This paper presents a new mesh generation method for a  simply connected curved  domain of a planar region whose  boundary is defined by one or more equations in polar coordinates and a positive integer  parameter  ‘n’,  ( n=2,4,6,8,......etc.) for specification of the wished degree of refinement.We first decompose this curved domain  into simple sub regions in the shape of curved triangles. These simple regions are then triangulated to generate a fine mesh of   linear triangles in the interior and curved triangles or linear triangles  near to the boundary of curved domain.These simple regions are then triangulated  to create 6-node triangles  by inserting midside nodes to these triangles. Each isolated 6-node triangle is then split into four triangles according to the usual scheme, that is, by using straight lines to join the midside nodes. To preserve the mesh conformity a similar procedure is also applied to every triangle of the domain to fully discretize the given starshaped curved domain  or cracked convex  curved  domains   into all triangles,and quadrilaterals