Cubic Hermite Splines in Interpolation

• Cubic Hermite splines are piecewise cubic C^1 interpolants that use both function and derivative values to ensure smooth, tangent-controlled curves.
• They provide optimal local support and precise polynomial reproduction, making them effective for geometric modeling and numerical simulation.
• Enhanced variants employ energy minimization, monotonicity limiters, and area-preserving formulations to improve interpolation accuracy in practical applications.

A cubic Hermite spline is a piecewise polynomial interpolant defined by both function values and derivative values (tangents) at each nodal point, with each segment represented as a cubic polynomial on its subinterval. Distinct from B-splines or Lagrange interpolants, Hermite splines provide explicit interpolation of both values and first derivatives, resulting in $C^1$ (continuously differentiable) global curves. Their canonical form is widely used in geometric modeling, computer graphics, and interpolation tasks where tangent control is essential.

1. Mathematical Formulation and Basis Construction

Given an ordered sequence of nodes $\{x_i\}_{i=0}^n$ with associated values $\{y_i\}$ and derivatives $\{m_i\}$, the standard cubic Hermite interpolant $S(x)$ is constructed piecewise on each interval $[x_i, x_{i+1}]$ as follows. Using the local variable $t = \frac{x-x_i}{h_i}$ with $h_i = x_{i+1} - x_i$,

$$S_i(x) = h_{00}(t) y_i + h_{10}(t) m_i h_i + h_{01}(t) y_{i+1} + h_{11}(t) m_{i+1} h_i,$$

where the basis functions are: $$\begin{aligned} h_{00}(t) &= 2t^3-3t^2+1, \ h_{10}(t) &= t^3-2t^2+t, \ h_{01}(t) &= -2t^3+3t^2, \ h_{11}(t) &= t^3-t^2. \end{aligned}$$ This construction ensures $C^1$ continuity at all knots, with $S_i(x_i) = y_i$, $S_i'(x_i) = m_i$, $S_i(x_{i+1}) = y_{i+1}$, and $S_i'(x_{i+1}) = m_{i+1}$ (Aràndiga et al., 2021, Benoit, 2010).

For cardinal cubic Hermite splines on a uniform ($T$-spaced) integer grid, the generators $\{\phi_{1}(t),\phi_{2}(t)\}$ satisfy compact support and minimality: $\phi_{1}$ is even, interpolates values, and $\phi_{2}$ is odd, interpolates slopes, both vanishing outside $[-1,1]$ (Fageot et al., 2019).

2. Support, Reproduction, and Approximation Properties

Hermite splines achieve minimal joint support among all two-generator schemes that reproduce both quadratic and cubic B-splines: $$|\mathrm{supp} \ \phi_1| + |\mathrm{supp} \ \phi_2| = 4$$, attaining the theoretical lower bound (Fageot et al., 2019). Polynomial reproduction formulas guarantee that any cubic polynomial $P(t)$ can be exactly represented by

$$P(t) = \sum_{k \in \mathbb{Z}} \left(P(k)\phi_1(t-k) + P'(k)\phi_2(t-k)\right).$$

Approximation powers are quantified in Sobolev spaces: $$\|f - \tilde{f}_T\|_{L^2} \sim \frac{1}{72 \sqrt{70}} \|f^{(4)}\|_{L^2} T^4, \qquad \|f' - \tilde{f}_T'\|_{L^2} \sim \frac{1}{12 \sqrt{210}} \|f^{(3)}\|_{L^2} T^3,$$ demonstrating fourth-order $L^2$ convergence for function values and third-order for derivatives, asymptotically matching cubic B-spline interpolation (Fageot et al., 2019).

3. Tangent Selection and Energy Minimization

The choice of nodal derivatives (tangents $m_i$) critically determines spline smoothness, shape, and physical properties. Minimum-energy schemes optimize tangents to minimize the integral of squared curvature, suppressing oscillations and spurious extrema. For a sequence of control points $\{P_j\}$, the tangent at $P_i$ is constructed by embedding $P_{i-1}, P_i, P_{i+1}$ in a unique quadratic interpolant $r(t)$ through the three points, with internal parameter $T_i$ chosen so that

$E(T) = \int_0^1 \kappa(t)^2 \, dt$

is minimized, where $\kappa(t)$ is the curvature. The resulting $V_i = \dot{r}_i(T_i)$ is then used as the Hermite tangent at $P_i$ for both adjacent segments. This approach guarantees $G^1$ continuous, globally low-energy cubic Hermite splines, with segment energies often half or less compared to Catmull–Rom, Cardinal, and Kochanek–Bartels splines (Benoit, 2010).

