This thesis is the research of the geometric transformstion between Lissajous and trochoidal curves. Lissajous and trochoidal curves are important in geometric curves. However there is not any connection between Lissajous and trochoidal curves in early researches. Firstly, we start from the simple harmonic motions, and the solution is found as Lissajous parametric curves. Furthermore, by means of the transformation, the matrix SU(2) in group theory , a series of curves between Lissajous and trochoids are demonstrated. They have not been discussed until now. Through the concept of SU(2), we obtained the intriguing geometric curves. Importantly, the physics of the transformation is worthy to discussed further in the future.