Slerp: Geodesic Interpolation on Spheres

• Spherical Linear Interpolation (Slerp) is defined as the interpolation between two unit vectors along the great-circle arc, ensuring geodesic accuracy on spherical manifolds.
• Its formulation employs trigonometric functions to blend vectors in a numerically stable way, preserving unit-norm constraints even in small-angle or antipodal scenarios.
• Slerp underpins applications in computer graphics, robotics, numerical methods, and deep learning by providing smooth, manifold-preserving interpolants for tasks like rigid body motions and model merging.

Spherical linear interpolation (Slerp) is a canonical method for interpolating between two unit-norm vectors on a sphere, traversing the shortest great-circle arc at constant angular speed. Unlike Euclidean linear interpolation, which generates straight line segments in $\mathbb{R}^n$ and leaves the sphere unless the vectors are collinear with the origin, Slerp stays strictly on the spherical manifold, preserving the unit-norm constraint and geodesic character. Slerp has become foundational in computational geometry, computer graphics, numerical algorithms on manifolds, model merging in deep learning, and the geometric analysis of structured data.

1. Mathematical Definition and Geometric Interpretation

Given distinct unit vectors $u, v \in S^2 \subset \mathbb{R}^3$, let $\theta = \arccos(\langle u, v\rangle)$ denote their central angle, with $0 < \theta \leq \pi$. For parameter $t\in[0,1]$, spherical linear interpolation is defined as

$$\mathrm{Slerp}(u,v;t) = \frac{\sin((1-t)\theta)}{\sin\theta} u + \frac{\sin(t\theta)}{\sin\theta} v.$$

By construction, $\mathrm{Slerp}(u,v;0) = u$, $\mathrm{Slerp}(u,v;1) = v$, and $\|\mathrm{Slerp}(u,v;t)\| = 1$ for all $t$. The path swept out is the great-circle connecting $u$ and $v$, parameterized at constant angular velocity $\theta$, with geodesic distance $\theta$ on $S^2$. This parametric form guarantees that the interpolant remains exactly on the unit sphere and moves at uniform speed with respect to arc length, as verified analytically in standard treatments (Roychowdhury, 7 Nov 2025).

This generalizes naturally to higher-dimensional spheres: for $x, y \in S^{n-1} \subset \mathbb{R}^n$, $\mathrm{Slerp}(x,y;t)$ is defined analogously, and for unit quaternions $q_1, q_2$ (representing 3D orientations), the formula operates on $S^3$ via quaternion interpolation (Goncharova et al., 2024).

2. Derivations and Analytical Properties

Several equivalent viewpoints underpin the Slerp formula:

• Plane projection: $u$ and $v$ span a plane through the origin. Any intermediate vector on the geodesic can be represented as a linear combination in an orthonormal basis $\{e_1, e_2\}$ (with $e_1=u$, $e_2=(v-\cos\theta\, u)/\sin\theta$). The Slerp formula arises by expressing $v$ as $v = \cos\theta\, e_1 + \sin\theta\, e_2$ and parametrizing the arc as $\gamma(t) = \cos(t\theta) e_1 + \sin(t\theta) e_2$ (Roychowdhury, 7 Nov 2025, Leung, 22 Mar 2025).
• Exponential map/logarithm: Slerp can be seen as $\mathrm{Slerp}(x,y;t) = \exp_x(t\log_x y)$, where $\exp_x$ is the Riemannian exponential map on the sphere, and $\log_x y$ maps $y$ onto the tangent space at $x$ (Leung et al., 2024).
• Quaternion interpolation: On $S^3$, for unit quaternions $q_1$, $q_2$, the shortest-arc interpolation is $$\mathrm{Slerp}(q_1, q_2; t) = \frac{\sin((1-t)\Theta)}{\sin\Theta}q_1 + \frac{\sin(t\Theta)}{\sin\Theta}q_2$$, with $\Theta = \arccos(q_1\cdot q_2)$ (Goncharova et al., 2024).

Key analytical properties:

• Unit-norm preservation ($\|\mathrm{Slerp}(\cdot)\|=1$).
• Smooth, analytic dependence on $t$ (except at ill-posed endpoints $\theta=0, \pi$).
• Constant-speed traversal: The derivative norm is constant ($\|\gamma'(t)\| = \theta$).
• For $t$ outside $[0,1]$, Slerp extends beyond the original arc.

3. Implementation Considerations and Edge Cases

Slerp requires robust handling of numerical and geometric edge cases:

• Small-angle limit ($\theta\to 0$): $\sin\theta\approx\theta$, causing potential cancellation. Standard practice replaces Slerp with the normalized linear blend, $(1-t)u + t v$, for sufficiently small $\theta$ (Roychowdhury, 7 Nov 2025, Leung et al., 2024).
• Antipodal points ($\theta=\pi$): $\sin\theta=0$, so the formula degenerates. All great circles through antipodes are valid interpolants; implementations select an arbitrary orthogonal direction or perturb the endpoints slightly (Roychowdhury, 7 Nov 2025, Leung, 22 Mar 2025).
• Numerical stability: Inner products are clamped to $[-1,1]$ before calling arccos. The trigonometric coefficients are computed using cached $\theta$, and for high-dimensional vectors, normalization is applied to avoid norm drift (Leung et al., 2024, Gaikwad et al., 3 Aug 2025).
• Computational cost: Minimal—one inner product, one arccos, two sines, two scalar multiplications, and a vector addition per call (Leung et al., 2024).

