Implementation of virtual ruler for interactive object design system

基于三维数据采集系统的研究结果表明：曲面可以根据高斯曲率或平均曲率进行分类，而最大曲率K1和最小曲率K2分别是较好的分类指标。基于K1和K2的符号分类表现出曲面在面积大小上按凸面、超曲面和凹面的顺序出现的特征。如果K1和K2接近零，通过引入圆柱曲面，适当的阈值可能有助于极大地减少三角面片的数量。对于应该由虚拟尺子生成的曲线，进一步的结果是：摆线曲线的曲率变化是线性的，适合作为设计目的的导向曲线，但对于曲线一般不通过任意给定的三点有很大的限制。我们不可避免地要使用复合曲线，这使得设计Pr…更多的尴尬。贝塞尔(子)段可能很容易被虚拟标尺利用，但由于它们的曲率发生不必要的变化，所获得的段不能直接用于设计目的。作为回旋线和贝塞尔曲线之间的折衷曲线，贝塞尔螺线是有用的。这条曲线最适合于虚拟标尺曲线。为了实现与广泛使用的样条尺的一致性，将贝塞尔螺线应用于设计时，阶数应为3(或控制点的数量应为4)。如果将第二控制点设置在第一和第三控制点之间的中间位置，则可以解析得到存在贝塞尔螺线的区域。如果第二控制点在第一和第三控制点之间(一般不在中间点)，则可以数值推导出可能的贝塞尔螺线。这些螺旋线是虚拟标尺的目标曲线，这些结果产生了应该在交互设计工作中使用的虚拟标尺的原型。较少

The investigation based on the 3D data acquisition system brought the following results :Curved surfaces might be classified depending on the Gaussian or Mean curvature, while the maximum and minimum curvatures, k1 and k2, respectively, serves as the better index for the classification.The classification based on the sign of k1 and k2 shows the characteristics that the convex, hyper and concave surfaces appear in this order regarding the size of areas.If k1 and k2 are closed to zero, appropriate threshold value might help greatly reduce the number of triangular patches by introducing the cylindrical surface.Further results are regarding on the curve that should be generated by our virtual rulers :The curvature change of Clothoid curves is linear and appropriate as the guide curves for the design purpose, while there is heavy restriction that the curves do not in general pass through the arbitrarily given three points. We inevitably have to use composite curves, this makes the design pr … More ocess awkward.Bezier (sub) segment may be easily utilized by a virtual ruler, the obtained segment could not be used directly for the design purpose due to their unwanted change in their curvature.As the trade off curve between Clothoid and Bezier curves, Bezier spiral is useful. This curve is the best for virtual ruler curves.To apply Bezier spirals for the design, the order should be three (or the number of control points should be four) to realize the consistency with the widely used spline rulers.If we set the second control point in the middle position between the first and the third control points, the area where Bezier spiral exist may be analytically obtained.If the second control point is between the first and the third control points (not in the middle point in general) the possible Bezier spiral may be numerically derived. These spirals are the target curves by virtual rulers.These results yield the prototype of the virtual ruler that should be used in interactive design work. Less