The generalization of this lemma permits opening of the angles beyond , as far reflex as they were originally convex. The conclusion remains the same: e cannot shorten.
This theorem can be derived from Chern's proof1 of a theorem of Axel Schur2, employing differential geometry, or, independently, by induction [O'R00].
The turn angle ranges are displayed in green, and the "forbidden shoulder circle" is drawn in blue. The user may then turn any joint of the chain by double-clicking on it and dragging the next link:
The theorem says that no reconfiguration within the allowable turn angle ranges will permit the hand to enter the forbidden circle:
A corollary says that every joint has a correspoding forbidden circle, which can be viewed by selecting:View all concentric circles
For more details, see the paper [O'R00] or view Help on applet.
|Internet Explorer||Netscape Communicator|
|PC||Windows 9x, WindowsNT4||IE v4: No reported problems on above resolution.||NC v4.7: No reported problems on above resolution.|
|Red Hat Linux 6.1||Not tested.||Netscape v4.61 for Linux. No reported problems.|
|MAC||MacOS 8.5||IE v4: No reported problems at above resolutions.||Some Mac versions of Netscape do not support the Java1.2 event model, which is used extensively on the applet. Therefore, IE is recommended for the Mac|
|SGI||Irix 6.x||Not tested.||Tested on Netscape 4.07. Reported problems are: