The Envelope Of A Family Of Spheres In ??? Corresponding To A Curve In P(L^4,2)

Lie geometry is the study of invariant and invariant properties under the Lie sphere trans-formation, which maps Lie spheres (including oriented hyperspheres, points and hyperplanes) to Lie spheres (including oriented hyperspheres, points and hyperplanes) and preserves orient-ed contact of Lie spheres in RN. The points of Lie quadric P(LN+1,2) are in a one-to-one correspondance with Lie spheres in RN.In this paper, we study the Lie quadric P(L4,2) when N= 3. Firstly, we give a parametriza-tion of P(L4,2) in RA,2 and get the Gauss-Weingarten Equations with respect to (r;r1, r2,r3, r4, n1, over P(L4?2). Then we give an explicit representation of the geodesies in P(L4,2) and envelope of a family of spheres in R3 corresponding to a geodesic of P(L4,2). Specific as follows: Proposition 3.2. Let C be a smooth unit speed curve in P(L4,2) with the parameterized equation:r(s)= (cos ?1 cos ?2, cos?1sin?2, sin?1cos ?3, sin?1sin?3, cos ?4, sin?4)T. ?1,?2,?3,?4 are functions respect to arc-length parameter s. If C is a geodesic, then r= UVa, with Where s = C3s - C4 and ?1, ?2,?3,?, ? and p are constants. Moreover 0???1 and (?,?) lies in the shadow range in Fig 3.1. Theorem 4.1. A geodesic of P(L4,2) corresponds to a family of spheres in R3, whose envelope is a torus.At last, by using analytic geometry we obtain that the envelope of a family of spheres in R3 corresponding to any smooth curve in P(L4,2) is a cyclide and get the parametric equation of this cyclide.