Cubic Hermite Splines with Temporal Regularization

• The paper introduces a method for constructing C¹-continuous cubic Hermite splines using minimum-energy quadratic tangents derived from prescribed control points.
• It minimizes both bending energy and curvature variation with respect to time, offering substantially smoother and less oscillatory interpolants than classical schemes.
• The approach leverages explicit solutions for energy minimization, supporting efficient implementation in applications like animation and keyframe interpolation.

Cubic Hermite splines with temporal curvature regularization are a class of $C^1$-continuous parametric spline interpolants designed for planar curves that pass through prescribed control points and possess minimized bending energy and curvature variation with respect to a temporal parameter. The method integrates exact energy-minimizing quadratic curve construction into the tangent calculation at each control point, inducing globally smoother and less oscillatory interpolants compared to classical schemes based on finite differences. This approach eliminates the need for hand-tuned trade-offs between bending energy and curvature variation by leveraging explicit minimization results for quadratic arcs, and has been shown to yield spline curves of lower elastic energy in most practical test cases (Benoit, 2010).

1. Bending-Energy and Curvature-Variation Functionals

The mathematical foundation rests on two fairness measures for planar curves $r(t) = (x(t), y(t))$:

• Bending energy: $E = \int_{t_0}^{t_1} \kappa(t)^2\,dt$,
• Curvature variation: $V = \int_{t_0}^{t_1} (\dot{\kappa}(t))^2\,dt$,

where the curvature $\kappa(t)$ is

$$\kappa(t) = \frac{\dot{x}(t)\ddot{y}(t) - \dot{y}(t)\ddot{x}(t)}{(\dot{x}(t)^2 + \dot{y}(t)^2)^{3/2}}.$$

For quadratic curves, minimizing $E$ also minimizes $V$, and thus a single-objective criterion suffices [(Benoit, 2010), Lemma 3.1]. Both functionals, interpreted temporally, provide the regularization properties sought for temporal interpolation.

2. Determining the Minimum-Energy Quadratic Through Three Points

Given three non-colinear points $p_1$, $p_2$, $p_3$ and a time parameter $t \in [0,1]$, the minimal-energy quadratic is parameterized as

$r(t) = a_1 t^2 + a_2 t + a_3,$

with $r(0) = p_1$, $r(T) = p_2$, $r(1) = p_3$ for some $T \in (0,1)$. The coefficients are computed by solving

$$\begin{cases} a_3 = p_1, \ a_1 T^2 + a_2 T + p_1 = p_2, \ a_1 + a_2 + p_1 = p_3, \end{cases}$$

yielding

$$a_1 = \frac{p_2-p_1 - (p_3-p_1)T}{T^2 - T},\qquad a_2 = p_3 - p_1 - a_1.$$

The instantaneous tangent at $t=T$, $v = r'(T) = 2 a_1 T + a_2$, is used for Hermite construction. If a nonuniform parameterization is used, the tangent is scaled by dividing by $(t_{i+1}-t_{i-1})$.

3. Selection of the Intermediate Parameter $T$ via Energy Minimization

To explicitly compute the minimum-energy quadratic, a normalization is applied: $p_1 \mapsto (0,0)$, $p_3 \mapsto (1,0)$, $p_2 \mapsto q_2$. The unique $T$ in $(0,1)$ minimizing bending energy solves the cubic equation

$$T^3 - \frac{3}{2}T^2 + (q_{2x} - |q_2|^2)T + \frac{1}{2}|q_2|^2 = 0.$$

The explicit Cardano solution is provided in the published work, with the middle root always lying in $(0,1)$. In practice, $T$ is computed for every local triplet of points to determine the associated minimum-energy tangent.

4. Construction and Evaluation of the Cubic Hermite Spline

With $n$ control points $\{p_1, p_2, \ldots, p_n\}$ and increasing knots $\{t_1 < t_2 < \cdots < t_n\}$, the spline construction algorithm proceeds as follows:

• For each $i=2,\ldots,n-1$, compute $T_i$ by solving the cubic equation for $\{p_{i-1}, p_i, p_{i+1}\}$.
• Calculate $(a_1^{(i)}, a_2^{(i)})$ and the time-scaled tangent

$$v_i = \frac{2 a_1^{(i)} T_i + a_2^{(i)}}{t_{i+1} - t_{i-1}}.$$

• For endpoints, use either one-sided differences or duplicate the nearest interior tangent.
• On $[t_i, t_{i+1}]$, the cubic Hermite segment is

$$r_i(s)=H_{00}(s)\,p_i+H_{10}(s)\,v_i+H_{01}(s)\,p_{i+1}+H_{11}(s)\,v_{i+1},\quad s=\frac{t-t_i}{t_{i+1}-t_i};$$

where

$$H_{00} = 2s^3 - 3s^2 + 1,\ H_{10} = s^3 - 2s^2 + s,\ H_{01} = -2s^3 + 3s^2,\ H_{11} = s^3 - s^2.$$

5. Arc-Length Computation for Minimum-Energy Quadratics

The exact arc-length of the minimum-energy quadratic interpolant between $p_1$ and $p_3$ (with $p_2$ as the intermediate point at $t=T$) is computed as follows. Define $\mathbf{s}_3 = p_3 - p_1$, $\mathbf{s}_2 = p_2 - p_1$, and

$$\begin{aligned} &r_1 = T\,\mathbf{s}_3 - \mathbf{s}_2,\ &r_2 = T^2\,\mathbf{s}_3 - \mathbf{s}_2,\ &\cos\theta = \frac{r_1 \cdot r_2}{|r_1||r_2|},\ &\rho = \sqrt{4|r_1|^2-4|r_1||r_2|\cos\theta + |r_2|^2}. \end{aligned}$$

Then, the arc-length is

$$l = \frac{1}{4 |r_1| (T - T^2)} \left[ |r_2|^2 \cos\theta + \left(2|r_1| - |r_2|\cos\theta\right)\rho + |r_2|^2 \sin^2\theta \ln \frac{2|r_1| - |r_2|\cos\theta + \rho}{|r_2|(1-\cos\theta)} \right].$$

This formula is exact and applies for each quadratic arc joining consecutive triplets in the control sequence (Benoit, 2010).

6. Comparative Analysis Against Traditional Schemes

A summary of numerical and visual comparisons (see Tables and Figures 2–5 in (Benoit, 2010)) demonstrates that this minimum-energy-tangent (MET) approach yields splines of substantially reduced bending energy $E$ and curvature variation $V$ relative to Catmull–Rom, cardinal, and Kochanek–Bartels tangent heuristics. Representative reductions of $E$ and $V$ by up to an order of magnitude are typical. This manifests as visibly smoother and less oscillatory interpolating splines under the MET construction. Tangent selection for cubic Hermite splines via local minimal-energy quadratics thus optimally regularizes curvature temporally, achieving $C^1$ continuity, temporal fairness, and quantifiably superior spline fairness metrics (Benoit, 2010).

7. Implementation Considerations and Theoretical Implications

The proposed methodology requires a single pass through the control points to determine each local parameter $T_i$ and the associated tangent $v_i$. Endpoint treatment may use finite differences or tangent duplication. No additional weighting between $E$ and $V$ is necessary, as minimization of $E$ alone suffices for quadratic segments. A plausible implication is that, in practical applications such as animation paths or smooth keyframe interpolation, this approach provides principled temporal regularization without user-parameter tuning, reducing interpolation artifacts while guaranteeing mathematically optimal curvature behavior for each segment in the Hermite spline (Benoit, 2010).