We apply topological analysis to functionally based shape metamorphosis. The time-dependent shape is defined using homotopy. The advantage of this method is the automatic generation of the intermediate shapes between the key shapes of different topology types. To complete the method, we have to find a way to automatically detect the critical points on the time axis while the shape undergoes topological changes. These critical points can be later used for generation of non-linear time steps distribution along the time axis, for example, for providing ease-in/ease-out effects in animation. We present a new method for analysis of shape metamorphosis based on the Morse theory, oriented to analysis of a height function. Although we analyze the shape in an N-dimensional space, the height function is defined in the N+2 dimensional space with N point coordinates and two additional coordinates of the defining function and time values. We can analyze how the critical points are changing in the given height level, which takes only zero value of the shape defining function. In this paper, we present this method in comparison with typical Morse theory analysis using simple objects in 2D and 3D spaces.