Construction of G2G2 rounded corners with Pythagorean-hodograph curves

• Pythagorean-hodograph (PH) curves are used to round sharp corners.
• G2 continuity is possible with planar PH curves of degree 5 and 7.
• The corners admit exact offset curve and arc length computation.
• The PH curve corner constructions incur only quadratic equations.

The problem of designing smoothly rounded right-angle corners with Pythagorean-hodograph (PH) curves is addressed. A G1G1 corner can be uniquely specified as a single PH cubic segment, closely approximating a circular arc. Similarly, a G2G2 corner can be uniquely constructed with a single PH quintic segment having a unimodal curvature distribution. To obtain G2G2 corners incorporating shape freedoms that permit a fine tuning of the curvature profile, PH curves of degree 7 are required. It is shown that degree 7 PH curves define a one-parameter family of G2G2 corners, facilitating precise control over the extremum of the unimodal curvature distribution, within a certain range of the parameter. As an alternative, a G2G2 corner construction based upon splicing together two PH quintic segments is proposed, that provides two free parameters for shape adjustment. The smooth corner shapes constructed through these schemes can exploit the computational advantages of PH curves, including exact computation of arc length, rational offset curves, and real-time interpolator algorithms for motion control in manufacturing, robotics, inspection, and similar applications.