High accuracy geometric Hermite interpolation

Geometric Interpolation by Quartic Rational Spline Motions

“…In order to obtain a motion of degree 2d, the quaternion can be chosen as a polynomial of degree d with 4(d + 1) coefficients (where one of them can be eliminated by a normalization). The trajectory of the origin, constructed according to (2), then provides 3(2d + 1) = 6d + 3 degrees of freedom.…”

“…As an advantage, they generally require lower polynomial degrees than standard methods. For instance, a polynomial cubic in the plane can match two points with associated tangents and curvatures [2], while the interpolation of two points with associated first and second derivatives needs curves of degree five.…”

On geometric interpolation of circle-like curves

“…Furthermore, the optimal asymptotic approximation order 2n has been confirmed provided the interpolating polynomial curve exists. But its existence for general n has been an open challenge for quite a while since the pioneering work on geometric interpolation has appeared ( [2]). …”

The best uniform quadratic approximation of circular arcs with high accuracy

“…Circular arcs are commonly used in the fields of Computer Aided Geometric Design CAGD, Computer Graphics, and many other applications. Since circular arcs are represented by rational Bézier curves and cannot be represented by polynomial curves in explicit form, circular arc representations using polynomial Bézier curves have been developed by many researchers, see for example [4][5][6][7][8][9][10][11][12][13][14].…”