Classification of the relative positions between a small ellipsoid and an elliptic paraboloid

Application of the Concave Arc Surface Collision Detection Method to Virtual Assembly

“…Fortunately, Brozos-Vázquez et al [20][21] provided a method to detect the positional relationship between ellipsoid and an elliptic paraboloid, which has inspired our work. Furthermore, the splint grooves in the parallel groove clamp can be considered an arc surface due to the design and product.…”

Contact detection between an ellipsoid and a combination of quadrics

“…The work of Wang et al [24] was seminal in introducing this polynomial associated to the pencil of two ellipsoids to detect contact between them. These methods have been extended to other quadric surfaces [2,3] and exploited for practical uses, such as the detection of position for UAVs [4,8]. The analysis of the intersection of quadrics was initiated much earlier (see [18]) and continues to be an active research field (see [15,19,21,22,23,26] and references therein).…”

“…Generally, we consider an ellipsoid E and another quadric surface Q. While previous works as [2,3,24] treated particular quadrics, here we consider a wider class of surfaces. Along this work, the possible quadric surface Q is going to be one of the following: ellipsoid, hyperbolic or elliptic paraboloid, hyperboloid of one or two sheets, elliptic, parabolic or hyperbolic cylinder, or two planes.…”

“…The characteristic roots of P permitted to detect the relative position between two ellipsoids, an ellipsoid and a paraboloid, or an ellipsoid and a hyperboloid of one sheet in [24,2,3] in some instances. In particular, it was shown that if there exists two complex conjugate (non-real) roots of P then the quadric surfaces are in non-tangent contact.…”

“…We will see in Section 3 (Lemma 7) that the possibility of one isolated tangent point and a curve is also eliminated by the smallness condition. A similar definition was first given in [3] to solve the particular problem of an ellipsoid and an elliptic paraboloid, although the condition was slightly more restrictive, since two tangent points were not allowed as a possible intersection set.…”