The Geometry Continuity And Interpolation Method Of Gener Cubic Parametric Spline Curves

The geometric continuity of two regular curves at the connection point can be discussed from differential geometry and Algebra. We utilize the curvature κ, torsion r, Frenet frame in differential geometry. G² continuity is the same as continuity of curvature κ and Frenet Frame,G³ continuity is the same as continuity of curvature κ ,torsion r, derivative of curvature κ¹ and Frenet FrameThis paper considers the possibility about the quadratic geometric continuity of generic cubic parametric spline curve using Algebra. First of all, some excellent results are given. For example, the geometric meaning of the continuity of the G², G³ and some method of paper [5] and [14] .And then,Based on the definition in [3], We proof the equivalent theorem of any order geometric continuity about two regular curves segment at nodes and introduce the created method of generic cubic parametric spline curve. Generic cubic parametric spline curve which is G²,G³ at nodes need to satisfy The three tangent vector equation. We also give the sufficient condition that generic cubic parametric spline curve is G² at node when the tangent vectors at node arc given. Last but not the least,We write the contents in paper [32] and give the example that the figure of generic cubic parametric spline curve is changed when β_i2 is increased,proof P_s"(t) is equivalent at nodes.