I like to create interpolation functions for second order finite element meshes. For elements with straight edges all is good, but some of my elements may have curved edges as shown in the figure:
I am looking for references on how to test efficiently and robustly if a query point is inside a curved element. The interpolation itself is not a problem. One general way to do it is to find the mapping of the global element in $x$, $y$-coordinates to its mother element in $r$, $s$-coordinates, then map the query point and see if that mapped query point is within the mother element which has straight edges. This seems quite excessive and I am wondering if there are alternatives that you could point me to.
Update: This is for a general interpolation, to be used in plotting functions and the like. The query points can be 'random'. Even if one has a very good algorithm to find the closest element to a query point one needs to check if it is in the element. If that's not the case then a new potential element needs to be found and tested. This means that at a minimum at least one query needs to be done per query point. The expensive testing can be reduced if information if the element is curved or not can be used. In case the element is not curved then a cheap linear test can be used.