Energy minimization has been widely used for constructing curve and surface in the fields such as computer-aided geometric design, computer graphics and so on. However, xamples show that energy minimization does not optimize the shape of the curve sometimes. This paper studies the relationship between minimizing strain energy and curve shapes using a family of cubic Hermite curves. Each of the cubic Hermite curves interpolates positions and tangent vectors of two given endpoints. Computer simulation technique has become the one of the methods of scientific discovery, the study process is carried out by the numerical computation and computer simulation techniques. Our result shows that (1) cubic Hermite curves cannot be constructed by solely minimizing the strain energy; (2) by using a local minimum value of the strain energy, the shapes of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shapes, however. Based on our experiments and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shapes. The new model uses an explicit formula to compute the magnitudes of the two tangent vectors, and has the properties that (1) it is easy to compute; (2) it makes the cubic Hermite curves have satisfactory shapes while holding the good property of minimizing strain energy for some cases in curve construction. The comparison of the new model with the minimum strain energy model is included.