Can a tripod table be placed upon any surface?

Question

When I try to sleep at night, I find myself envisioning topological objects and wondering about the behavior of material in the abstract. Here's my latest quandary​:

Let an equilateral triangle be constrained so that one vertex lies on a surface, at some point. Is there a surface such that the other two points of the triangle cannot also be placed upon the surface?

More intuitively, is there any finite object upon which a pizza saver/tripod table cannot be set to rest?

More precisely, in 3-dimensional Euclidean space, is there a surface with a point at which one of an equilateral triangle's vertices can be placed, while the other two vertices cannot be be simultaneously also placed onto some other points of the surface?

Note that the the plane of the triangle may be pierced by the surface (alternatively, the length of the pizza saver's legs may be arbitrarily long).

Fig. 1 A trivial case. Credit

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Notes:

• The triangle's sides can be made arbitrarily small.

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After more careful thought:

Perhaps what I'm really wondering is What limitations should be set upon this scenario so that the problem is non-trivial?

Answer 1

Yes, there are such surfaces. For example, take the sphere that you pictured, and shrink it down so that the other two vertices are much farther apart than the edges of the sphere. Then, if one vertex has been constrained to lie on the sphere, then the other two can't, because the sphere just isn't big enough.

However, there are further versions of this which ARE true, in a sense. Look up the Wobbly Table Theorem.

Answer 2

Imagine 2 points on the surface, $d$ distance apart. Imagine a circle in the plane perpendicular to the segment joining them, centered at the midpoint of the segment and having radius $\frac{\sqrt{3}}{2}d$. If this circle intersects the surface, the point of intersection is the third point, otherwise we can make $d$ arbitrarily smaller until it intersects.