Spatial data structures allow to make efficient queries on Geographical Information Systems (GIS). Spatial queries involve the geometry of the data, such as points, lines, or polygons. For instance, a spatial query could poll for the nearest restaurants from a given location. Spatial queries can be solved exhaustively by going through the entire data, which is prohibitive as the number of data points increases. In this article, we are interested in making efficient queries on infinitely long geometrical shapes. For instance, angular sectors, defined as the intersection of two half-spaces, are infinitely long. However, regular spatial data structures are not adapted to these geometrical shapes. We propose a new method allowing to make efficient spatial queries on angular sectors (i.e. whether a point is inside an angular sector). It builds a R-tree from the dual space of angular sectors. An extensive evaluation shows our method is faster than using a regular R-tree.