The Ellipse Escape
OK, so the original problem was this - start with an ellipse. Starting at any point within the ellipse, you are allowed to travel in any direction, a distance of X (say, 1.0m). If you extend the major and minor axes of the original ellipse by that same X, does the extended ellipse fully encompass the range of movement? The answer is: no, it is possible to escape the outer ellipse.
In this image, one quadrant of an ellipse is drawn (note that computers draw the "Y-axis" inverted from how mathmaticians draw the "Y-axis," so this image is the first quadrant, but in computer-mode). The original ellipse is drawn in blue. The red circles are all centered on the points of the blue ellipse. The black ellipse is the outer ellipse whose major and minor axes have been extended by the same distance as the radii of the red circles. Notice how the red circles escape the black boundary towards the lower right (upper right of math people) of the ellipse.
Here is my super-fancy-shmancy source code for the image above. Written in 'C' for X11. M$-Windoze programmers are on their own, but this should still be useful...... I guess..... ellipses.c
gcc ellipses.c -L/usr/X11R6/lib -lX11 -lm -o ellipses