Spherical Essentially Non-oscillatory (SENO) Interpolation

Abstract

We develop two new ideas for interpolation on $\mathbb {S}^2$. In this first part, we will introduce a simple interpolation method named Spherical Interpolation of orDER n (SIDER-n) that gives a $C^{n}$ interpolant given $n \ge 2$. The idea generalizes the construction of the Bézier curves developed for $\mathbb {R}$. The second part incorporates the ENO philosophy and develops a new Spherical Essentially Non-Oscillatory (SENO) interpolation method. When the underlying curve on $\mathbb {S}^2$ has kinks or sharp discontinuity in the higher derivatives, our proposed approach can reduce spurious oscillations in the high-order reconstruction. We will give multiple examples to demonstrate the accuracy and effectiveness of the proposed approaches.

Appendix: Derivatives of SQUAD, SIDER2 and SIDER3

1.1 Time Derivatives of the Interpolants

Assume $t \in [0, 1]$, and $\mathbf {q_i}=(0,\mathbf {p_i})$ is the starting point. The derivatives of SLERP are given by

$$\begin{aligned} \dfrac{d}{dt}(\text {SLERP}(\mathbf {q_i}, \mathbf {q_{i+1}}, f(t)))^{\pm 1} = \pm \text {SLERP}(\mathbf {q_i}, \mathbf {q_{i+1}}, f(t)) \cdot \ln ((\mathbf {q_i})^{-1} \mathbf {q_{i+1}}) \cdot \dfrac{df(t)}{dt}. \end{aligned}$$

To simplify the expressions, we introduce the following notations.

$$\begin{aligned} \begin{array}{cccc} \text {Scheme} &{} \text {SIDER2} &{} \text {SQUAD} &{} \text {SIDER3} \\ f(t) &{} t &{} 2t(1-t) &{} t \\ g(t) &{} t &{} t &{} 3t/2 \\ h(t) &{} t &{} t &{} (3t-1)/2 \\ \textbf{p}(t) &{} \text {SLERP}(\mathbf {q_i}, \mathbf {d_{(i+1)a}}, t) &{} \text {SLERP}(\mathbf {q_i}, \mathbf {q_{i+1}}, t) &{} \text {SIDER2} (\mathbf {q_i}, \mathbf {q_{i+1}}, \mathbf {q_{i+2}}, t) \\ \textbf{s}(t) &{} \text {SLERP}(\mathbf {d_{(i+1)b}}, \mathbf {q_{i+2}}, t) &{} \text {SLERP}(\mathbf {s_i}, \mathbf {s_{i+1}}, t) &{} \text {SIDER2} (\mathbf {q_{i+1}}, \mathbf {q_{i+2}}, \mathbf {q_{i+3}}, t) \\ \end{array} \end{aligned}$$

Since

$$\begin{aligned} \begin{aligned}&(0, \textbf{p}(g(t))) \times ((0, -\textbf{p}(g(t))) \times (0, \textbf{s}(h(t))))^{f(t)} \\&\quad = (0, \mathbf {p_g}(t)) \times ((0, -\mathbf {p_g}(t)) \times (0, \mathbf {s_h}(t))^{f(t)}) \\&\quad = (0, \mathbf {p_g}(t)) \times (\mathbf {p_g}(t) \cdot \mathbf {s_h}(t), -\mathbf {p_g}(t) \times \mathbf {s_h}(t))^{f(t)} \\&\quad = (0, \mathbf {p_g}(t)) \times (\cos (\theta _{\textbf{ps}}(t)), \sin (\theta _{\textbf{ps}}(t)) \mathbf {a_{ps}}(t))^{f(t)} \\&\quad = (0, \mathbf {p_g}(t)) \times (\cos (f_{\theta , \textbf{ps}}(t)), \sin (f_{\theta , \textbf{ps}}(t)) \mathbf {a_{ps}}(t)), \end{aligned} \end{aligned}$$

