Double Tangent Cubic Splines

Abstract

We may construct a more general class of cubic space curve splines called double tangent splines by introducing two tangent vectors at each point p _i . Let m₁,…, m _n be the exit tangent vectors at p₁,…, p _n , and let l₁,…,l _n be the entry tangent vectors at p₁,…, p _n . We intend that our double tangent spline will asymptotically enter the point p _i along the line parallel to the vector l _i and exit from the point p _i , along the line parallel to the vector m _i . Now we define the double tangent cubic spline space curve segment x _i which defines the spline between p _i and p_i+1 as follows:

$$\eqalign{ & {x_i}(t) = {a_i} + {b_i}t + {c_i}{t^2} + {d_i}{t^3},{\rm{where}} \cr & {a_i} = {p_i}, \cr & {b_i} = {m_i}, \cr & {c_i} = 3({p_{i + 1}} - {p_i})/{t_i}^2 - (2{m_i} + {l_{i + 1}})/{t_i}, \cr & {d_i} = 2({p_i} - {p_{i + 1}})/{t_i}^3 + ({m_i} + {l_{i + 1}})/{t_i}^2 \cr} $$