Approximating bounded, non-orientable surfaces from points

We present an approach to surface approximation from points that allows reconstructing surfaces with boundaries, including globally non-orientable surfaces. The surface is defined implicitly using directions of weighted co-variances and weighted averages of the points. Specifically, a point belongs to the surface, if its direction to the weighted average has no component into the direction of smallest co-variance. For bounded surfaces, we require in addition that any point on the surface is close to the weighted average of the input points. We compare this definition to alternatives and discuss the details and parameter choices. Points on the surface can be determined by intersection computations. We show that the computation is local and, therefore, no globally consistent orientation of normals is needed. Continuity of the surfaces is not affected by the particular choice of local orientation. We demonstrate our approach by rendering several bounded (and non-orientable) surfaces using ray casting.