Trigonometric Angle-Based Features

• Trigonometric angle-based features are mathematically structured descriptors that utilize classical and generalized trigonometric functions to encode rotation-invariant geometric relationships.
• They derive from fundamental identities and employ multiple-angle and addition formulas to enable continuous tuning of feature shapes in various computational applications.
• Empirical studies, particularly in graph neural networks for 3D object detection, demonstrate that these features significantly enhance performance and robustness.

Trigonometric angle-based features are mathematically structured descriptors that utilize angular relationships—derived from classical or generalized trigonometric functions—within geometric or learned representations. Rooted in the properties of both classical and two-parameter generalized trigonometric functions—including new multiple-angle and addition formulas—these features provide a continuous, tunable family of periodic mappings for use in a variety of computational pipelines. Recent advancements demonstrate their impact, particularly regarding encoding rotation-invariant geometric relationships in high-dimensional data, as in graph neural networks (GNNs) for 3D object detection and feature engineering.

1. Generalized Trigonometric Functions and Fundamental Identities

Let $p, q \in (1,\infty)$ be fixed exponents. The generalized arcsine is defined as

$$\arcsin_{p, q}(y) = \int_0^y \frac{dt}{(1-t^q)^{1/p}},\quad y\in[0,1]$$

with half-period

$$T_{p, q} = 2 \int_0^1 \frac{dt}{(1-t^q)^{1/p}} = \frac{2}{q} B\left(\frac{1}{q},\,\frac{1}{p/(p-1)}\right).$$

Define $p^* = p/(p-1)$. The periodic continuation yields

$$\sin_{p,q}: \mathbb{R} \rightarrow [-1,1], \quad \cos_{p,q}(x) := \frac{d}{dx} \sin_{p,q}(x).$$

A key relation is the generalized Pythagorean identity: $|\cos_{p, q}(x)|^p + |\sin_{p, q}(x)|^q = 1.$ Derivatives satisfy

$$(\sin_{p, q} x)' = \cos_{p, q} x,\quad (\cos_{p, q} x)' = -|\cos_{p, q} x|^{2-p} |\sin_{p, q} x|^{q-1}.$$

These generalized functions recover classical trigonometric identities when $p=q=2$ and enable continuous transitions between various function shapes, facilitating feature shape tuning. The domain of both $\sin_{p,q}$ and $\cos_{p,q}$ is $\mathbb{R}$, with fundamental period $2T_{p,q}$ (Takeuchi, 2016).

2. Multiple-Angle and Duplication Formulas

Multiple-angle formulas for generalized trigonometric functions extend classical results. For all $x \in \mathbb{R}$ and $p>1$,

$$\sin_{2, p}\bigl(2^{2/p} x\bigr) = 2^{2/p} \sin_{p^*, p}(x) \left| \cos_{p^*, p}(x) \right|^{p^*-1},$$

$$\cos_{2, p}\bigl(2^{2/p} x\bigr) = \left| \cos_{p^*, p}(x) \right|^{p^*} - \left| \sin_{p^*, p}(x) \right|^{p} = 1-2\left| \sin_{p^*, p}(x) \right|^{p}.$$

The half-periods relate as $T_{p^*, p} = 2^{-2/p} T_{2, p}$. For each integer $n$ and by periodic continuation, closed-form expressions for $\sin_{2,p}(2^{2/p}x + nT_{2,p})$ and similar for cosine are available in terms of base functions (Takeuchi, 2016).

A slowly-convergent series expansion analogous to the Gregory-Leibniz series for $\pi/2$ exists for $T_{p^*,p}$: $$T_{p^*,p} = \sum_{n=0}^\infty \frac{(2/p)_n}{n!} \cdot \frac{(-1)^n}{pn+1}$$ where $(a)_n = \Gamma(a+n)/\Gamma(a)$.

3. Addition-Type Identities

While a full generalization of $\sin(x+y) = \sin x\,\cos y + \cos x\,\sin y$ is not available for all parameters, important addition-type results include:

• The $p$-Pythagorean identity: $|\cos_{p,q}x|^p + |\sin_{p,q}x|^q = 1$.
• Derivative identities enabling implicit integral-based addition relations.
• For $(p^*, p) = (4/3, 4)$ (Edmunds–Gurka–Lang), an explicit four-term addition theorem: $$\sin_{4/3,4}(2x) = \frac{4 \sin_{4/3,4} x\, \cos_{4/3,4}^{1/3} x}{1 + 4 \sin_{4/3,4}^{4/3} x\, \cos_{4/3,4}^{4/3} x}$$ (Takeuchi, 2016).

