DERIV-fillet-arc — Tangent points, center and SVG flags of a corner fillet

Used by: EN-006 (geometry), EN-007 (path data). Builds on DERIV-fillet-setback and DERIV-turning-angle.

Notation

Screen frame of SVG: x to the right, y down (REF-SVG2-COORDS §8.4).

Symbol	Definition
P, V, N	previous, current and next vertices of the contour
u_in, u_out	unit directions of the incoming edge V − P and the outgoing edge N − V
τ	turning angle at V (DERIV-turning-angle), in ]−π, π[; τ > 0 when the contour turns clockwise on screen
r	effective radius at V (after DERIV-local-radius-clamp), r ≥ 0
s	setback, s = r · tan(|τ| / 2) (DERIV-fillet-setback)
w⊥	w rotated by a quarter turn: (x, y)⊥ = (−y, x)

Step 1 — Unit directions

u = w / ‖w‖ with ‖w‖ = √(x² + y²) (REF-MATHWORLD-UNIT-VECTOR, REF-MATHWORLD-VECTOR-NORM). A zero-length edge (a size of 0, Q15) has no direction: its unit vector is taken as (0, 0). Its turning angle is then taken as 0 explicitly (DERIV-turning-angle: not left to atan2, which gives π for atan2(+0, −0)), so s = 0 and the vertex keeps no arc.

Step 2 — Tangent points

The fillet touches each edge at distance s from V (DERIV-fillet-setback, steps 1–4: the two tangent lengths are equal); a point at distance s along a unit direction is V + s · u (scalar multiplication and addition in components, REF-OPENSTAX-CALC3-VECTORS):

• on the incoming edge: T_in = V − s · u_in;
• on the outgoing edge: T_out = V + s · u_out.

Step 3 — Center

The radius to the point of contact is perpendicular to the tangent (Euclid III.18, REF-EUCLID-III18): the center C lies on the perpendicular to the incoming edge through T_in, at distance r. The quarter turn of a vector is (x, y) ↦ (−y, x), the rotation matrix with angle π/2 (REF-MATHWORLD-ROTATION-MATRIX).

The center is on the side of the angle's bisector (Euclid IV.4, REF-EUCLID-IV4), the side towards which the contour turns. Since perpDot(u_in, u_out) = u_in⊥ · u_out = sin τ (REF-MATHWORLD-PERP-DOT), the outgoing edge leaves on the u_in⊥ side when τ > 0 and on the other side when τ < 0:

C = T_in + sign(τ) · r · u_in⊥

Step 4 — Angle of the arc and large-arc flag

In the quadrilateral V T_in C T_out, the angles at T_in and T_out are right (step 3) and the angle at V is θ = π − |τ|. The angles of a quadrilateral sum to four right angles (two triangles, Euclid I.32, REF-EUCLID-I32), so the angle at C is 2π − π − θ = |τ|. As |τ| < π, the fillet is the smaller arc: large-arc-flag = 0 (REF-SVG2-IMPLNOTE B.2.1: fA = 1 only for an arc of more than 180°).

Step 5 — Direction of the arc and sweep flag

Along the arc from T_in to T_out, the tangent turns from u_in to u_out, by τ. The radius stays perpendicular to the tangent (Euclid III.18), so it turns by the same signed angle: the angle around C increases when τ > 0. SVG draws the arc "in a positive-angle direction" when the sweep flag is 1 (REF-SVG2-PATHS §9.3.8, REF-SVG2-IMPLNOTE B.2.1). Hence sweep-flag = 1 when τ > 0, 0 when τ < 0. With y down, a positive angle turns clockwise on screen: the convex corners of a clockwise contour (Q11) have τ > 0 and sweep 1.

Step 6 — Radius 0

r = 0 gives s = 0, T_in = T_out = V: no arc. SVG would also draw an arc of zero radius as a straight line (REF-SVG2-IMPLNOTE B.2.5, step 1); the library never writes one.

Step 7 — Starting point of a rounded contour

A contour starts at its top-left vertex (Q11). Once rounded, that vertex is no longer on the path; the path starts at T_out of the first vertex, so that its commands follow the vertices in order — edge to the second vertex first — and the first vertex's arc closes the path. This is a convention of the output, chosen for this order; starting at T_in would draw the same shape.

Pieces without length are dropped: a segment between two tangent points that meet (two fillets covering a whole edge), or the arc of a corner that keeps no fillet (T_in = T_out). A piece is kept when the distance between its ends is at least EPSILON (Q10). Consecutive pieces therefore meet within EPSILON, not exactly.

When the contour is written as path data, a final segment is left to closepath, which draws a straight line back to the start (REF-SVG2-PATHS §9.3.4): with every radius 0 and edges of positive length, the result is exactly the sharp path data M p₀ L p₁ … Z. Zero-length edges (a size of 0, Q15) are dropped, so a 0 × 50 rectangle is written M0 0 L0 50 Z: the same drawing with fewer commands.

Checks (test cases)

Rectangle 100 × 50 from the top-left vertex, clockwise: (0, 0), (100, 0), (100, 50), (0, 50), effective radius 10 everywhere. Every corner has τ = π/2, so s = 10 · tan(π/4) = 10.

Vertex	u_in	u_out	T_in	T_out	u_in⊥	C	flags
(0, 0)	(0, −1)	(1, 0)	(0, 10)	(10, 0)	(1, 0)	(10, 10)	0 1
(100, 0)	(1, 0)	(0, 1)	(90, 0)	(100, 10)	(0, 1)	(90, 10)	0 1
(100, 50)	(0, 1)	(−1, 0)	(100, 40)	(90, 50)	(−1, 0)	(90, 40)	0 1
(0, 50)	(−1, 0)	(0, −1)	(10, 50)	(0, 40)	(0, −1)	(10, 40)	0 1

Other cases:

Case	Computation	Expected
Concave corner: u_in = (1, 0), u_out = (0, −1), V = (10, 0), r = 5	perpDot = 1·(−1) − 0·0 = −1 → τ = −π/2, s = 5	T_in = (5, 0), T_out = (10, −5), C = (5, 0) − 5·(0, 1) = (5, −5), flags 0 0
Aligned vertex, r = 30	τ = 0 → s = 0	T_in = T_out = V, no arc
Square 100 × 100, r = 50	s = 50 on every corner	arcs meet: T_out of (100, 0) = (100, 50) = T_in of (100, 100); no segment of positive length
Zero-length edge (rectangle 0 × 50)	unit vector (0, 0), τ = 0	s = 0, no arc, no NaN

Each center is at distance r from both tangent points, e.g. ‖(90, 10) − (100, 10)‖ = 10.

Limits

• The turning angle must be in ]−π, π[. At a vertex where the contour turns back on itself (τ = ±π, a spike), tan(|τ|/2) is huge: the clamp then reduces the radius to almost 0 while the setback covers the whole edge, so the spike is consumed. This is the intended behavior (Q16, settled by the Product Owner); the sign of τ there also depends on the sign of a zero.
• At τ = 0, sign(τ) = 0 gives C = V; there is no arc, so no center is used.
• The angle between C − T_in and C − T_out equals |τ|; the library never needs the center to write the path (only radius, flags and end point), but keeps it for later uses (hit-testing, snapping).