Geometric Constraint Solving with Conics
用圆锥曲线求解几何约束问题

Geometric constraint solving has applications in a wide variety of fields, such as mechanical engineering, chemical molecular conformation, geometric theorem proving, and surveying. There are mainly three approaches to geometric constraint solving: numerical approach, symbolic approach and constructive approach. Since the constructive approach are simple and practicable, most parametric design take the constructive approach as the basic approach. In the light of the shortage of using only line and circle (rule and compass) as basic drawing tools in constructive approach, in this paper, we introduce a class of new drawing tools: conics. We prove that the class of diagrams within the drawing scope of this new tool is larger than that can be drawn with line and circle. Actually, we prove that a diagram can be drawn with conics if and only if this diagram can be described with a sequence of triangulated equations of degree less than or equal to four. This allows us to maintain the elegance of geometric constraint solving with ruler and compass, because the solutions of cubic and quartic equations can be written explicitly with radicals.