This thesis makes a broad and profound research on the key technique in Parametric and Variational design. Many methods is overviewed and many new techniques are also proposed.The graph-based geometric constraint solving method proposed by C.M. Hoffmann is introduced and analyzed detailedly. Based on this, we continue to give a theoretical geometry solution of the geometry template composed by 3 points and 3 planes in 3-dimension space. The experimental numerical solution verifies the correctness of the geometry solution. The analyzed template is more typical than the other templates in the former literatures because it includes not only the constraints appeared in the before literatures, but also the angle constraints between the unknown geometry elements never discussed before. Our research on the special constraint template enriches the graph-based constraint solving method greatly. During the reasoning process, we correct two mistakes in the former literatures.GASA (genetic annealing simulated algorithm) is introduced into GCS (geometry constraint solving). The over-constrained and under-constrained problem can be solved naturally in our approach because that a constraint problem is transformed into an optimal problem doesn't entail that the number of constraint equations equal to the one of constraint variables. GASA is characteristic of many advantages, such as the calculating robustness, implied inherent parallelism, global searching and local convergence. These advantages are integrated in GCS in our method and make the constraint problems solved robustly and efficiently. Based on such approach, the ecological niche ideal is further integrated with GASA. The integrated GCS can get all the solutions for the well-constraint problems with many solutions. Also many experimental data are provided in this thesis.The BFGS algorithm applied in GCS is analyzed and pointed out two shortcomings: (1) always trapped in local optimal solution; (2) can not pass through the critical point. These shortcomings make the BFGS algorithm can not find the global best solution. Based on the knowledge of multi-variation extreme value solving and chaos technique, we use the chaos algorithm to overcome these shortcomings. The chaos algorithm is embedded in the BFGS algorithm, which helps the BFGS algorithm jump over the trap of local optimal solution and find the global optimal solution. On the other hand, the compounded algorithm is also filled with theadvantages of the BFGS algorithm: the over- and under-constraint problem can be disposed naturally.The thesis also discusses the GCS in assembly. We analyze the assembly model and propose a mathematics model and tree-representing model of the assembly who not only has a small data size, but also can be manipulated easily. This thesis not only gives the virtual representing concept the instance representing one, but also defines the concept of assembly constraint such as coupling, being alignment, being coaxial and being equidirectional. Based on such concepts, a revised Newton-Raphson iterative algorithm is proposed, which can deal with the singularity and ill-condition of Jacobi matrix.