Many applications of geometry modeling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a smooth surface. We show that, when a geodesic polygon P is closely inscribed on a $C^2$-regular curve, the discrete geodesic curvature of P estimates the geodesic curvature of C. This result allows us to evaluate the geodesic curvature of discrete curves on surfaces. In particular, we apply such result to planar and spherical 4-point angle-based subdivision schemes. We show that such schemes cannot generate in general $G^2$-continuous curves. We also give a novel example of $G^2$-continuous subdivision scheme on the unit sphere using only points and discrete geodesic curvature called curvature-based 6-point spherical scheme.
翻译:几何建模和计算机图形的许多应用必然对平面或方块上的曲线进行精确的曲度估计。 在本文中, 我们定义了平滑表面的大地测量多边形离地的曲度概念。 我们显示, 当一个大地测量多边形P 紧紧地刻在$C$2$-正则曲线上时, 离地测量模型和计算机图形图形的曲度参数是P 估计C 的大地测量曲度的离地测算。 这个结果使我们能够评估表面离地曲线的大地测量曲度。 特别是, 我们将这种结果应用到平面和球四点角角的角子剖面图中。 我们显示, 这样的方案无法产生一般的$G$2$- 持续曲线。 我们还举了一个新颖的例子, 仅使用点和离地地质判分的被称为基于曲线的6点的外观组合。