A homotopy model for cup lifting

Introduces two new theoretical tools - homotopy and cellular structured spaces - for visualization. Any object is represented by a filtration space, which is a sequence of skeletons that are topological spaces. Using an attaching function that attaches n-1 dimensional balls to the boundaries of n-dimensional balls, a filtration space is composed inductively and step-by-step, by increasing the dimensions. The space obtained by this process is called a cellular structured space, which is composed of cells. The cellular structured space preserves invariant properties of entities. On the other hand, traditional polygonalization has difficulty in preserving invariant properties. A change from one situation represented by a cellular structured space to another situation of a cellular structured space is represented by a homotopy if the change is continuous. Using homotopy and cellular structured spaces, invariant properties are preserved while very large data compression is achieved.