When a geometric constraint system can not be fully decomposed

numerical solving methods are used

in which Newton-Raphson iteration method is the most popular. However

Newton-Raphson iteration method is not stable. To improve the stability of numerical geometric constraint solving

an homotopy method

named under constrained homotopy

is advanced in the paper especially for under-constrained geometric system. It can be combined with the decomposition of geometric constraint system and can be used together with other solving methods easily

and thus helps to the solving ability of geometric constraint solver. Some key problems of under constrained homotopy

such as construction of the homotopy function

homotopy path tracing and singularity analysis of homotopy path

are discussed in the paper. A pure homotopy method for under-constrained geometric systems is not very effective. To solve this problem

a hybrid Newton-Homotopy method is proposed. It makes use of both the fastness of Newton-Raphson iteration method and the stability of homotopy method and thus improves both the ability and the efficiency of the geometric constraint solver.