Geometric constraint solving with conics and linkages

Abstract

Most geometric construction methods of geometric constraint solving systems use line and circle (rule and compass) as basic drawing tools. In this paper, by introducing conics and linkages, we provide a set of complete drawing tools for the construction approach of geometric constraint solving. Using these tools, we may enlarge the drawing scope of the construction approach and still keep the elegant style of geometric solutions to geometric constraint solving. As applications, we obtain pure geometric solutions to three sets of well-known constraint problems: the 10 Apollonian drawing problems, the 13 cases of a smallest tri-connected constraint graph, and constraint problems with distance constraints only.

Introduction

Geometric constraint solving (GCS) is the central topic in much of the current work of developing intelligent and parametric CAD systems. It also has applications in chemical molecular modeling, linkage design, computer vision and computer aided instruction [8]. GCS algorithms accept the declarative description of geometric diagrams or engineering drawings as input and output a drawing procedure for the diagram. There are four main approaches to GCS: the graph analysis approach [19], [20], [26], [29], the rule-based approach [2], [9], [14], [23], [27], [30], [31], the numerical computation approach [18], [25], [28], and the symbolic computation approach [4], [10], [24]. In practice, combinations of these approaches are often used to obtain the best result. The work in Ref. [15] is such a successful attempt.

The graph analysis and the rule-based approaches are also called geometric construction approach. In this approach, a pre-treatment is carried out to transform the constraint problem into a constructive form that is easy to draw. In most cases, we construct the diagram sequentially with ruler and compass. This approach follows the tradition of engineering drawings, where ruler and compass are used as basic drawing tools. The construction approaches are the first choices for most GCS systems because they may provide complete, fast and stable solutions. However, the drawing scope of ruler and compass are restrictive. It is well known that using ruler and compass alone, we can draw diagrams that can be described by a sequence of triangular equations (see Section 2.1) of degree less than or equal to two. As shown in Ref. [10], we may find very simple design problems that cannot be solved with ruler and compass. The aim of this paper is to introduce new drawing tools to enlarge the drawing scope of the construction approach and still keep its elegant solution style.

We will consider two kinds of tools: conics and linkages. We prove that the diagrams that can be drawn with conics are those that can be described by a sequence of triangular equations of degree less than or equal to four. Therefore, the drawing scope by introducing conics is essentially larger than using lines and circles only. Since equations with degree ≤4 can be solved explicitly by radicals, the computation procedure is still complete and efficient as in the case of ruler and compass construction. Furthermore, for some diagrams that can be drawn with ruler and compass, we may give much simpler construction procedures by using conics. As an example, simple solutions are obtained for the 10 Apollonian drawing problems.

Linkages are one of the most basic kinds of mechanisms. Here, we will use them as drawing tools. Based on a result of Kempe, we prove that linkages as tools are complete in a certain sense. We also give an algorithm to find linkages in a constrained diagram. As a consequence, we prove that all constraint problems involving distance constraints only can be solved sequentially with linkages.

To solve the equations raised from linkage constructions, besides the often used numerical and symbolic computation methods, we introduce a geometric method which is to use the linkages to generate loci and then to find the intersection of these loci by searching the points on the loci. This geometric method is based on dynamic generation of geometric loci which are widely used in dynamic geometry software [12].

As an application, we show that a simplest constraint graph which is beyond the scope of Hoffmann and Owen's triangle decomposition methods can be transformed to pure geometric constructive form if linkages are allowed as construction tools. The linkages used in the construction are three kinds of four-bar linkages. One of the four-bar linkages is the traditional one. The other two kinds of four-bar linkages are newly introduced.

The rest of this paper is organized as follows. In Section 2, we show how to solve constraint problems with conics. In Section 3, we introduce linkages as a drawing tool. In Section 4, we present the solution to the smallest tri-connected constraint graph. In Section 5, we present the conclusion.