4.1 Geometry 221yz= 2 λx
a^2,
xz= 2 λ
y
b^2,
yz= 2 λz
c^2,
x^2
a^2+
y^2
b^2+
z^2
c^2= 1.
Multiplying the first equation byx, the second byy, and the third byz, then summing up
the three equations gives
3 xyz= 2 λ(
x^2
a^2+
y^2
b^2+
z^2
c^2)
= 2 λ.Henceλ=^32 xyz. Then multiplying the first three equations of the system together, we
obtain
(xyz)^2 = 8 λ^3
xyz
a^2 b^2 c^2=
27 (xyz)^4
a^2 b^2 c^2.
The solutionxyz=0 we exclude, since it does not yield a maximum or a minimum.
Otherwise,xyz=±√abc 27. The equality with the plus sign is the maximum off; the other
is the minimum. Substituting in the formula for the volume, we find that the smallest
volume is
√ 3
2 abc. Example.Find the nature of the surface defined as the locus of the lines parallel to a given
plane and intersecting two given skew lines, neither of which is parallel to the plane.
Solution.We will work in affine coordinates. Call the planeπand the two skew linesl 1
andl 2. Thex- andy-axes lie inπand thez-axis isl 1. Thex-axis passes throughl 2 ∩π.
They-axis is chosen to makel 2 parallel to theyz-plane. Finally, the orientation and the
units are such thatl 2 is given byx=1,y=z(see Figure 29).
A line parallel toπand intersectingl 1 andl 2 passes through( 1 ,s,s)and( 0 , 0 ,s),
wheresis some real parameter playing the role of the “height.’’ Thus the locus consists
of all points of the formt( 1 ,s,s)+( 1 −t)( 0 , 0 ,s), wheresandtare real parameters.
The coordinates(X,Y,Z)of such a point satisfyX=t,Y=ts,Z=s. By elimination
we obtain the equationXZ=Y, which is a hyperbolic paraboloid like the one from
Figure 30. We stress once more that the type of a quadric is invariant under affine
transformations.
A surface generated by a moving line is called a ruled surface. Ruled surfaces are
easy to build in real life. This together with its structural resistance makes the hyperbolic
paraboloid popular as a roof in modern architecture (see for example Felix Candela’s roof