What is the proper name of the geometrical form of a hot dog.  Clearly
it is not a cyliner, as it has two hemispherical ends on it.  I just
need the name, I do not need much background.

This educational site refers to such a shape as a "cylinder with
hemispheres on both ends":

Mississippi State University
http://grapevine.abe.msstate.edu/~fto/tools/vol/outcylinder.html

Here a technical paper refers to the shape as "a truncated cylinder
capped on each end by hemispheres":

Brooks Air Force Base
http://www.brooks.af.mil/AFRL/HED/hedr/reports/handbook/chp5-1-1.htm

The shape is referred to as a "cylindrical capsule" here:

Nature
http://www.nature.com/nature/journal/v394/n6695/fig_tab/394764a0_F1.html

"Cylindrical capsule" is specifically used to describe the "shape of a
hot dog" in this newsgroup post:

"Alternatively, as described in David Eberly's book '3D Game Engine
Design', it would be easier if you allowed your cylinder to have
hemisphere caps (shape of a hot dog). Since then you only need to
compute the distance between the triangle and the center line segment
of the cylindrical capsule and compare it with the radius of the
capsule."

Post to comp.games.development.programming.algorithms newsgroup
http://groups.google.com/groups?selm=3B6E4992.635205D6%40dtecta.com&oe=UTF-8&output=gplain

I believe the best, most concise descriptive term is "cylindrical
capsule." "Circular cylindrical capsule" would be a more precise term.

Thanks for a most interesting question! I hope this meets your needs.
If anything is unclear, or if a link does not function, please request
clarification; I'll be glad to offer further assistance before you
rate my answer.

Search terms used:

"cylinder" + "hemispheres"
"hot dog" + "shape"
"cylindrical capsule"

Best regards,
pinkfreud

Thanks, a great answer.