The discrete wavelet transform is a well-known mathematical tool for shape recognition. In this study, a wavelet descriptor with two shape signatures, i.e., centroid distance and triangular centroid area, is developed for shape synthesis for path generation of planar mechanisms. This study is the first to apply the discrete wavelet transform for direct shape synthesis with the two-phase synthesis method for path generation problems. The problem of invariance in the similarity transform for the proposed wavelet descriptor is introduced. The proposed wavelet descriptor contains not only the approximation coefficients but also all the detail coefficients up to a prescribed decomposition level. A geometric-based approach which has been discussed in the literature is used for the scale-rotation-translation synthesis. The differential evolutionary algorithm with a combined mutation strategy is used to solve the optimization problems for shape synthesis. The proposed wavelet-based shape synthesis method is applied to solve five difficult path generation problems for the special paths generated by geared five-bar mechanisms discussed in the literature. The effects of the Haar, Daubechies-1, Daubechies-2 and Daubechies-3 wavelets as well as the number of levels from 1 to 3 on the synthesis results are investigated. Finally, the proposed wavelet-based shape synthesis method successfully solves the path generation problems. The generating coupler curves approximate the desired paths quite well except for one of the five paths. If only the approximation coefficients or only the detail coefficients are employed, the obtained optimum shape may appear similar but does not match the prescribed pattern, e.g. has an incorrect aspect ratio. The most suitable decomposition level depends on the nature of discrete shape signatures and the type of wavelet. The present synthesis results are compared to those obtained with other methods described in the literature.
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