4. Applications in Geometry, Analysis, and Quantization

Slerp is canonical in manifold optimization, differential geometry, and quantization:

• Spherical quantization: Uniform partitioning and centroid computation along geodesic arcs on the sphere are achieved via Slerp. The midpoint $t=1/2$ yields the normalized average $(u + v)/\|u+v\|$. Slerp enables explicit construction of optimal quantizers and Voronoi regions along great and small circles, respecting the manifold's geometry (Roychowdhury, 7 Nov 2025).
• Geodesic parameterization: Slerp expresses geodesics in closed form, underpins subdivision and uniform sampling along spherical curves, and enables geometric constructions such as cell centroids and intrinsic distances.
• Differential geometry: Slerp-based convex combinations generalize Euclidean barycenters to the sphere, serving as the building block in manifold-valued interpolants and averaging (Leung et al., 2024).
• Scientific computing: Slerp is crucial for path interpolation on Riemannian manifolds, especially when geometric fidelity and constraint preservation are paramount (e.g., rigid body orientation, robot motion planning).

5. Slerp in Numerical Methods for Spherical Manifolds

Recent advancements leverage Slerp as a core operation in manifold-preserving integrators:

• Explicit and Implicit ODE Integrators: Slerp replaces Euclidean convex combinations in geometric integration of ODEs with values on $S^n$. In "SLERP-TVDRK" methods, convex combination steps in Runge–Kutta schemes are implemented by Slerp, ensuring all iterates remain on the sphere (Leung et al., 2024). Similarly, the spherical Crank–Nicolson integrator employs Slerp for midpoint evaluation in symplectic integrators without need for explicit projection (Leung, 22 Mar 2025).
• Performance and accuracy: Empirical results show that Slerp-based integrators (e.g., second- and third-order STVDRK) exhibit reduced global error constants, exact maintenance of unit-norm constraints, and improved time-reversibility in Hamiltonian systems. Attempts to generalize directly to higher-order (e.g., fourth-order) integrators confront obstacles due to the non-associativity of Slerp (i.e., barycentric combinations of more than two points are undefined in this framework) (Leung et al., 2024).

6. Model Merging and Representation Geometry in Deep Learning

Slerp has been adapted for geometric interpolation of neural network weights and high-dimensional embeddings:

• Latent space interpolation: In variational autoencoders and other generative models, Slerp ensures interpolants traverse the high-probability manifold region in latent space, avoiding linear-blend paths that stray into unsupported areas (Gaikwad et al., 3 Aug 2025). This yields smoother, more realistic morphing in generative tasks, as demonstrated by the visual and statistical metrics of microstructure prediction.
• Vision-language fusion: In composed image retrieval, Slerp is used to merge image and text embeddings lying on a shared hypersphere. Slerp-based merging provides geodesic interpolants that respect the representation distributions and avoids norm distortion. In the presence of modality gaps (e.g., CLIP, BLIP), Text-Anchored Tuning (TAT) enhances Slerp by reducing the mean geodesic angle between images and texts, improving retrieval performance (Jang et al., 2024).
• Model parameter merging: For merging base-model and adapted (e.g., LoRA-fine-tuned) weights, Slerp offers an angular-preserving blending that maintains the pretrained representational geometry—unlike naive averaging ("model soup"), which can distort learned directions and degrade generalization. Quantitative clustering benchmarks demonstrate improved Silhouette and Davies–Bouldin indices for Slerp-merged models, with best-practice guidelines emphasizing normalization and layerwise interpolation (Kabane, 16 Nov 2025).

7. Quaternion Slerp and Rigid Body Motions

For rigid 3D rotations, Slerp is implemented on the 4D unit quaternion sphere $S^3$:

• Molecular and solid-state modeling: In nudged-elastic band calculations of molecular transitions and polymorphs, Slerp on quaternions enables smooth, collision-free interpolation of orientations. When combined with linear interpolation of cell parameters and centers of mass, this hybrid approach outperforms naive Cartesian interpolation, reducing the number of convergence steps and eliminating atomic overlaps (Goncharova et al., 2024).
• Numerical protocol: Rotations are expressed in quaternion form, an optimal Kabsch rotation is extracted to get initial and final orientations, and intermediate states are generated by Slerp at uniformly spaced $u$, with subsequent conversion back to rotation matrices for atom mapping.
• Performance: In the tested molecular systems, Slerp-based path generation yields convergence benefits of $\sim$12% fewer steps with energy barriers matching reference results to $<0.1$ kJ mol⁻¹, demonstrating the method's physical fidelity and efficiency (Goncharova et al., 2024).

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Slerp is thus established as the canonical tool for geodesic interpolation on spheres and related homogeneous manifolds. Its combination of geometric exactness, analytical tractability, and robustness to numerical pitfalls ensures continued foundational status across computational geometry, numerical analysis, and high-dimensional representation learning (Roychowdhury, 7 Nov 2025, Leung, 22 Mar 2025, Leung et al., 2024, Goncharova et al., 2024, Jang et al., 2024, Gaikwad et al., 3 Aug 2025, Kabane, 16 Nov 2025).