This is a really nice question. You should start by drawing a wine glass on its side. There are a number of different styles to choose from such as flutes, goblets, sherry glasses, etc...
(Why on its side? The edges can be described with functions.)
If you orient the glass so that the vertical line of symmetry runs along the x-axis it simplifies things.
Here is an example:
(1) x=6 for -1<y<1 This is the bottom of the glass. If the glass is rotated slightly, you could use an ellipse with a vertical major axis.
(2) `y=.25(.5)^x, -2<x<2 ` This is the upper side of the stem.
(3) `y=-.25(.5)^x,-2<x<2 ` This is the lower side of the stem. Note that using the x-axis as the line of symmetry simplifies finding this equation as it is the reflection across the axis. Vertically shifting both equations the same amount will raise the "glass" above the axis if desired.
(4) `y=sqrt(x-2)+1/16,2<x<6 ` This is the upper edge of the "flute".
(5) `y=-sqrt(x-2)-1/16;2<x<6 ` . This is the lower edge of the flute. Again note the use of the reflection.
(6) x=6 for -2<y<2 This is the top of the glass seen edge on. Again, if the glass is slightly rotated you could use an ellipse.
The top and bottom of the glass are lines while the edges are curves. I elected to use exponentials and square roots, but it might work better for certain types of glass to use polynomials, etc...