I have a condition in my program where I have to combine a server (e.g. http://server1.my.corp/) that may or may not have an ending slash with a relative path (
Take a look at the constructors for the Uri class. You need to specify a base Uri and a relative path to get the canonized behavior. Try something like this:
var server = new Uri("http://server1.my.corp/");
var resource = new Uri(server, "/Apps/TestOne/");