Computer Aided Engineering Design

176 COMPUTER AIDED ENGINEERING DESIGN

Case 1. M^2 – LN < 0: For any departure dv from point P, there is no real value of du. This implies
that the tangent plane at P does not intersect the surface at any other point, or P is the only point
common between the tangent plane and the surface and that the surface lies only on one side of the
tangent plane. Point P is called the elliptic point of the surface. All points on an ellipsoid, a sphere,
an elliptic paraboloid and hyperboloid of two sheets are elliptic points.

Case 2.M^2 – LN = 0, L^2 + M^2 + N^2 > 0: There are no double roots and du = – (M/L)dv. For P
(u 0 ,v 0 ) and R (u,v), the result implies u–u 0 = – (M/L) (v–v 0 ) which is the equation of a straight
line in u and v. Thus, the tangent plane intersects the surface along the aforementioned straight
line. P is then called a parabolic point. A cylinder or a truncated (frustrum) cone consists entirely
of parabolic points. All regular points of any developable surface (covered later) are parabolic
points.

Case 3. M^2 – LN > 0: There exist two real roots, and the tangent plane at P intersects the surface along
two lines passing through P. For du = u–u 0 and dv = v–v 0

Figure 6.10 Classification of points on a surface

n

P

(c) Hyperbolic Point (M^2 – LN > 0)

Tangent

plane at P

Tangent plane

SurfaceS

(d) Flat Point (L = M = N = 0)

(a) Elliptic point P (M^2 – LN < 0)

Tangent

plane at P

n

P

(b) Parabolic Point (M^2 – LN = 0)

n

P