Biarc Approximation Of Algebraic Curve And Offset Construction

Based on the algorithm for computing singular points and inflection points of algebraic curve an new algorithm for computing discrete singular points of an algebraic surface is proposed with the help of Octree data structure and interval arithmetic. The algorithm guarantee to find all discrete singular points of an algebraic surface with any degree in any given cuboid within any given precision that is allowed by machine precision limit.Based on the above work, we develop a new biarc and offset approximation algorithm of planar algebraic curve. Firstly, we separate the algebraic curve into several parts according to the singular points, inflection points and extreme points, and then we calculate tangents at the two end points of each algebraic curve segment to obtain the intersection point, therefore the control triangle from these three points is formed, we then select the joint of the biarc, In this way the shape of biarc which approximate the original algebraic curve segment is constructed. If the approximation error does not meet the requirement, we bisect the algebraic curve segment at its middle point and repeat the above process until it meets. Finally we calculate the offsets of biarc to approximate the offsets of algebraic curve. Several examples are given to show the effectiveness of the proposed algorithms. The idea of biarc approximation algorithm is quite simple, and biarc is widely used in the industries like NC machining because of its unique qualities. The biarc approximation curve keeps the important qualities of the original algebraic curve such as the monotonicity, concavity and convexity, and Gl continuity. Any approximation accuracy can be achieved by recursive call of the algorithm.