Spherical Linear Interpolation¶
Interpolation along a selected great-circle arc whose angular distance from the starting point varies linearly with the interpolation parameter.
Core Idea¶
Spherical linear interpolation, or slerp, is the geodesic analogue of linear interpolation on a unit sphere. Given two non-antipodal endpoints and an arc choice, it uses sine-weighted endpoint vectors so parameter t advances a fraction t of the total spherical angle.
Applied to unit quaternions, the path represents constant-angular-velocity change between 3D orientations. Because q and −q denote the same rotation, sign and branch choices decide whether interpolation takes a short or long path; degenerate endpoints require explicit handling.
Structural Signature¶
Sig role-phrases:
- Unit-sphere endpoints — Supply normalized start and end points. It is required input. Counterfactual: Non-normalized endpoints do not define the intended spherical geodesic without preprocessing.
- Endpoint angle — Measures geodesic separation and parameterizes weights. It is geometric parameter. Counterfactual: At zero or pi, the usual formula is singular or nonunique.
- Arc choice — Selects short, long, or otherwise oriented great-circle segment. It is branch condition. Counterfactual: Antipodal endpoints admit infinitely many great circles.
- Linear parameter — Maps t to a proportional fraction of angular distance. It is interpolation rule. Counterfactual: Nonlinear timing changes speed even on the same arc.
- Sine-weighted combination — Produces points on the chosen great-circle arc. It is defining operation. Counterfactual: Ordinary linear interpolation cuts through the sphere and needs renormalization.
- Quaternion rotation interpretation — Maps unit-quaternion points to orientations under a double cover. It is application semantics. Counterfactual: Failing to handle q and minus q may choose an unintended long rotation.
What It Is Not¶
- It is not ordinary linear interpolation in the ambient vector space.
- It is not normalized lerp, which generally has nonuniform angular speed.
- It is not unique for antipodal endpoints without an additional arc choice.
- Quaternion slerp does not identify q and minus q automatically unless the implementation handles the double cover.
- Closest near-miss. Normalized linear interpolation stays on the sphere after renormalization but generally does not move at constant angular speed.
Scope of Application¶
- Computer animation. Interpolates orientation keyframes smoothly.
- Robotics and navigation. Produces geodesic attitude transitions.
- Geometry. Interpolates unit vectors on spheres of any dimension.
- Signal and shape processing. Blends normalized directional or rotational states when branch conditions are controlled.
Clarity¶
State endpoint normalization, dot product and angle, arc branch, quaternion sign convention if used, parameter range, and treatment of coincident and antipodal cases. Distinguish uniform parameter speed from uniform angular speed.
Manages Complexity¶
Slerp converts curved-state interpolation into endpoint, angle, branch, and one parameter. It preserves spherical norm and constant angular progress while making explicit the degeneracies that Euclidean interpolation hides.
Abstract Reasoning¶
- Normalize and type the endpoints.
- Compute their dot product and angular separation.
- Choose the intended great-circle branch and quaternion lift.
- Apply sine-weighted interpolation or a stable limiting formula.
- Verify unit norm and endpoint recovery.
- Check angular distance versus t and handle zero or antipodal limits.
Knowledge Transfer¶
Slerp transfers literally to any unit-sphere state where geodesic distance and branch are defined. Interpolation on a general manifold requires its own geodesics and exponential map; visual smoothness alone does not make a curve slerp.
Examples¶
Canonical¶
Two non-antipodal unit vectors with angle Omega are combined using sin((1-t)Omega)/sin(Omega) and sin(tOmega)/sin(Omega); at t=½ the result lies halfway along the chosen great-circle arc.
Mapped back: endpoints → unit vectors; angle → Omega; operation → sine weights; parameter → half angular distance.
Applied / In Practice¶
Animation negates the destination unit quaternion when the dot product is negative, then applies slerp so the represented orientation follows the shorter constant-angular-velocity rotation.
Mapped back: endpoints → unit quaternions; branch → sign-corrected short path; semantics → 3D rotation; output → uniform angular motion.
Structural Tensions¶
T1 — Shortest Arc versus Chosen Orientation Path. The short path is usually desired, but an application may intentionally require the long rotation.
Diagnostic: Which lift and arc are part of the intended motion?
T2 — Geometric Fidelity versus Numerical Conditioning. Exact sine weights preserve uniform angle but become ill-conditioned near coincident or antipodal endpoints.
Diagnostic: Which limiting or fallback method preserves the intended branch?
Structural–Framed Character¶
Slerp is strongly structural as a metric-geometric construction; implementation conventions enter through branch, quaternion cover, and numerical limiting cases.
Structural Core vs. Domain Accent¶
The skeleton is constant-speed geodesic interpolation. Sphere geometry supplies great circles, sine weights, angular distance, and quaternion rotation semantics, keeping the named method mathematical and computational.
Instantiates / Related Primes¶
This entry presupposes Continuity.
-
Approved root. This exact constant-speed spherical rule has no frozen parent edge.
-
Related — interpolation, geodesic, quaternion rotation, and trilinear interpolation. They supply the broader task, geometry, application, or a different coordinatewise method.
Relationships to Other Abstractions¶
Current abstraction Spherical Linear Interpolation Domain-specific
Parents (1) — more general patterns this builds on
-
Spherical Linear Interpolation presupposes Continuity Prime
Spherical Linear Interpolation presupposes Continuity because the interpolation traverses a selected great-circle arc continuously with linearly varying angle.Every reviewed Spherical Linear Interpolation instance depends on the parent role: the interpolation traverses a selected great-circle arc continuously with linearly varying angle. Removing that role makes the frozen child identity undefined or changes it into a different abstraction. Continuity can occur without Spherical Linear Interpolation, so the relation is dependency rather than subsumption.
Hierarchy paths (2) — routes to 2 parentless roots
- Spherical Linear Interpolation → Continuity → Neighborhood → Topology
- Spherical Linear Interpolation → Continuity → Invariance
Neighborhood in Abstraction Space¶
Spherical Linear Interpolation sits in a crowded region of the domain-specific corpus (36th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Coordinate Systems & Spatial Measures (29 abstractions)
Nearest neighbors
- Equivalent Radius — 0.89
- Central angle — 0.89
- Chamberlin Trimetric Projection — 0.88
- Hypercycle (Geometry) — 0.88
- Elliptic Cylindrical Coordinates — 0.87
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Linear interpolation. Tell: Traverses the chord rather than the spherical arc.
- Normalized lerp. Tell: Renormalizes chord interpolation but usually varies angular speed.
- Quaternion lerp. Tell: Does not intrinsically preserve constant angular velocity.
- Geodesic interpolation on a general manifold. Tell: Requires a different metric and construction.
References¶
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Spherical_linear_interpolation (revision 1340311860).
- Preserved source candidate: https://dl.acm.org/profile/81100026146
- Preserved source candidate: https://hal.inria.fr/inria-00073318
- Preserved source candidate: https://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf
- Preserved source candidate: http://web.mit.edu/2.998/www/QuaternionReport1.pdf
- Preserved source candidate: https://web.archive.org/web/20170830031501/http://web.mit.edu/2.998/www/QuaternionReport1.pdf
- Preserved source candidate: http://number-none.com/product/Understanding%20Slerp,%20Then%20Not%20Using%20It
- Preserved source candidate: https://web.archive.org/web/20170825184056/http://number-none.com/product/Understanding%20Slerp,%20Then%20Not%20Using%20It/
- Preserved source candidate: http://theory.org/software/qfa/writeup/node12.html
The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.