I very strongly recommend reading Alfred Gray's Tubes:
Gray, Alfred, Tubes, Progress in Mathematics (Boston, Mass.) 221. Basel: BirkhauserBirkhäuser (ISBN 3-7643-6907-8/hbk). xiii, 280~p. (2003). ZBL1048.53040.
(for bonus points, how do I get BirkhauserBirkhäuser to typeset properly?)