11.3.3 Self-intersection of offsets of implicit quadratic surfaces

The second order algebraic surfaces (i.e. quadric surfaces) are widely used in mechanical design. Especially the natural quadrics, i.e. sphere, circular cone and circular cylinder result from machining operations such as rolling, turning, filleting, drilling and milling [149]. The offsets of the natural quadrics are also natural quadrics. Implicit quadrics such as ellipsoids, elliptic cones and elliptic cylinders are commonly found in die cavities and punches and are manufactured by NC machining [60]. Although Salmon [362] discussed the offsets of quadrics more than a century ago, this was not widely known in the CAGD literature until recently. Maekawa [249] showed that self-intersection curves of offsets of all the implicit quadratic surfaces are planar implicit conics and their corresponding curve on the progenitor surface can be expressed as the intersection curve between an ellipsoid, whose semi-axes are proportional to the offset distance, and the implicit quadratic surfaces themselves.

The equations of implicit quadrics including ellipsoids, hyperboloids of one and two sheets, elliptic cones, elliptic cylinders and hyperbolic cylinders can be expressed in a standard form (3.74). In the sequel we assume without loss of generality.

The components , , of the position vector of the offset of implicit surface can be expressed as

where satisfy and , , are the , components of . Note that is zero at the apex of a cone and in this case the normal vector is not defined. Substituting (3.74) into (11.30), (11.31), (11.32) yields

It is obvious from (11.33) to (11.35) that the offsets of implicit quadratic surfaces in a standard form are symmetric with respect to , and -planes.

Self-intersection of offsets of implicit surfaces can be formulated by seeking pairs of distinct points on the progenitor surface such that

Hence using (11.33) to (11.35), the equations for self-intersection reduce to

The self-intersection curve of an offset can be considered as a locus of the center of a sphere, whose radius is the offset distance, rolling on the progenitor surface with two contact points. Because of the symmetry of the offsets of implicit quadratic surfaces, the center of rolling sphere must move only on the planes of symmetry and hence the self-intersection curves are on the planes of symmetry. In other words, a pair of points and on the progenitor surface are located symmetrically with respect to , or -plane and their offsets meet on the , or -plane.

When the offsets self-intersect in the -direction, the self-intersection curve will lie on the -plane. In such case we can set , , and , thus and hence (11.39) reduces to

while (11.40) and (11.41) reduce to identities. Similarly we obtain

for the and -directions respectively. Since all the left hand sides are positive, the right hand sides , and must be also positive. By referring to Table 3.1, we can easily find the offsetting direction (sign of ) to have self-intersection. By squaring both hand sides of (11.42) to (11.44), and summarizing the results of this section we have:

Theorem 11.3.1. The offset of an implicit quadratic surface self-intersects in -direction if the progenitor surface intersects the following ellipsoid provided that the offset distance is taken such that is positive, namely

Similarly the self-intersections in and -directions occur if the progenitor surface intersects the following ellipsoids

provided that and are positive, respectively. The intersection curves between the progenitor surface and each ellipsoid are the foot point curves of the self-intersection curves of offsets [249].

Remark 11.3.1. When one of the coefficients , , is zero, the progenitor surface reduces to either an elliptic cylinder or hyperbolic cylinder. Also the three ellipsoids (11.45), (11.46), (11.47) reduce to two elliptic cylinders.

We will not go into the details of quadric-surface intersection problems, but rather refer to many papers on this problem [233,234,367,104,443,268]. Using (3.74), (11.33) to (11.35) and (11.42) to (11.44), it is easy to show that the self-intersection curves in the , and -directions are implicit conics in the , and -planes given by

Example 11.3.1. Cylindrical surfaces include elliptic cylinders and hyperbolic cylinders. Here we will only examine the hyperbolic cylinder with , and ,

since the rest of the cases for cylindrical surfaces (see Table 3.1) can be derived in a similar way. The curvatures of the hyperbolic cylinder (11.51) based on the curvature sign convention (a) are given in (3.79) and (3.80). For the curvature sign convention (b), we have

where . The extrema of the minimum principal curvature can be computed by using the Lagrange multiplier technique described in Sect. 8.4 (see (8.89), (8.90)), which yields points . The corresponding minimum principal curvature is .

The three ellipsoids in (11.45), (11.46) and (11.47) reduce to the following two elliptic cylinders

Since must be positive to have self-intersection in -direction (see (11.42)), is forced to be negative, while must be positive to have self-intersection in -direction to satisfy (see (11.43)). Now let us consider the self-intersection in -direction which is illustrated in Fig. 11.14 (a). According to Theorem 11.3.1, hyperbolic cylinder (11.51) must intersect the elliptic cylinder (11.52) to have self-intersection in the -direction. It is apparent that these two surfaces will intersect if , since the minor axis of the elliptic cylinder is and hyperbolic cylinder intersects the -axis at . This self-intersection is due to the global distance function property (constriction) of the hyperbola. Similarly the self-intersection in -direction occurs if , which corresponds to the maximum concave radius of curvature as obtained above.

