Hermite spline curves



              
Hermite polynomials of degree 3 that form a basis for Hermite curves, see the formulas from expression (4.19) in Blending techniques in Curve and Surface constructions

A Hermite curve is expressed by the following formula:  \( C(t) = \displaystyle\sum_{i=0}^{3} H_{i}(t)\ c_i \)  where the polynomial degree is 3, and where,   \(c_0 = C(0)\), \(c_1 = C(1)\)  and  \(c_2 = C'(0)\), \(c_3 = C'(1)\), and where \(H_{0}(t) = 2t^3-3t^2+1\)\(H_{1}(t) = -2t^3+3t^2\)\(H_{2}(t) = t^3-2t^2+t\)  and  \(H_{3}(t) = t^3-t^2\)  are Hermite basis functions. In the figure above, a set of 4 3rd degree Hermite basis functions is plotted. Note that the parameter domain is \([0,\ 1]\), but can be anything else, but then with a reparameterization.
In the figure below, we have one set of \(n+1\) control points (grey pentagons) with associated 1st derivatives (red vectors). The number \(n\) represents the number of curve intervals and can be set to one of the numbers \(\{1, 2, 3, 4, 5, 6, 7 \} \) in the box at the top right. In each interval it then plotted a Hermite curve from the points at the two ends and with the associated 1st derivatives. This construction is called a Hermite spline.

The plot is programmed according to chapter 5.6 in the book.

A Hermite spline curve is a connected sequence of Hermite curves, with a common point and 1st derivatives at the knots.

To create a Hermite spline curve, we need a knot vector (cf. B-splines og Hermite interpolation from divided differences).  In the curve above, a uniform knot vector   \(\mathbf{t} = \{t_0, t_1, t_2, t_3, ... \} = \{0, 1, 2, 3, ... \}\) has been used.

The formula for each interval \(i\) er: \( c_i(t) = c(t_i) H_{i,0}(t)+c(t_{i+1}) H_{i,1}(t)+ c'(t_i) H_{i,2}(t)+c'(t_{i+1}) H_{i,3}(t) \)  where:    \(
\begin{array}[c]{lll}
H_{i,0}(t) & \hspace{-7pt} = \hspace{-7pt} & 2w_{i}(t)^3-3w_{i}(t)^2+1, \\
H_{i,1}(t) & \hspace{-7pt} = \hspace{-7pt} & -2w_{i}(t)^3+3w_{i}(t)^2, \\
H_{i,2}(t) & \hspace{-7pt} = \hspace{-7pt} & \Delta t_i\left(w_{i}(t)^3-2w_{i}(t)^2 + w_{i}(t)\right), \\
H_{i,3}(t) & \hspace{-7pt} = \hspace{-7pt} & \Delta t_i\left(w_{i}(t)^3-w_{i}(t)^2\right),
\end{array}
\)  ((5.13) i boka)  where \( w_{i}(t)= \frac{t - t_i}{t_{i+1} - t_i}\)   and  \( \Delta t_i  =  t_{i+1} - t_i  = \frac{1}{w'_i}\).   Furthermore, the derivatives are:   \( \begin{array}[c]{lll}
H'_{i,0}(t) & \hspace{-7pt} = \hspace{-7pt} & 6\frac{(w_{i}(t)^2-w_{i}(t))}{\Delta t_i}, \\
H'_{i,1}(t) & \hspace{-7pt} = \hspace{-7pt} & -6\frac{(w_{i}(t)^2-w_{i}(t))}{\Delta t_i}, \\
H'_{i,2}(t) & \hspace{-7pt} = \hspace{-7pt} & 3w_{i}(t)^2-4w_{i}(t) + 1, \\
H'_{i,3}(t) & \hspace{-7pt} = \hspace{-7pt} & 3w_{i}(t)^2-2w_{i}(t).
\end{array}
\)

If we look at the Hermite basis of degree 3 over the domain \( [0, 1]\)  we have:  \(2t^3 -3t^2 +1\),    \( -2t^3 + 3t^2\),     \(t^3-2t^2+t\)    and    \(t^3 -t^2\). Arranged in matrix form, the Hermite polynomials are:  \( \begin{bmatrix} 1 & 0 & -3 & 2\\ 0 & 0 & 3 & -2\\ 0 & 1 & -2 & 1\\ 0 & 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} 1 \\  t\\ t^2 \\ t^3 \end{bmatrix} \).  This matrix is then the basis change matrix from a monominal basis of degree 3 to the Hermite basis of degree 3. See table 4.1 in the blending book.  table 4.1 in the book.