Solution In the figure 7-10 the point Prepresents a general point on the space curve.
Let the coordinates of this point be denoted by (x(t∗), y (t∗), z (t∗)) where t 0 ≤t∗≤t 1
and t∗is held constant. Construct the vector r Pfrom the origin to the point P and
construct the position vector r 0 from the origin to the fixed point (x 0 , y 0 , z 0 )on the
axis of rotation. A unit vector in the direction of the axis or rotation is described
by
ˆeB=^1
|b|
b=b^1 ˆe√^1 +b^2 eˆ^2 +b^3 eˆ^3
b^21 +b^22 +b^23
=B 1 ˆe 1 +B 2 ˆe 2 +B 3 ˆe 3 (7 .40)
where ˆeB·ˆeB=B^21 +B^22 +B 32 = 1. Also construct the vector r P−r 0 from the point
(x 0 , y 0 , z 0 )to the point Pas illustrated in the figure 7-10.
Figure 7-10. Space curve revolved about line to form surface of revolution.
Consider a line perpendicular to the axis of rotation and passing through the
point P. Denote by Qthe point where this line intersects the axis of rotation. The
distance sfrom the point (x 0 , y 0 , z 0 )to the point Qis given by the projection of the
vector r P−r 0 onto the unit vector ˆeB. This projection gives the distance
s=ˆeB·(r P−r 0 ) (7 .41)
The point Qcan be described by the position vector
r Q=r 0 +sˆeB (7 .42)
The distance from P to Qrepresents the radius of the circle of revolution when the
point P is revolved about the axis of rotation. This distance, call it R, is given by
