隐式曲线、曲面的几何不变量及几何连续性

THE GEOMETRIC INVARIANTS OF IMPLICIT CURVE AND SURFACE AND GEOMETRIC CONTINUITY BETWEEN IMPLICIT SURFACES

首先给出了隐式曲线的曲率计算公式的隐式表达形式，进而给出了隐式曲面的高斯曲率公式以及平均曲率公式的隐式形式．其次利用曲面的几何不变量高斯曲率和平均曲率的连续性得到了一种代数曲面之间的过渡曲面的构造方法，使得过渡曲面也是代数曲面，且能使过渡曲面的次数尽可能低，并且过渡曲面与给定的曲面能达到 Ｇ２ 连续．最后给出一个应用实例．

Algebraic curves/surfaces are typical implicit curves/surfaces. They are used widely in CAD and computer graphics. For example, quadratic curve and quadric surface are used as primitive element in most modeling/CAD system. The implicit formulae for computing the intrinsic geometric invariant——Gaussian and mean curvature of an implicit surface are derived in the first part of this paper. The formulae for them are represented by determinant(s) over a simple term, the elements in which are all the derivatives of order one or two of the surface function. The intrinsic geometric invariant is used to study the geometric continuity between two implicit surfaces and the G 2 continuity conditions for two surfaces along their adjacent bound are derived in the second part of the paper. A G 2 blending surface between two given algebraic surfaces are constructed with Gaussian and mean curvature continuity in the last part of the paper, the blending surface itself is also algebraic surface with minimum order.