Figures 11.14 (a) (b) show the cross section of self-intersections of offsets of a hyperbolic cylinder (with , ) in -direction with and in -direction with . The thick and thin solid lines represent the hyperbolic cylinder and its offset. The thick dashed dot lines represents the elliptic cylinders (11.52) and (11.53). Four thin dashed lines emanating from the intersections points and intersecting at the self-intersection points of the offset are the vector . The four intersection points between the hyperbolic cylinder (11.51) and elliptic cylinder (11.52) in Fig. 11.14 (a), and the four intersection points between hyperbolic cylinder (11.51) and the elliptic cylinder (11.53) in Fig. 11.14 (b) are given by

Figure 11.14: Cross sections of self-intersecting offsets of a hyperbolic cylinder (adapted from [249]): (a) -direction, (b) -direction

Figure 11.15 shows the self-intersecting offsets of an elliptic cylinder (with , , and ) in the -direction (a) with and in the -direction (b) with . The four intersection points for both cases are given by

Figure 11.15: Cross sections of self-intersecting offsets of an elliptic cylinder (adapted from [249]): (a) -direction, (b) -direction

Example 11.3.2. Consider an ellipsoid (with ) of the form

The curvatures based on the curvature sign convention (a) are given in (3.82) and (3.83). For the curvature sign convention (b) we have

where .

The critical points of both principal curvatures can be obtained by using the Lagrange multiplier technique described in Sect. 8.4 (see (8.89), (8.90)). Since we are assuming , the maximum principal curvature has a global minimum at , a local maximum at and a global maximum at , while the minimum principal curvature has a global minimum at , a local minimum at and a global maximum at . Figure 11.16 shows the locations of extrema (black square), umbilics (white circle), the maximum principal curvature lines (solid line) and the minimum principal curvature lines (dotted line).

It is apparent from (11.42) to (11.44) that must be negative to have self-intersections in the offset. First we consider the case of self-intersection in -direction. The two ellipsoids and will not intersect when is inside , or is inside . This leads to the conclusion that and intersect if

The magnitude of the upper bound corresponds to the smallest concave radius of curvature at , while the magnitude of the lower bound corresponds to the smallest semi-axis. With a similar discussion, we can derive the conditions for the self-intersection in the and -directions as

-direction:
-direction:

Figure 11.17 shows two ellipsoids (with , , ) and ((11.45) with =-0.45) which is equal to the maximum principal radius of curvature at , intersecting each other. This is a degenerate intersection of two ellipsoids, consisting of two ellipses, which have the rational parametrization given by

for . The self-intersection curve of the offset in the -plane is an ellipse given by

which is obtained by substituting into (11.48). Figures 11.18 show the wireframe of the ellipsoid , the intersection curves of two ellipsoids (two ellipses) and pairs of vectors emanating from the intersection curves and intersecting in the -plane from two different view points. The locus of these intersecting points in is the ellipse (11.58).

Figure 11.16: Locations of extrema of principal curvatures (black square), umbilics (white circle) and line of curvatures of ellipsoid (a=0.6, b=0.8, c=1.0) (adapted from [249])

Figure 11.17: Two intersecting ellipsoids (adapted from [249]): intersection curves, which comprise of two ellipses, represent the footpoint curve of the self-intersection curve of offset of ellipsoid

Figure 11.18: Wireframe of the ellipsoid , the intersection curves of two ellipsoids (two ellipses) and pairs of vectors emanating from the intersection curves and intersecting on the self-intersection curves, adapted from [249]: (a) view parallel to the -plane, (b) view parallel to the -plane

Example 11.3.3. Consider an elliptic cone ( , and ) of the form

The curvatures of the elliptic cone (11.59) based on the curvature sign convention (a) are given in (3.85) and (3.86). For the sign convention (b) they are

where except at the apex (0,0,0).

Since the Gaussian curvature is zero everywhere, the elliptic cone is a developable surface and hence the minimum principal curvature lines are in the ruling direction (see Sect. 9.7.1). The maximum principal curvature lines, which are orthogonal to the minimum principal curvature lines, are thus orthogonal to the ruling directions. Therefore as a point on a ruling approaches the apex, the maximum principal curvature monotonically increases and will become infinite at the apex.

It is apparent from (11.42) to (11.44) that the offset of the elliptic cone self-intersects in the and -directions if the offset distance is negative, while it self-intersects in the -direction if the offset distance is positive. Unlike the case for ellipsoids, all the three ellipsoids intersect with the elliptic cone for all nonzero , provided the correct sign is chosen. This observation agrees with the result that the maximum principal curvature has an infinite value at the apex.

Figure 11.19 (a) shows the elliptic cone ( =0.6, =0.8 and =1.0) intersecting the ( =-0.45). The self-intersection curve in the -plane is given by setting and into (11.48)

which is a hyperbola. Figure 11.19 (b) shows the wireframe of the elliptic cone , the intersection curves of and and pairs of vectors emanating from the intersection curves and intersecting in the -plane. The locus of these intersecting points is the hyperbola (11.62).

Theorem 11.3.1 provides a generalized method for obtaining the self-intersection curves of offsets and the corresponding foot point curves on the progenitor implicit quadratic surfaces. The theorem is useful for tool path generation for NC machining and other engineering applications.

Figure 11.19: (a) Intersection between ( , , ) and ( =-0.45), (b) wireframe of the elliptic cone , the intersection curves of and and pairs of vectors emanating from the intersection curves and intersecting on the self-intersection curves (adapted from [249])