SlerpEdit
Slerp, short for spherical linear interpolation, is the standard method for smoothly transitioning between two orientations in three-dimensional space. By operating on unit quaternions, slerp traces a great-circle path on the 3-sphere, delivering a rotation that changes at a constant rate and avoids the distortions that can come from simpler interpolation schemes. This makes it a workhorse technique in computer graphics, animation, and simulation where dependable, predictable orientation changes are essential.
The technique was popularized in the computer-graphics literature by Ken Shoemake, who introduced quaternion-based approaches to interpolation and animation. Since then, slerp has become a core building block in many graphics pipelines and game engines, appearing in Unity (game engine), Unreal Engine, and a wide range of math libraries such as GLM and graphics APIs like OpenGL and DirectX. Its prominence rests on a clean geometric interpretation and on the practical benefits it delivers for real-time rendering and motion synthesis.
Mathematical foundations
Quaternions and rotations
Rotations in three dimensions can be represented by unit quaternions, objects that live on a 4-dimensional sphere. Interpolating rotations therefore becomes a problem of moving along the surface of this sphere in a way that preserves the unit-norm constraint. The quaternion representation is compact, avoids gimbal lock, and composes cleanly under multiplication. For a unit quaternion q, the corresponding rotation is well-defined, and the set of all such quaternions captures the entire space of 3D orientations. See quaternion for a broader mathematical treatment.
Geodesics on the 3-sphere
When interpolating between two orientations, the most natural path is the geodesic on the unit sphere in four dimensions. This path is the shortest route on the sphere and translates into a rotation that progresses with constant angular velocity. The spherical nature of the path is what gives slerp its name and its smooth, uniform motion characteristics. See great-circle for an accessible discussion of geodesics on spheres.
The slerp formula and its interpretation
Let q1 and q2 be unit quaternions representing the starting and ending orientations, and let t ∈ [0,1] be the interpolation parameter. Define dot = q1 · q2 (the inner product). If dot < 0, negate q2 to ensure the path takes the shortest arc. Let omega = arccos(dot). Then the SLERP result is
slerp(q1, q2, t) = (sin((1−t)·omega)/sin(omega))·q1 + (sin(t·omega)/sin(omega))·q2.
When sin(omega) is very small (the quaternions are almost identical), it is customary to fallback to a linear interpolation followed by normalization (often called NLERP) to avoid numerical issues. See Ken Shoemake for the historical development of these ideas and quaternion for the broader algebraic context.
Algorithms and implementation details
The standard SLERP computation
The classic SLERP computation uses the omega angle between q1 and q2 and builds a convex combination of q1 and q2 that lies on the great circle of unit quaternions. The method is robust and yields constant angular velocity, which is why it is preferred for keyframed rotations and smooth camera motion.
Shortest-arc normalization
To ensure the interpolation travels along the shorter path on the sphere, implementers commonly adjust q2 sign so that q1 · q2 ≥ 0 before computing omega. This small safeguard prevents unintended long arcs and flip-flopping of rotation directions during animation. See quaternion for details on dot products and sign considerations.
Numerical stability and near-singular cases
If omega is very small (the start and end orientations are nearly the same) or if the angle between quaternions is close to π (opposite quaternions), the denominator sin(omega) can approach zero and the B-splitting form becomes unstable. In these cases, a stable alternative is NLERP or a guarded SLERP that switches to a normalized linear interpolation when the angle falls below a chosen threshold. See normalized linear interpolation for a practical variant.
Variants and alternatives
- NLERP (normalized linear interpolation) provides a cheaper alternative when the exact constant-speed property is not strictly required. It linearly blends the quaternion coefficients and re-normalizes the result. See NLERP for more on this approach.
- Matrix and axis-angle alternatives exist, but they often suffer from cumulative drift, require re-orthogonalization, or introduce non-constant-speed motion. For many engines, quaternions with SLERP offer the most robust balance of accuracy and performance.
Performance considerations
SLERP requires trigonometric evaluations (sin and arccos), which are more expensive than simple linear blends. In performance-critical pipelines, developers may choose NLERP or even precomputed interpolation tables for static sequences, depending on the application’s tolerance for small deviations from perfect spherical motion. See GLM or engine-specific math libraries for concrete performance profiles.
Applications and practical considerations
- Real-time rendering and animation often rely on SLERP for character and camera orientations, ensuring predictable motion during keyframe interpolation and skeletal animation. See Unity and Unreal Engine documentation for typical usage patterns.
- Robotics and aerospace also employ quaternion interpolation to smoothly transition between attitude setpoints, with SLERP providing a stable geometric path between orientations. See quaternion-driven control literature for related methods.
- In networked simulations and multiplayer games, compact quaternion representations and straightforward interpolation help minimize bandwidth while preserving smooth attitude updates. See discussions of serialization and state synchronization in OpenGL-based or DirectX-driven graphics stacks.
Controversies and debates
The central debates around slerp in practice tend to be practical engineering choices rather than ideological disputes. On one side, SLERP offers mathematically exact, constant-speed interpolation along the shortest path, which makes it the default choice in many pipelines, especially where visual fidelity and determinism are priorities. On the other side, some teams prefer cheaper alternatives like NLERP or even simple lerp on rotation matrices when budgets are tight or when the application can tolerate small deviations from perfect spherical motion. In modern engines, the decision often comes down to a trade-off between computational cost and perceptual quality, with SLERP favored when the highest degree of rotational realism is required and NLERP or matrix-based methods reserved for performance-critical cooldowns or simpler visuals.
There is also discussion about portability and standardization. Because SLERP is well-understood and robust, it tends to be favored in cross-platform graphics pipelines and in open-source libraries, where predictable behavior and well-defined edge cases are valued. Critics who push for non-standard, problem-specific interpolation schemes may point to marginal gains in niche scenarios, but the mainstream consensus remains that SLERP provides a reliable baseline for quaternion-based rotation interpolation.