Method	Energy $E$	Curvature Variation $V$
Minimal-energy quadratic	$\approx 6.8$	$\approx 4.7 \times 10^2$
Catmull–Rom	$\approx 13.5$	$\approx 1.7 \times 10^3$
Cardinal ($\tau=0.5$)	$\approx 71.5$	$\approx 4.1 \times 10^4$

4. Monotonicity and Limiter Algorithms

Cubic Hermite splines do not intrinsically guarantee monotonicity of interpolated data, leading to overshoots (Gibbs phenomena) near discontinuities or strong gradients. Monotonicity-preserving conditions, such as the Fritsch–Carlson criteria, require scaled nodal derivatives (relative to data-slopes $\Delta_i = (y_{i+1}-y_i)/h_i$) to satisfy $0 \leq \alpha_i, \beta_i \leq 3$ for all intervals. Violations are corrected using nonlinear “limiter” formulas:

• Fritsch–Butland: $O(h^2)$, simple averages
• Brodlie (PCHIP): $O(h^3)$ on uniform meshes
• Arndiga–Yáñez: $O(h^3)$ even on nonuniform meshes

Implementation can follow “maximal-order” (only adjust violating nodes) or “maximal-regularity” (split and re-solve spline) strategies, producing robust monotone interpolants, with the Arndiga–Yáñez limiter providing superior nonuniform grid accuracy (Aràndiga et al., 2021).

5. Area-Preserving and Geometric Variants

Area-preserving cubic Hermite interpolation sets additional constraints so that the signed area between the interpolant and the chord matches a prescribed value (e.g., to conserve physical quantities in PDEs). The construction employs a Bernstein-Bézier representation: $$B(t) = P_0 (1-t)^3 + 3 (P_0 + r_1 \alpha/3) (1-t)^2 t + 3 (P_1 - r_2 \beta/3) (1-t) t^2 + P_1 t^3,$$ with endpoint derivatives in directions $\alpha$, $\beta$ but unknown magnitudes $r_1, r_2$. These are determined by a quadratic constraint stemming from the area preservation condition. The resulting error order is $O(h^5)$ (one order higher than standard) when endpoint curvature does not vanish; otherwise $O(h^4)$. Optimization over $(r_1, r_2)$ minimizes the prefactor of the error, improving accuracy without sacrificing the area constraint (McGregor et al., 2018).

6. Computational Realizations and Quasi-Interpolation

Cubic Hermite quasi-interpolation operators combine Hermite data with B-spline bases to construct $C^1$ piecewise-cubic splines, even when derivatives are only approximately known. For function $f$ sampled at nodes, the operator $Q_3^{(BS)}(f)$ is

$$Q_3^{(BS)}(f)(x) = \sum_j \mu_j^{(BS)}(f)\, B_j(x),$$

with each $\mu_j$ a local combination of function and derivative values. If derivatives are not given, they are approximated using finite-difference schemes, with global spline accuracy depending on the order of difference used. Tensor-product extensions generalize the method to surfaces and volumes. The open-source library QIBSH++ implements these algorithms in C++ and Matlab/MEX, delivering efficient construction, evaluation, and derivative computation for high-dimensional data (Bertolazzi et al., 2022).

7. Applications, Comparisons, and Implications

Cubic Hermite splines are integral to computer graphics, geometric design, scientific visualization, and numerical simulation, where control over both value and slope interpolation at knots is essential. Their minimal support and closed-form expressions make them attractive for localized approximation. Hermite splines are asymptotically equivalent to cubic B-splines in approximation order, with the added benefit of pointwise derivative control and sparser, optimally-localized basis functions (Fageot et al., 2019).

Energy-minimizing Hermite splines suppress artificial undulations better than standard parameterizations. Area-preserving Hermite constructions offer higher order accuracy when conservation principles are required, with computationally efficient determination of free parameters (McGregor et al., 2018). Monotonicity corrections render Hermite splines suitable for edge-preserving interpolation in numerically sensitive contexts (Aràndiga et al., 2021). Quasi-interpolation techniques expand their practical role in high-dimensional and large-scale settings, as in QIBSH++ (Bertolazzi et al., 2022).

In summary, cubic Hermite splines are a flexible and rigorously-characterized class of piecewise-cubic $C^1$ interpolants, combining value and tangent interpolation, minimal-support optimality, near-best approximation, and a rich suite of enhancements (energy, monotonicity, area) for high-precision geometric and numerical modeling.