Martini Glass solution

Start with a rectangular co-ordinate system tied to the glass, with the origin at the the conical vertex, z being the vertical direction, and +x the direction of tilt. Assume for now that the sides of the glass are 45 degrees off horizontal.

Consider the shape the water makes with and without tilt in this co-ordinate system. What we want is a linear transformation that transforms the upright shape into the tilted shape. There are two hard requirements for such a linear transformation.

1. It must preserve volume.
2. It must respect the conical glass boundary.

If we examine the second requirement, we see that it implies that vectors which are angled 45 degrees off horizontal transform into vectors which preserve that angle, but are perhaps rotated. If we momentarily interpret the z-axis as a time direction, it implies that vectors with some specific speed remain at that speed after transformation. This resembles a postulate of Special Relativity, and immediately suggests the Lorentz transform as a candidate.

Going back to our original interpretation, the Lorentz transform has this matrix for our purposes.

    [ y   0  -yb ]
L = [ 0   1   0  ]
    [-yb  0   y  ]

Here b is some number to be determined in the interval (-1, 1) and y = 1/sqrt(1 - b^2). This transformation respects the conical boundary by virtue of the original Lorentz transformation preserving the space-time interval.  One can also confirm directly that the transformation preserves volume.

One nice thing about this transformation is that the vectors (1, 0, 1) and (-l, 0, 1) are both eigenvectors of L. These represent the directions of maximum and minimum 'height' along the wall of the glass when tilted.

Now consider a vector along the glass wall in the direction of tilt to the untilted surface. The length of this vector is kp, where k is some constant. When we apply transformation L, the length of the transformed vector must be k. Therefore, the eigenvalue of the vector must be 1/p. It can be calculated that this eigenvalue is also equal to sqrt((1+b)/(1-b)), so b must be chosen so that this quantity equals 1/p.

At the same time, the vector that goes to the untilted surface along the glass wall away from the tilt is also an eigenvector with length kp. Its eigenvalue can be calculated to be sqrt((1-b)/(1+b)), which is the reciprocal of the previously calculated eigenvalue. Therefore, the length of the transformed vector is kp^2, and the height up the glass opposite the direction of tilt must be p^2.

If the angle of the glass walls off horizontal is not 45 degrees, one can use a modified transformation M = VL(V^-1), where V is a simple vertical scaling that transforms a 45 degree slope cone to the desired slope. This does not change the result of the calculation.