When defining a geometric object as a "
locus" of points
based based upon a
metric, it is always interesting to consider what
sorts of figures appear when applying other metrics than the well-known
Pythagorean metric.
For example, when applying the "Manhattan" or taxicab
metric to R2, the locus definition of an "ellipse" always gives
us an octagon. For example, in the following diagram,
. . . . . . . . . . .
. . . . ._._. . . . .
. . . ./. . .\. . . .
. . ./. . . . .\. . .
. ./. . . . o . .\. .
. | . . . . . . . | .
. | . . . . . . . | .
. | . . o . . . . | .
. .\. . . . . . ./. .
. . .\. . . . ./. . .
. . . .\._._./. . . .
. . . . . . . . . . .
The foci are the two o characters. The distance between
the two foci (5) and the two distances from both foci to any point on the
octagonal figure always add up to 16.
These octagonal ellipses have some very odd traits:
-
All four diagonal sides are always the same length.
-
The two vertical sides are always the same length as the "y" part of the distance
between the two foci.
-
The two horizontal sides are always the same length as the "x" part of the distance
between the two foci.
-
Since a circle is an ellipse whose foci are coincident, a "circle" using
our metric is a square turned 45 degrees. Talk about squaring
the circle!
An ellipse defined using the
box metric is still an octagon,
but this time, the horizontal and vertical sides are the same length, and
the diagonal sides grow as the foci move apart from each other. A
"circle" using the box metric looks like a
rectangle. See
if you can draw the ellipse using the same two foci and total distance
as above, but using the box metric instead.
Hint: the foci are
3 units apart this time.
Answer