Out of My Depth

You are in a pitch-black room. But don't be alarmed.

Someone is pouring liquid at a slow, uniform speed into a thin, transparent vessel; it is pooling at the bottom. The fluid itself is dark, but when in direct contact with the atmosphere it glows purple. So what you see is precisely the evolving shape of the fluid's surface. It seems that you stand to learn something: the shape of the inside of the vessel will be gradually revealed in cross-sections.

But wait--this seems to rely on the assumption that the vessel itself is not moving. Suppose, instead, that the vessel is almost perfectly weightless, and may roll as the heavy liquid's center of gravity shifts.

Can you reconstruct the vessel's shape in this case?

I'm curious to know, but probably underequipped to solve this question, which was the result of a momentary hallucination during a long concert. There are some potential messy issues with cusps, and questions of whether the vessel is open at the top, etc., to which I advise that one assumes whatever makes the problem interesting or tractable.