For the smooth surface M, the explicit construction of local flattening map p is given, where the bijection projection p is the local flattening map from a smooth surface M to a plane. By virtue of the inverse projection p-1, the local wavelet transform on M can be generated from wavelet transform on a plane. Take the torus T2 for example, by using the local flattening map p of torus, the construction of the local dilation on the torus is systematically studied, the local wavelet transform formula on the torus is offered and the inverse transform formula of the local wavelet transform, that is, the reconstruction formula is also offered. Finally, we show the graphical representation of the local wavelet on the torus.

In this paper, a simple method is given in order to construct an area preserving mapping from a developable surface M to a plane. Based on the area preserving projection, we give some important formulas on M , and define a multi-resolution analysis on L 2 (M ). We provide the conditions to further discuss the continuous wavelet transform and discrete wavelet transform on developable surface. At the same time, we derived two-scale equations that the scaling function and wavelet function on developable surface satisfied, we also define and discuss the orthogonality, and several important theorems are given. Finally, we construct the numerical examples. The focus of this paper is the area preserving mapping that from developable surface M to a plane, and the discrete wavelet transform on developable surface.