When a fluid interface is subjected to a strong viscous flow, it tends to develop near-conical ends with pointed tips so sharp, their radius of curvature is undetectable. In microfluidic applications, tips can be made to eject fine jets, from which micron-sized drops can be produced. Here we show theoretically that the opening angle of the conical interface varies on a logarithmic scale as function of the distance from the tip, owing to non-local coupling between the tip and the external flow. Using this insight we are able to show that the tip curvature grows like the exponential of the square of the strength of the external flow, and to calculate the universal shape of the interface near the tip. Our experiments confirm the scaling of the tip curvature as well as of the interface’s universal shape. Our analytical technique, based on an integral over the surface, may also have far wider applications, for example treating problems with electric fields, such as electrosprays.