In the elastic mechanics

the general form of mapping function

which represents the area external to a given underground cavern by the area outside the unit circle in another plane

is Laurent series

hence to resolve the coefficients of Laurent series is one of the key points to make analytical solution to underground cavern. A new method for calculating mapping function is put forward by means of searching mapped points on the boundary of underground cavern. While employing this method

one set of mapping relation should be initialized

by which the initial mapping function is obtained; thus corresponding mapped points and mapped shape of the given cavern are achieved. According to the distances of adjacent mapped points in the mapped boundary

the relevant realistic point coordinates in the real boundary are obtained in terms of the principle of equal rate of distance. Then

based on this improved mapping relation

the first iterative mapping function is calculated. In light of this algorithm

the final mapping function

whose accuracy is in the controlling range

can be achieved. Applying this method

the number of series and iteration can be easily controlled; and the accuracy of the mapping function can also be flexibly dominated. The mapping function of single underground cavern with arbitrary shape can be rapidly calculated. Especially with regard to these common engineering caverns with complicated shape

quite accurate mapping function can be obtained by this method