Three Dimensional Sketch for a Landscape Using Morse Theory and Reeb Graphs

The theoretical procedure for making a three dimensional landscape sketch is proposed in this paper. At the first stage, the critical points such as summits, ravine bottoms and passes are allocated by a user. These critical points construct fundamental properties of the mountain that a user tries to draw. Applying Morse theory to these critical points, the landscape of the mountain is defined as a sequence of a CW-complex that explains how the surface structure of the mountain changes when the mountain is formed from the bottom to the top step by step. The CW-complex is then concreted by obtaining skeletons from the Reeb graph of the mountain. The skeleton consisting of contour lines is modified without changing its topological structure by giving ranges and ravines. These lines constitute a series of continuous critical points and become invariants as well as summits, ravine bottoms and passes. At the final step, the system automatically generates a proposed landscape from the skeleton using Nurbs lines and surfaces or polygon meshes. By rendering the landscape, a user can change its shape by modifying ranges and ravines until the user satisfies it.