the first and the second time derivatives are given by

$$\begin{aligned}{} & {} \dfrac{d}{dt} \left[ (0, \textbf{p}(g(t))) \times ((0, \textbf{p}(g(t))) \times (0, \textbf{s}(h(t))))^{f(t)}\right] \\{} & {} \quad = (0, (\mathbf {p_g})'(t)) \times (\cos (f_{\theta , \textbf{ps}}(t)), \sin (f_{\theta , \textbf{ps}}(t)) \mathbf {a_{ps}}(t)) - \mathbf {p_g}(t) \cdot \sin (f_{\theta , \textbf{ps}}(t)) \cdot (f_{\theta , \textbf{ps}})'(t) \\{} & {} \quad \quad + (0, \mathbf {p_g}(t)) \times (0, \cos (f_{\theta , \textbf{ps}}(t)) \cdot (f_{\theta , \textbf{ps}})'(t) \times \mathbf {a_{ps}}(t) + \sin (f_{\theta , \textbf{ps}}(t)) \times (\mathbf {a_{ps}})'(t)) \end{aligned}$$

and

$$\begin{aligned}{} & {} \dfrac{d}{dt} \left[ \dfrac{d}{dt} \left[ (0, \textbf{p}(g(t))) \times ((0, -\textbf{p}(g(t))) \times (0, \textbf{s}(h(t))))^{f(t)}\right] \right] \\{} & {} \quad = (0, -(\theta _{\textbf{p}})^2(\mathbf {p_g}(t))) \times (\cos (f_{\theta , \textbf{ps}}(t)), \sin (f_{\theta , \textbf{ps}}(t)) \mathbf {a_{ps}}(t)) \\{} & {} \quad \quad - 2(\mathbf {p_g})'(t) \cdot \sin (f_{\theta , \textbf{ps}}(t)) \cdot (f_{\theta , \textbf{ps}})'(t) \\{} & {} \quad \quad + 2 \cdot (0, (\mathbf {p_g})'(t)) \times (0, \cos (f_{\theta , \textbf{ps}}(t)) \cdot (f_{\theta , \textbf{ps}})'(t) \times \mathbf {a_{ps}}(t) + \sin (f_{\theta , \textbf{ps}}(t)) \times (\mathbf {a_{ps}})'(t)) \\{} & {} \quad \quad - \mathbf {p_g}(t) \cdot (\cos (f_{\theta , \textbf{ps}}(t)) \cdot ((f_{\theta , \textbf{ps}})'(t))^2 + \sin (f_{\theta , \textbf{ps}}(t)) \cdot (f_{\theta , \textbf{ps}})''(t)) \\{} & {} \quad \quad + (0, \mathbf {p_g}(t)) \times (0, (\cos (f_{\theta , \textbf{ps}}(t)) \cdot (f_{\theta , \textbf{ps}})''(t) - \sin (f_{\theta , \textbf{ps}}(t)) \cdot ((f_{\theta , \textbf{ps}})'(t))^2) \times \mathbf {a_{ps}}(t)) \\{} & {} \quad \quad + (0, \mathbf {p_g}(t)) \times (0, 2 \cos (f_{\theta , \textbf{ps}}(t)) \cdot (f_{\theta , \textbf{ps}})'(t) \times (\mathbf {a_{ps}})'(t) + \sin (f_{\theta , \textbf{ps}}(t)) \times (\mathbf {a_{ps}})''(t)) \end{aligned}$$

