Hermite 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.

In the menu at the top right, you can determine the number of intervals \(n\).
In the figure below, you can move the control points by pressing the left mouse button when the cursor is over one of the gray pentagons that represent a control point. The pentagon moves as long as you hold the button down. When you release the button, the pentagon is locked and the curve is updated. The 1st derivative arrow follows the control point.
You can change the 1st derivative by pressing the left mouse button when the cursor is over one of the red arrowheads. The tip of the red arrow moves as long as you hold the button down. When you release the button, the arrow (1st derivative) is locked and the curve is updated.

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.

Hermite spline curves