5 Conclusions and Further Re- Search

A novel family of curves was described, which allows to construct arc length parameterized curves between two arbitrary points, with given slopes at the curves' two endpoints. A numerical routine for nding the curves was developed and implemented. Future work may include solving more general problems, such as extending the concepts introduced here to three-dimensional curves. References 1] B. A. Barsky and T. D. Rose. Parametric curves tutorial: Geometric continuity of parametric curves: Three equivalent characterizations. 8 the same point loci. We can therefore assume, without loss of generality, that <() = 0 and =() = L. Thus, r 0 (t) = e ii t + iL t ? iL : (46) Integration yields r(t) = e ii (t + 2iL ln(t ? iL)) + C : (47) In the interpolation problem we are given p 0 and p 1 , points in the plane, and tangents' slopes 0 and 1. We are to nd t 0 , t 1 , L, and C such that the following boundary conditions are satissed: r(t 0) = p 0 (48) r(t 1) = p 1 (49)