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Splines over triangulations and splines over quadrangulations (tensor productsplines) are two common ways to extend bivariate polynomials to splines.However, combination of both approaches leads to splines defined over mixedtriangle and quadrilateral meshes using the isogeometric approach. Mixed meshesare especially useful for representing complicated geometries obtained e.g.from trimming. As (bi-)linearly parameterized mesh elements are not flexibleenough to cover smooth domains, we focus in this work on the case of planarmixed meshes parameterized by (bi-)quadratic geometry mappings. In particularwe study in detail the space of $C^1$-smooth isogeometric spline functions ofgeneral polynomial degree over two such mixed mesh elements. We present thetheoretical framework to analyze the smoothness conditions over the commoninterface for all possible configurations of mesh elements. This comprises theinvestigation of the dimension as well as the construction of a basis of thecorresponding $C^1$-smooth isogeometric spline space over the domain describedby two elements. Several examples of interest are presented in detail.

$C^1$-smooth isogeometric spline functions of general degree over planar mixed meshes: The case of two quadratic mesh elements