the magnitude of the vector r P−r Qand one can determine this distance from the
dot product relation
R^2 = (r P−r Q)·(r P−r Q) (7 .43)
Construct the unit vector ˆeApointing from the point Qto the point Pby expanding
the equation
ˆeA=
r P−r Q
|r P−r Q|=
r P−r Q
R (7 .44)
The unit vector ˆeC which is perpendicular the unit vectors eˆAand ˆeBcan be con-
structed using the cross product
ˆeC=ˆeA׈eB (7 .45)
Note that when the point Pis revolved about the axis of rotation, the circle generated
lies in the plane of the vectors ˆeAand ˆeCand a point on this circle can be described
using the equation
r =r Q+Rcos θˆeA+Rsin θˆeC, 0 ≤θ≤ 2 π (7 .46)
Recall that the point Prepresents a general point on the space curve and so the
vector in equation (7.46) is really a function of the two variables tand θ and one
can express the equation (7.46) as the two parameter surface described by
r =r (t, θ) = r Q+Rcos θˆeA+Rsin θˆeC, 0 ≤θ≤ 2 π (7 .47)
where the vectors r Q,eˆAand ˆeCare all calculated in terms of the parameter tand
can be constructed using the equations (7.40), (7.41), (7.42), (7.43), (7.44), (7.45).
That is, to construct the surface of revolution, one can construct a circle for each
value of the space parameter tvarying between two fixed values, say t 0 ≤t≤t 1.
An alternative method for constructing the circles of revolution of each point P
on the space curve for t 0 ≤t≤t 1 is as follows. First, assume P is fixed and construct
the sphere centered at the point (x 0 , y 0 , z 0 )which passes through the point P. This
sphere has a radius given by r=|r P−r 0 |and this radius is a function of the parameter
t∗ used to describe the point P. The equation of the sphere centered at (x 0 , y 0 , z 0 )
and passing through the point Pis given by
(x−x 0 )^2 + (y−y 0 )^2 + (z−z 0 )^2 =r^2 (7 .48)