where

$$\begin{aligned} \theta _{\textbf{p}}= & {} \arccos (\mathbf {p_i} \cdot \mathbf {c_{(i+1)a}} \text { for SIDER2 or } \cdot \mathbf {p_{i+1}} \text { for SQUAD}), \\ \mathbf {a_p}= & {} (-\mathbf {p_i} \times \mathbf {c_{(i+1)a}} \text { for SIDER2 or } \times \mathbf {p_{i+1}} \text { for SQUAD}) / \sin (\theta _{\textbf{p}}), \\ (0, \mathbf {p_g}(t))= & {} (0, \textbf{p}(g(t))) = (\cos (g(t) \cdot \theta _{\textbf{p}}))\mathbf {p_i} + (\sin (g(t) \cdot \theta _{\textbf{p}}))(\mathbf {a_p} \times \mathbf {p_i}), \nonumber \\ (0, \mathbf {s_h}(t))= & {} (0, \textbf{s}(h(t))), \\ (\mathbf {p_g})'(t)= & {} (\theta _{\textbf{p}})(\mathbf {a_p} \times \mathbf {p_g}(t)), (\mathbf {p_g})''(t) = -(\theta _{\textbf{p}})^2(\mathbf {p_g}(t)) \\ \theta _{\textbf{ps}}(t)= & {} \arccos (\mathbf {p_g}(t) \cdot \mathbf {s_h}(t)), \\ (\theta _{\textbf{ps}})'(t)= & {} -((\mathbf {p_g})'(t) \cdot \mathbf {s_h}(t) + \mathbf {p_g}(t) \cdot (\mathbf {s_h})'(t)) / \sin (\theta _{\textbf{ps}}(t)), \\ {(\theta _{\textbf{ps}})''(t)}= & {} -((\mathbf {p_g})''(t) \cdot \mathbf {s_h}(t) + 2 \cdot (\mathbf {p_g})'(t) \cdot (\mathbf {s_h})'(t) + \mathbf {p_g}(t) \cdot (\mathbf {s_h})''(t)\\{} & {} + \cos (\theta _{\textbf{ps}}(t)) \cdot ((\theta _{\textbf{ps}})'(t))^2) / \sin (\theta _{\textbf{ps}}(t)), \nonumber \\ \mathbf {a_{ps}}(t)= & {} (-\mathbf {p_g}(t) \times \mathbf {s_h}(t))/\sin (\theta _{\textbf{ps}}(t)), \\ (\mathbf {a_{ps}})'(t)= & {} (-(\mathbf {p_g})'(t) \times \mathbf {s_h}(t) + (-\mathbf {p_g}(t)) \times (\mathbf {s_h})'(t) \\{} & {} - \cos (\theta _{\textbf{ps}}(t)) \times (\theta _{\textbf{ps}})'(t) \times \mathbf {a_{ps}}(t))/\sin (\theta _{\textbf{ps}}(t)), \nonumber \\ (\mathbf {a_{ps}})''(t)= & {} (-(\mathbf {p_g})''(t) \times \mathbf {s_h}(t) + 2 \cdot (-(\mathbf {p_g})'(t)) \times (\mathbf {s_h})'(t) + (-\mathbf {p_g}(t)) \times (\mathbf {s_h})''(t) \nonumber \\{} & {} + \sin (\theta _{\textbf{ps}}(t)) \times ((\theta _{\textbf{ps}})'(t))^2 \times \mathbf {a_{ps}}(t) - \cos (\theta _{\textbf{ps}}(t)) \times (\theta _{\textbf{ps}})''(t) \times \mathbf {a_{ps}}(t) \\{} & {} - 2 \cdot \cos (\theta _{\textbf{ps}}(t)) \times (\theta _{\textbf{ps}})'(t) \times (\mathbf {a_{ps}})'(t))/\sin (\theta _{\textbf{ps}}(t)), \nonumber \\ f_{\theta , \textbf{ps}}(t)= & {} f(t) \cdot \theta _{\textbf{ps}}(t), \quad \dfrac{d^n f_{\theta , \textbf{ps}}(t)}{dt^n} = \sum _{r = 0}^n \dfrac{d^r f(t)}{dt^r} \cdot \dfrac{d^{n-r} \theta _{\textbf{ps}}(t)}{dt^{n-r}}. \text { (product rule}) \, . \end{aligned}$$

The derivatives of $\mathbf {s_h}(t)$ follows the same manner as $\mathbf {p_g}(t)$, but they are not explicitly written out because $\textbf{p}(t), \textbf{s}(t), f(t), g(t)$ and/or h(t) of SQUAD, SIDER2 and SIDER3 are not the same.

1.2 Angular Derivatives of the Interpolants

With references to [5], if the interpolant is $\textbf{q}(t)$, then we can deduce that

$$\begin{aligned} \begin{aligned} \mathbf {\omega }(t)&= 2 \textbf{q}'(t) \times (\textbf{q}(t))^{-1}, \\ \mathbf {\alpha }(t)&= (2\textbf{q}''(t) - \mathbf {\omega }(t) \times \textbf{q}'(t)) \times (\textbf{q}(t))^{-1}, \text { and} \\ \mathbf {\zeta }(t)&= (2\textbf{q}'''(t) - 2 \mathbf {\alpha }(t) \times \textbf{q}'(t) - \mathbf {\omega }(t) \times \textbf{q}''(t)) \times (\textbf{q}(t))^{-1}. \end{aligned} \end{aligned}$$