DERIV-fillet-setback — Setback of a corner fillet

Used by: EN-004, DERIV-local-radius-clamp step 1. Replaces the unverified REF-GG-FILLET with steps each backed by a verified source.

Notation

Symbol	Definition
B	corner vertex
θ	interior angle at B, between the two edges
τ	turning angle at B (DERIV-turning-angle), θ = π − |τ|
r	fillet radius, r ≥ 0
D	center of the fillet
E	tangent point on one edge
s	setback: distance BE from the vertex to the tangent point

Step 1 — Center on the bisector

Take D on the bisector of the angle at B, at distance r from one edge. Bisecting the angle gives two congruent right triangles, so the perpendiculars from D to both edges are equal (Euclid IV.4, REF-EUCLID-IV4): D is at distance r from both edges.

Step 2 — Tangency

The line drawn at right angles to a radius from its end touches the circle (Euclid III.16, porism, REF-EUCLID-III16), so the circle of center D and radius r is tangent to both edges at the feet of the perpendiculars. Conversely the radius to the point of contact is perpendicular to the tangent (Euclid III.18, REF-EUCLID-III18): the triangle BED is right-angled at E.

Step 3 — Angles of the triangle BED

• Angle at B: θ / 2 (bisector).
• Angle at E: π / 2 (step 2).
• Angle at D: π − π/2 − θ/2 = (π − θ) / 2 = |τ| / 2, since the angles of a triangle sum to two right angles (Euclid I.32, REF-EUCLID-I32).

Step 4 — Setback

In the right triangle BED, the tangent of an angle is the opposite side over the adjacent side (REF-MATHWORLD-TANGENT). At D, the opposite side is BE = s, the adjacent side is DE = r:

s = r · tan(|τ| / 2), equivalently s = r / tan(θ / 2) (same triangle, angle at B).

Cross-check: road design uses the same relation for the tangent length of a circular curve, T = R tan(Δ/2), Δ being the supplement of the interior angle between the tangents, i.e. |τ| (REF-WIKIBOOKS-HORIZONTAL-CURVES, level-6 source, agreement only).

Checks (test cases)

r	|τ|	s
10	π/2 (right corner)	10 · tan(π/4) = 10
30	0 (aligned)	0: no arc
10	π/3	10 · tan(π/6) = 10 / √3 ≈ 5.773503
0	π/2	0: sharp corner
10	−π/2 (counter-clockwise turn)	10: only |τ| matters

Limits

• |τ| → π (very sharp spike): s → ∞; the clamp of DERIV-local-radius-clamp bounds it.