Index: Lib/ntpath.py
===================================================================
--- Lib/ntpath.py	(revision 85168)
+++ Lib/ntpath.py	(working copy)
@@ -616,7 +616,7 @@
     path_abs = abspath(normpath(path))
     start_drive, start_rest = splitdrive(start_abs)
     path_drive, path_rest = splitdrive(path_abs)
-    if start_drive != path_drive:
+    if normcase(start_drive) != normcase(path_drive):
         error = "path is on mount '{0}', start on mount '{1}'".format(
             path_drive, start_drive)
         raise ValueError(error)
@@ -626,7 +626,7 @@
     # Work out how much of the filepath is shared by start and path.
     i = 0
     for e1, e2 in zip(start_list, path_list):
-        if e1 != e2:
+        if normcase(e1) != normcase(e2):
             break
         i += 1
 
Index: Lib/posixpath.py
===================================================================
--- Lib/posixpath.py	(revision 85168)
+++ Lib/posixpath.py	(working copy)
@@ -447,8 +447,8 @@
     if start is None:
         start = curdir
 
-    start_list = abspath(start).split(sep)
-    path_list = abspath(path).split(sep)
+    start_list = [x for x in abspath(start).split(sep) if x]
+    path_list = [x for x in abspath(path).split(sep) if x]
 
     # Work out how much of the filepath is shared by start and path.
     i = len(commonprefix([start_list, path_list]))
Index: Lib/test/test_ntpath.py
===================================================================
--- Lib/test/test_ntpath.py	(revision 85168)
+++ Lib/test/test_ntpath.py	(working copy)
@@ -238,6 +238,7 @@
         tester('ntpath.relpath("/", "/")', '.')
         tester('ntpath.relpath("/a", "/a")', '.')
         tester('ntpath.relpath("/a/b", "/a/b")', '.')
+        tester('ntpath.relpath("c:/foo", "C:/FOO")', '.')
 
     def test_sameopenfile(self):
         with TemporaryFile() as tf1, TemporaryFile() as tf2:
Index: Lib/test/test_posixpath.py
===================================================================
--- Lib/test/test_posixpath.py	(revision 85168)
+++ Lib/test/test_posixpath.py	(working copy)
@@ -394,6 +394,15 @@
                              "../"+curdir+"/a/b")
             self.assertEqual(posixpath.relpath("a", "b/c"), "../../a")
             self.assertEqual(posixpath.relpath("a", "a"), ".")
+            self.assertEqual(posixpath.relpath("/foo/bar/bat", "/x/y/z"), '../../../foo/bar/bat')
+            self.assertEqual(posixpath.relpath("/foo/bar/bat", "/foo/bar"), 'bat')
+            self.assertEqual(posixpath.relpath("/foo/bar/bat", "/"), 'foo/bar/bat')
+            self.assertEqual(posixpath.relpath("/", "/foo/bar/bat"), '../../..')
+            self.assertEqual(posixpath.relpath("/foo/bar/bat", "/x"), '../foo/bar/bat')
+            self.assertEqual(posixpath.relpath("/x", "/foo/bar/bat"), '../../../x')
+            self.assertEqual(posixpath.relpath("/", "/"), '.')
+            self.assertEqual(posixpath.relpath("/a", "/a"), '.')
+            self.assertEqual(posixpath.relpath("/a/b", "/a/b"), '.')
         finally:
             os.getcwd = real_getcwd
 
@@ -412,6 +421,15 @@
                              b"../"+curdir+b"/a/b")
             self.assertEqual(posixpath.relpath(b"a", b"b/c"), b"../../a")
             self.assertEqual(posixpath.relpath(b"a", b"a"), b".")
+            self.assertEqual(posixpath.relpath(b"/foo/bar/bat", b"/x/y/z"), b'../../../foo/bar/bat')
+            self.assertEqual(posixpath.relpath(b"/foo/bar/bat", b"/foo/bar"), b'bat')
+            self.assertEqual(posixpath.relpath(b"/foo/bar/bat", b"/"), b'foo/bar/bat')
+            self.assertEqual(posixpath.relpath(b"/", b"/foo/bar/bat"), b'../../..')
+            self.assertEqual(posixpath.relpath(b"/foo/bar/bat", b"/x"), b'../foo/bar/bat')
+            self.assertEqual(posixpath.relpath(b"/x", b"/foo/bar/bat"), b'../../../x')
+            self.assertEqual(posixpath.relpath(b"/", b"/"), b'.')
+            self.assertEqual(posixpath.relpath(b"/a", b"/a"), b'.')
+            self.assertEqual(posixpath.relpath(b"/a/b", b"/a/b"), b'.')
 
             self.assertRaises(TypeError, posixpath.relpath, b"bytes", "str")
             self.assertRaises(TypeError, posixpath.relpath, "str", b"bytes")