66 Vectors and geometry in three dimensions
are determined by P and P as in the figure, we see that(1) IQPI =IQP'I =b+acos6
and that(2) OP = I QPI (cos 451 + sin 4)j)
so(3) QP = (b + a cos B) cos 4)i + (b + a cos B) sin 4)j.
Letting r be the vector running from the origin to the point P on T, we see that(4) r = QP +OQ= QP + a sin ok
and hence that(5) r = (b + a cos 0) cos 4,i + (b + a cos 0) sin 4)j + a sinBk.Thus a vector r having its tail at the origin has its tip on the torus T if and only if
there exist angles 0 and B for which (5) holds. Thus (5) is a vector equation of the
torus. This implies that P(x,y,z) lies on the torus T if and only if there exist
angles 4) and 0 such that(6) x(b+acos0)cos4), y=(b+acos0)sin 4), z = a sin 0.
These are parametric equations of the torus, the parameters being ¢ and B.
The torus is the graph of the parametric equations.
23 Show that the x, y, z equation of the torus of the preceding problem can
be put in one or the other of the two equivalent forms(1) (a2 + b2 - x2 - Y2 - z2)2 = 4b2(a2 - z2)
(2) (b2 - a2 + x2 + y2 + a2)2 = 4b2(x2 + y2).
Hint: One way to start is to square and add the first two of the equations (6)
of the preceding problem. Remark: In case 0 < b 5 a, the equations of this
and the preceding problem are not equations of a torus but they are equations
of a surface.
24 Sketch the visible edges of the solid that remains when a cube having
edges of length a/2 is removed from the upper right front corner of a cube having
edges of length a. Remark: This figure can easily become a favorite doodle,
and perfectly normal people can become very much interested in it.
25 Mr. T., a topologist, claims that a given right-handed rectangular coordi-
nate system in Ea can, when considered to consist of three stiff wires rigidly
welded together at their origin, be moved around in Ea in such a way that it will
coincide with any other such system but cannot be similarly moved into coinci-
dence with a left-handed system. It is not always easy to understand such
assertions and'to give proofs of them. It is sometimes easy to wave some arms
around and give some more or less convincing arguments that could not be called
proofs. The difficulty here is that those who wave arms and produce more or
less convincing arguments sometimes reach erroneous conclusions. Anyone who
wishes to do so may think about this matter.