planar bodies. The planarity requirement limits our investigation to the 2-dimentional case. The compactness requirement is in agreement with our intuition and essentially means that the body is bounded and closed. In other words the body is a bounded 2-dimentional surface without holes. Additional assumptions are made to simplify the problem:

1. The interior of the shape is nonempty. This assumption avoids degenerate bodies without mass.
2. The approximate position of the body is known. More precisely, it is assumed that the origin of the coordinate system falls within the bounds of the shape. This assumption implies that the position of the body is known but its exact shape is unknown.
3. The shape or boundary of the body is represented by a polygon with 3 or more vertices. This assumption imposes limits on the smoothness of the body's boundary.
4. The bounding polygon is convex. A convex polygon has the property that any line that joins two of its vertices is completely inside the polygon.

From now on our discussion will refer to the body under investigation as a polygon. Discovering the nature of the bounding polygon is equivalent to discovering the shape of the body. The following three shapes are valid. The interior of the shapes is gray and their boundary (polygon) is black.

 valid pentahedron valid triangle valid hexagon

The following figures show three shapes that are not handled in our definition of the problem. The first shape is a non-convex polygon and thus not covered by our discussion. The second shape is a line and cannot be accepted because the number of vertices is less than 3. The third shape is a circle and as such cannot be represented faithfully as a polygon. Note however than a polygon approximating a circle would be acceptable.

 non-convex polygon (invalid!) line (invalid!) perfect circle (invalid!)

 Problem Definition Probe