DERIV-turning-angle — Turning angle at a contour vertex

Used by: EN-003, DERIV-local-radius-clamp (notation τᵢ). Sources: REF-MATHWORLD-VECTOR-ADDITION, REF-MATHWORLD-VECTOR-DIFFERENCE, REF-MATHWORLD-DOT, REF-MATHWORLD-PERP-DOT, REF-MATHWORLD-ROTATION-MATRIX, REF-MDN-ATAN2.

Notation

Cyclic contour v₀ … vₙ₋₁; indices are taken modulo n (Q11): the vertex before v₀ is vₙ₋₁, the vertex after vₙ₋₁ is v₀.

Symbol	Definition
a = vᵢ − vᵢ₋₁	incoming edge at vᵢ
b = vᵢ₊₁ − vᵢ	outgoing edge at vᵢ
τᵢ	turning angle at vᵢ: angle from a to b

Step 1 — Edge vectors

A vector difference is the sum with the second vector reversed, A − B = A + (−B) (REF-MATHWORLD-VECTOR-DIFFERENCE), and vectors add component-wise (REF-MATHWORLD-VECTOR-ADDITION). Hence a = (aₓ, a_y) = (xᵢ − xᵢ₋₁, yᵢ − yᵢ₋₁).

Step 2 — Cosine and sine of the turn

• Dot product: a · b = aₓbₓ + a_y b_y = |a| |b| cos τ (REF-MATHWORLD-DOT).
• Perp dot product: a⊥ · b = |a| |b| sin τ, τ measured from a to b, with a⊥ = a rotated a quarter turn to the left (REF-MATHWORLD-PERP-DOT). The rotation matrix with θ = π/2 gives its components, a⊥ = (−a_y, aₓ) (REF-MATHWORLD-ROTATION-MATRIX), hence a⊥ · b = aₓ b_y − a_y bₓ (dot product in components, REF-MATHWORLD-DOT).

Step 3 — Angle

atan2(y, x) returns the angle, in [−π, π], between the positive x-axis and the point (x, y), counter-clockwise (REF-MDN-ATAN2). It only depends on the direction of (x, y), so for |a| |b| > 0:

τ = atan2(a⊥ · b, a · b) = atan2(|a||b| sin τ, |a||b| cos τ).

Frame

The formulas hold in the mathematical frame (y up). In the SVG frame (y down) the same numbers describe the mirrored drawing: a positive τ is a clockwise turn on screen. A contour listed clockwise on screen (Q11) therefore has positive turning angles at its convex vertices. The fillet computations only use |τ| (DERIV-local-radius-clamp).

Checks (test cases)

Vertices vᵢ₋₁, vᵢ, vᵢ₊₁	a, b	a⊥·b, a·b	τ
(0, 0), (100, 0), (100, 50)	(100, 0), (0, 50)	5000, 0	π/2 (clockwise on screen)
(0, 50), (0, 0), (100, 0) — first vertex of the rectangle, previous = last	(0, −50), (100, 0)	5000, 0	π/2
(0, 0), (5, 0), (10, 0)	(5, 0), (5, 0)	0, 25	0 (aligned)
(0, 0), (10, 0), (10, −10)	(10, 0), (0, −10)	−100, 0	−π/2 (counter-clockwise on screen)
(0, 0), (10, 0), (0, 0)	(10, 0), (−10, 0)	0, −100	π (back-turn)

Limits

• A zero-length edge (repeated vertex, e.g. a rectangle of width 0, Q15) has no direction: both products are 0 and τ is taken as 0 — no turn, hence no fillet. This is stated explicitly, not left to atan2(0, 0): with signed zeros, atan2(+0, −0) = π in IEEE 754 (found by the EN-005 audit, e.g. a = (−10, −10), b = (0, 0)). The sign of that zero is not meaningful.