ben green
(Ben Green)
#1
Next construct the plane which passes through the point P and is perpendicular to
the axis of rotation. The equation of this plane is given by
(r −r P)·ˆeB= 0 or (x−x(t∗))B 1 + (y−y(t∗))B 2 + (z−z(t∗))B 3 = 0 (7 .49)
This plane is the plane of rotation of the point P and it intersects the sphere in the
circle described by the point Pas it moves around the axis of rotation. To obtain the
equation for the surface of revolution one must eliminated the parameter t∗from the
equations (7.48) and (7.49). This elimination is not always an easy task to perform.
Ruled Surfaces
A surface r =r (u, v )or z=f(x, y )or F(x, y, z ) = 0 is called a ruled surface if it has
the following property. Through each point on the surface it is possible to draw a
straight line which lies entirely on the surface. For example, consider the set of all
straight lines which pass through a fixed point V and which intersect a fixed curve
C, which is not a straight line through V. The surface generated is called a general
cone with the point V called the vertex of the cone, the curve C being called the
directrix of the cone and the lines on the surface of the cone are called the generating
lines. Some example cones are illustrated in the figure 7-11.
Figure 7-11. A cone is an example of a ruled surface.
Another example of a ruled surface are general cylindrical surfaces which can be
described as a collection of straight lines all parallel to a given direction.
