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squeezing Rn∗
pahio†
2013-03-21 23:02:49
Squeezing the vector space Rn in the direction of one coordinate axis, i.e.
multiplying a certain component xi of all vectors by a non-zero real number k,
is a linear transformation of Rn .
A concrete example of such squeezing and its results is obtained if we squeeze
in R2 , i.e. in the Euclidean plane formed by all pairs (x, y) of real numbers,
every y-coordinate by a positive number k = ab where a > b > 0. We may look
how this procedure acts on the circle
x2 + y 2 = a2 .
(1)
b
which is less
a
than 1, we must must multiply the new y in equation (1) by the inverse number
a
in order to keep the equation satisfied; then the new y no longer denotes the
b
ordinate of the circle, but rather the ordinate of the squeezed circle. Thus, the
equation of the squeezed curve is
Since all ordinates of this equation are shrinked by the factor
x2 +
a
b
·y
2
= a2 .
Simplifying we first obtain
x2 +
a2 y 2
= a2 ,
b2
and dividing all terms by a2 yields
x2
y2
+
= 1.
a2
b2
(2)
So the resulting curve is an ellipse with semiaxes a and b.
∗ hSqueezingmathbbRni created: h2013-03-21i by: hpahioi version: h39479i Privacy setting: h1i hTopici h15-00i h15A04i
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1
In the picture below, the circle x2 +y 2 = a2 is drawn in red and the ellipse
y2
x
+
= 1 in blue. The angle t is the eccentric anomaly at the point P of
a2
b2
the ellipse, which has the parametric presentation x = a cos t, y = b sin t.
y
2
a
b
P
t
a
x
Squeezing R3 in the directions of the y-axis and z-axis one can get from the
sphere
x2 +y 2 +z 2 = a2
the ellipsoid
x2
y2
z2
+ 2 + 2 = 1.
2
a
b
c
2