4. Algorithmic Construction of Angle-Based Features

A broad class of angle-based features in signal processing and machine learning applications are constructed from periodic base functions. Given explicit duplication and multiple-angle formulas, one can:

• Select $$\varphi_1(\theta) = \sin_{p^*,p}(\theta),\ \varphi_2(\theta) = \cos_{p^*,p}(\theta)$$ as base features.
• For higher harmonics, form features like $$\sin_{2,p}(2^{2/p} \theta) = 2^{2/p} \varphi_1(\theta) |\varphi_2(\theta)|^{p^*-1}$$, without requiring numerical inversion or quadrature.
• Treat $p$ and $q$ as tunable hyperparameters for optimizing feature shape, seamlessly interpolating between classical sinusoids $(p=q=2)$ and other geometries (e.g., super-circles, astroids).

Application contexts include filter-bank design with band shapes adapted by $(p,q)$ and geometric modeling via parameterizations $$x(\theta) = \cos_{p,q}(\theta),\ y(\theta) = \sin_{p,q}(\theta)$$ (Takeuchi, 2016).

5. Rotation-Invariant Trigonometric Angular Features in Deep Learning

Within 3D point cloud analysis and graph neural network (GNN) architectures, trigonometric angle-based features are constructed as follows. For points $P_i, P_j \in \mathbb R^3$,

$$n_i = \frac{P_i}{\|P_i\|},\quad n_j = \frac{P_j}{\|P_j\|},\quad r_{ij}=P_j-P_i,\quad \hat{r}_{ij} = \frac{r_{ij}}{\|r_{ij}\|}.$$

Define angular features: $$\theta_{ij}^{(1)} = \arccos(n_i \cdot n_j), \quad \theta_{ij}^{(2)} = \arccos(\hat r_{ij} \cdot n_j), \quad \theta_{ij}^{(3)} = \pi - (\theta_{ij}^{(1)} + \theta_{ij}^{(2)}).$$ The triple $$(\theta_{ij}^{(1)}, \theta_{ij}^{(2)}, \theta_{ij}^{(3)})$$ forms a mathematically rotation-invariant descriptor for each point-pair.

These are encoded in GNN edge features as either angle-only or concatenated with relative offsets:

• Angle-only: $$[\theta_{ij}^{(1)}, \theta_{ij}^{(2)}, \theta_{ij}^{(3)}, \mathrm{intensity}_i]$$
• Angle+Relative: $$[\theta_{ij}^{(1)}, \theta_{ij}^{(2)}, \theta_{ij}^{(3)}, x_i-x_j, y_i-y_j, z_i-z_j, \mathrm{intensity}_i]$$.

Empirical analysis shows that angle-based encodings confer strong robustness to global rigid rotations, as angular quantities $\arccos((R v)\cdot(R w)) = \arccos(v\cdot w)$ for any rotation $R \in SO(3)$, while raw offsets do not (Ansari et al., 2021).

6. Comparative Impact and Quantitative Results

Empirical results from KITTI 3D object detection benchmarks demonstrate marked improvements in performance using trigonometric angle-based features within GNNs.

Feature Encoding	Car mAP (E/M/H)	Cyclist mAP (E/M/H)	Pedestrian mAP (E/M/H)
Euclidean-only	30.23/25.58/22.02	15.15/11.06/7.47	8.93/15.47/8.93
Absolute-only	35.56/28.66/25.41	20.71/18.92/16.17	32.67/29.91/22.41
Relative-only	62.23/49.57/42.42	45.64/40.14/38.19	51.28/47.42/42.56
Angle-only	85.64/75.66/67.69	50.55/41.07/38.27	71.42/63.12/55.47
Angle+Relative	90.12/88.86/79.53	54.23/48.67/41.21	80.61/62.41/58.01

The Angle+Relative encoding achieves the highest mean average precision while incurring only marginal runtime overhead compared to the baseline (Ansari et al., 2021).

7. Applications and Theoretical Significance

Trigonometric angle-based features provide a mathematically principled, computationally tractable, and robust methodology for constructing invariant geometric descriptors. Their generalization to $(p,q)$-parametric families enables function shape optimization for diverse domains:

• Machine learning: periodic feature engineering via learnable harmonic shapes.
• Signal processing: design of frequency bases and filter-banks beyond the classical Fourier family.
• Geometric modeling: smooth shape interpolation and parameterization of generalized curves.

The explicit algebraic structure of angle-based features rooted in generalized trigonometric identities enables their systematic deployment in mathematical and practical pipelines (Takeuchi, 2016, Ansari et al., 2021).