The identification, tracking, and characterisation of ocean eddies using observational and numerical data is essential for understanding eddy dynamics and their global climate impacts. Eulerian (point-based) and Lagrangian (trajectory-based) schemes are widely used to detect and track ocean eddies. However, these methods typically do not provide information about the spatial structure of eddies or properties such as vorticity, deformation, and vertical tilt, especially when using sparse data. Here, we describe a new efficient and robust geometrical approach for mapping the three dimensional structure of ocean eddies by fitting (partial) velocity data to a simplified elliptical streamfunction model with a small number of parameters. The flexibility of the approach is demonstrated through three variants of the method adapted to different velocity sampling patterns: single and double sections (from ship transects or a numerical grid) and scattered data (e.g. from surface drifters). We validate and demonstrate these new geometric methods on idealised, axisymmetric and non-axisymmetric Gaussian eddies, as well as numerical and observational datasets. We conclude that elliptical streamfunction parametrisation offers a versatile and effective method for research into ocean eddy characteristics.