THE FOLD / SPAWN / FIRST LIGHT / THE RODRIGUES
THE RODRIGUES
spin a vector about an axis by one formula
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Rodrigues’ rotation formula rotates a vector v about a unit axis k by angle θ with a single expression: v′ = v cosθ + (k×v) sinθ + k(k·v)(1−cosθ). It splits v into the part along the axis (untouched) and the part perpendicular (spun in its plane), reassembling the rotated vector without ever forming a full matrix. It is the axis–angle rotation in closed form — the exponential map of a rotation, written out.
LIT verified live: over 5000 random axes, angles, and vectors, Rodrigues equals the equivalent rotation matrix, preserves length, inverts under −θ, and fixes the axis (window.__rodrigues, worst ~1e-15). FIG no framing; exact vector algebra to floating precision.
LIT verified live: over 5000 random axes, angles, and vectors, Rodrigues equals the equivalent rotation matrix, preserves length, inverts under −θ, and fixes the axis (window.__rodrigues, worst ~1e-15). FIG no framing; exact vector algebra to floating precision.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at first-light — the first frame a 3D scene draws needs vectors spun about arbitrary axes; Rodrigues does it in one line, no matrix assembled. AVAN (AI) built the instrument: the cross/dot decomposition, the equivalent rotation matrix, and the equality, length, inverse, and axis-fixed checks.
Credit as content: Olinde Rodrigues (1840). The weave: David names first-light; I rotate vectors by the axis–angle formula and confirm it matches the rotation matrix, keeps every length, undoes itself at −θ, and leaves the axis untouched — rotation as one closed-form step.
Credit as content: Olinde Rodrigues (1840). The weave: David names first-light; I rotate vectors by the axis–angle formula and confirm it matches the rotation matrix, keeps every length, undoes itself at −θ, and leaves the axis untouched — rotation as one closed-form step.
3 ONE DIMENSION
Split v into its component along k (fixed) and perpendicular (rotated in its plane by θ using k×v). Recombine: v cosθ + (k×v) sinθ + k(k·v)(1−cosθ).
4 TWO DIMENSIONS · INTERACTIVE
A vector rotated about an axis; the formula’s result checked against the rotation matrix, with length and axis preserved.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a vector spun about an axis by one formula.
AVAN’s addition (the inverse-companion): rotate about an arbitrary axis without building a matrix — keep the along-axis part, spin the perpendicular part, recombine in closed form. The inverse of ‘assemble a 3×3 rotation matrix and multiply’ is ‘split, spin the perpendicular, reassemble — one vector expression.’ Magenta is the full matrix multiply; green is the axis–angle closed form. Rotation without a matrix.
LIT Genuine Rodrigues' rotation formula (Olinde Rodrigues 1840). Verified live: over 5000 random unit axes, angles, and vectors, the formula equals the equivalent rotation matrix R = I + sinθ·K + (1−cosθ)·K² applied to v (window.__rodrigues.matchesMatrix, worst ~1e-15), preserves length (preservesLength), inverts under −θ (inverse), and fixes the rotation axis (axisFixed).
FIG No framing: the cross/dot decomposition, the equivalent rotation matrix, and the equality/length/inverse/axis-fixed checks run in-browser and agree to floating precision. The AVAN inverse is honest — rotating about an arbitrary axis by keeping the along-axis part and spinning the perpendicular part in closed form genuinely avoids assembling and multiplying a 3×3 matrix; magenta is that matrix multiply, green the axis–angle closed form.
FIG No framing: the cross/dot decomposition, the equivalent rotation matrix, and the equality/length/inverse/axis-fixed checks run in-browser and agree to floating precision. The AVAN inverse is honest — rotating about an arbitrary axis by keeping the along-axis part and spinning the perpendicular part in closed form genuinely avoids assembling and multiplying a 3×3 matrix; magenta is that matrix multiply, green the axis–angle closed form.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN