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A particular class of planar two-patch geometries, called bilinear-like$G^{2}$ two-patch geometries, is introduced. This class includes the subclassof all bilinear two-patch parameterizations and possesses similar connectivityfunctions along the patch interface. It is demonstrated that the class ofbilinear-like $G^2$ two-patch parameterizations is much wider than the class ofbilinear parameterizations and can approximate with good quality given generictwo-patch parameterizations.  We investigate the space of $C^{2}$-smooth isogeometric functions over thisspecific class of two-patch geometries. The study is based on the equivalenceof the $C^2$-smoothness of an isogeometric function and the $G^2$-smoothness ofits graph surface (cf. [12, 20]). The dimension of the space is computed and anexplicit basis construction is presented. The resulting basis functions possesssimple closed form representations, have small local supports, and arewell-conditioned. In addition, we introduce a subspace whose basis functionscan be generated uniformly for all possible configurations of bilinear-like$G^{2}$ two-patch parameterizations. Numerical results obtained by performing$L^{2}$-approximation indicate that already the subspace possesses optimalapproximation properties.

Dimension and basis construction for $C^{2}$-smooth isogeometric spline spaces over bilinear-like $G^{2}$ two-patch parameterizations