A new adaptive tessellation method for general Catmull-Clark subdivision surfaces is presented. Development of the new method is based on the observation that optimum adaptive tessellation for rendering purpose is a recursive error evaluation and globalization process. The adaptive tessellation process is done by generating an inscribing polyhedron to approximate the limit surface for each individual patch. The inscribing polyhedron is generated through an adaptive subdivision on the patch's parameter space driven by a recursive error evaluation process. This approach generates less faces in the resulting approximating mesh while meeting the given precision requirement. The crack problem is avoided through globalization of new vertices generated in the adaptive subdivision process of the parameter space. No crack-detection or crack-elimination is needed in the adaptive tessellation process. Therefore, no mesh element splitting to eliminate cracks is necessary. The new adaptive tessellation method can precisely measure the error for every point of the limit surface. Hence, it has complete control of the accuracy of the tessellation result.