Use BiArc-curves for contour description of the turbine profiles

В работе рассмотрена возможность использования BiArc-кривых для математического описания турбинных профилей. Приведены элементы теории построения BiArc-кривых. Рассмотрены особенности описания контуров выпуклой и вогнутой частей турбинных профилей с помощью BiArc-кривых. Описан алгоритм автоматического построения контуров турбинных профилей с использованием геометрического критерия качества. Приведены примеры построения турбинных профилей с использованием BiArc-кривых с различными исходными данными.The article analyzes the possibility of using BiArc-curves for the mathematical description of turbine profiles. Confirmed the relevance of the application BiArc-curves in the manufacture of turbine blades. The article presents some of the elements of the theory of building BiArc-curves. The algorithm for determining the coordinates of conjugate points, radii and angles of opening arcs BiArc-curve is shown in the work. Strong influence on the quality of BiArc-curve, provide the coordinates of the point arc connection. Also, the description of certain structural features of convex and concave contours profiles turbines using BiArc-curves. Here the algorithm of automatic construction of a turbine profile using geometric criteria of quality. Optimization problem is solved for each element BiArc-curve using a quadratic fit DSC-Powell. Method of construction of turbine profiles using BiArc-curves programmed in the language c++. Screenshot of dialog boxes and graphics programs can be found in the article. The examples of constructing profiles turbines using BiArc-curves with different initial data. Received confirmation of the possibility to describe the contours of turbine profiles using BiArc-curves

Approximating smooth planar curves by arc splines

AbstractWhen a smooth curve is used to describe the path of a computer-controlled cutting machine, the path is usually approximated by many straight line segments. It is preferable to describe the cutting path as an arc spline, a tangent continuous piecewise curve made of circular arcs and straight line segments. This paper presents an algorithm for finding an arbitrarily close arc spline approximation of a smooth curve

Surface-Surface-Intersection Computation using a Bounding Volume Hierarchy with Osculating Toroidal Patches in the Leaf Nodes

We present an efficient and robust algorithm for computing the intersection curve of two freeform surfaces using a Bounding Volume Hierarchy (BVH), where the leaf nodes contain osculating toroidal patches. The covering of each surface by a union of tightly fitting toroidal patches greatly simplifies the geometric operations involved in the surface-surface-intersection computation, i.e., the bounding of surface normals, the detection of surface binormals, the point projection from one surface to the other surface, and the intersection of local surface patches. Moreover, the hierarchy of simple bounding volumes (such as rectangle-swept spheres) accelerates the geometric search for the potential pairs of surface patches that may generate some curve segments in the surface-surface-intersection. We demonstrate the effectiveness of our approach by using test examples of intersecting two freeform surfaces, including some highly non-trivial examples with tangential intersections. In particular, we test the intersection of two almost identical surfaces, where one surface is obtained from the same surface, using a rotation around a normal line by a smaller and smaller angle θ = 10−k degree, k = 0, · · · , 5. The intersection results are often given as surface subpatches in some highly tangential areas, and even as the whole surface itself, when θ = 0.00001◦

A New Four Point Circular-Invariant Corner-Cutting Subdivision for Curve Design

A 4-point nonlinear corner-cutting subdivision scheme is established. It is induced from a special C-shaped biarc circular spline structure. The scheme is circular-invariant and can be effectively applied to 2-dimensional (2D) data sets that are locally convex. The scheme is also extended adaptively to non-convex data. Explicit examples are demonstrated

Movimientos simétrico lineales esféricos segmentados para interpolación de orientaciones en planificación de trayectorias de herramienta en CNC de 5 Ejes

RESUMEN: Este artículo emplea biarcos cuaterniónicos para interpolar un conjunto de orientaciones con restricciones de velocidad angular. La curva cuaterniónica resultante representa un movimiento simétrico lineal esférico segmentado con continuidad C1 . El propósito de este esfuerzo es poner en uso los movimientos simétrico lineales desde el punto de vista de aproximación e interpolación de movimiento y presentar su potencial aplicación en la simulación de mecanizado por Control Numérico Computarizado (CNC) y planeación de trayectorias de herramienta. Los biarcos cuaterniónicos pueden ser usados para aproximar curvas B-spline cuaterniónicas que representan movimientos esféricos racionales, los cuales tienen aplicaciones en planeación de trayectorias de robots, en CAD/CAM y en gráficas por computador.ABSTRACT: This paper employs quaternion biarcs to interpolate a set of orientations with angular velocity constraints. The resulting quaternion curve represents a piecewise line-symmetric spherical motion with C1 continuity. The purpose of this effort is to put line-symmetric motions into use from the viewpoint of motion approximation and interpolation, and to present their potential applications in Computerized Numerical Control (CNC) machining simulation and tool path planning. Quaternion biarcs may be used to approximate B-spline quaternion curves that represent rational spherical motions that have applications in robot path planning, CAD/CAM and computer graphics

Precise Hausdorff distance computation for freeform surfaces based on computations with osculating toroidal patches

We present an efficient algorithm for computing the precise Hausdorff Distance (HD) between two freeform surfaces. The algorithm is based on a hybrid Bounding Volume Hierarchy (BVH), where osculating toroidal patches (stored in the leaf nodes) provide geometric properties essential for the HD computation in high precision. Intrinsic features from the osculating geometry resolve computational issues in handling the cross-boundary problem for composite surfaces, which leads to the acceleration of HD algorithm with a solution (within machine precision) to the exact HD. The HD computation for general freeform surfaces is discussed, where we focus on the computational issues in handling the local geometry across surface boundaries or around surface corners that appear as the result of gluing multiple patches together in the modeling of generic composite surfaces. We also discuss how to switch from an approximation stage to the final step of computing the precise HD using numerical improvements and confirming the correctness of the HD computation result. The main advantage of our algorithm is in the high precision of HD computation result. As the best cases of the proposed torus-based approach, we also consider the acceleration of HD computation for freeform surfaces of revolution and linear extrusion, where we can support real-time computation even for deformable surfaces. The acceleration is mainly due to a fast biarc approximation to the planar profile curves of the simple surfaces, each generated by rotating or translating a planar curve. We demonstrate the effectiveness of the proposed approach using experimental results