Conformal rotation vector

Represent a three-dimensional orientation by stereographically projecting its unit quaternion to a three-component modified Rodrigues vector, with an explicit shadow-set switch around the chart singularity.

Version

v2 · 2026-08-30 · History

Domain-specific #

1536

Origin domain

attitude kinematics

Subdomain

three dimensional rotation parameterizations

Core Idea

For a unit quaternion \(q=(q_0,\mathbf q)\), one common conformal-rotation-vector convention uses the modified Rodrigues vector \(\boldsymbol\sigma=\mathbf q/(1+q_0)=\hat{\mathbf e}\tan(\phi/4)\), the stereographic image of (q) in the pure-imaginary quaternion hyperplane. Stereographic projection replaces the four quaternion components plus a unit-norm constraint with three rational coordinates; because the antipodal quaternions encode the same physical rotation, a shadow vector can replace a large principal vector before the chosen projection pole is reached.

Scope of Application

Conformal rotation vector applies when the analyst can specify an orientation in the rotation group SO(3), represented first by one of the two unit quaternions on the three-sphere that encode it and then by a vector in a stereographic coordinate chart and establish that the three coordinates arise from a declared stereographic projection of a unit quaternion, reconstruct the same SO(3) attitude under the declared convention, and are managed with the corresponding singularity or shadow-set rule. The identity applies only after the quaternion, angle, frame, and projection conventions are fixed; formulas copied between scalar-first and scalar-last, active and passive, or opposite-pole conventions can represent the inverse or a different attitude.

Clarity

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because modified Rodrigues parameters, conformal rotation vector, and Wiener–Milenkovic parameters are overlapping names, while rotation vector and Rodrigues parameters often denote two different coordinate maps. The disciplined statement is that the object counts as Conformal rotation vector exactly when the three coordinates arise from a declared stereographic projection of a unit quaternion, reconstruct the same SO(3) attitude under the declared convention, and are managed with the corresponding singularity or shadow-set rule

Manages Complexity

The abstraction compresses principal and shadow modified Rodrigues parameters, Wiener–Milenkovic terminology, attitude and attitude-error coordinates, continuous and sampled propagation, spacecraft, robotics and computer-vision conventions, and alternative projection-pole choices into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Abstract Reasoning

1. Type the carrier. Establish an orientation in the rotation group SO(3), represented first by one of the two unit quaternions on the three-sphere that encode it and then by a vector in a stereographic coordinate chart and reject examples from a different problem. 2.

Knowledge Transfer

Transfer within attitude kinematics is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from A spacecraft attitude estimator propagates modified Rodrigues parameters while their norm is moderate and replaces them by the shadow vector when the chosen chart approaches its 360-degree singular orientation to A vision pipeline encodes camera orientations with modified Rodrigues parameters so local optimization uses three variables and rational quaternion conversion rather than enforcing a four-component unit constraint at every iterate demonstrates that continuity.

Relationships to Other Abstractions

Current abstraction Conformal rotation vector Domain-specific

Parents (1) — more general patterns this builds on

• Conformal rotation vector is a kind of Representation Prime

  The proposed strict upward parent is prime:representation.

Hierarchy path (1) — routes to 1 parentless root

• Conformal rotation vector → Representation → Abstraction

Neighborhood in Abstraction Space

Conformal rotation vector sits in a sparse region of the domain-specific corpus (65^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Geodesy, Orbits & Coordinate Frames (25 abstractions)

Nearest neighbors

• Three-dimensional space — 0.87
• Quaternionic representation — 0.86
• Trimetrogon — 0.85
• 3D projection — 0.85
• Angular displacement — 0.85

Computed from structural-signature embeddings · 2026-09-08