Lecture 1 3
Figure 1.2. Three ellipsoids.
squeezing the sphere along the three coordinate axes produces an
ellipsoid given by the equation
(1.4)
x2
a2
+
y2
b2
+
z2
c2
= 1,
where a, b, and c are parameters which depend on the degree of
stretching or squeezing. Of the three surfaces above, the overall shape
and crude properties of an ellipsoid (its topology) are most similar
to that of a sphere, and are quite different from that of a cylinder
or a cone; its geometry, however, displays many differences from the
geometry of a
sphere.1
For example, the sphere has many symmetries
(that is, rigid motions of the space which leave the sphere as a whole
in place), while a triaxial ellipsoid (one for which all three numbers
a, b, and c in (1.4) are different, such as the third shape shown in
Figure 1.2) has only a few.
Exercise 1.2. Find all the symmetries for
(1) a triaxial ellipsoid;
(2) an ellipsoid of revolution for which a = b = c (such as the
second ellipsoid in Figure 1.2).
Consider separately the symmetries which can be effected by a contin-
uous motion of the space and those which cannot, such as reflections
with respect to planes.
1For
the time being, we rely on intuitive ideas of what constitutes a general shape.
For a reader steeped in mathematical rigor, we refer to notions of homeomorphism and
diffeomorphism, which will be introduced later in Lectures 4 and 17, respectively, and
say that two surfaces have similar shapes if they are homeomorphic, or diffeomorphic
in the case of smooth surfaces.
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