Spatial reference · version 1

Circular tangent-joined profile

A reconstructible reading of a general Omega engineering profile: circular crown, circular return fillets, and straight feet, joined with a common tangent. It is a proposed model. It is not a recovered copy of missing equations, and the illustrative dimensions are not the archive’s marked reference geometry. No stress or electromagnetic field is solved.

RCrown radius, greater than 0.rFillet radius, greater than 0.alpha, degreesBetween 0 and 90. Not the seed angle.LLength of each foot.bExtrusion depth.tWall thickness. Not a stress input.a_wWire radius. Not the archive scale a.B_revRevolution radius. Not archive B.

Illustrative values use R = 1 and the stated fractions of R. They are display choices, not measured archive values. Cases 2 and 3 change only R, r, α, and L.

Construction

View

Front of the profile plane. Solid strokes are the five pieces, left to right. Dashed circles are the defining circles; the omitted lower crown sector is not part of the curve.

Derived from the current fields. Binary64, shown to six significant figures. Not an exact certificate and not archive measurements.
Quantity	Value
d = (R+r) sin α	0.676820
h = r + (R+r) cos α	1.14660
Height H = h + R	2.14660
Envelope width W = 2 max(R, d+L)	2.75364
Crown span 2π − 2α	5.06145
Crown span / π	1.61111
Crown arc length R(2π − 2α)	5.06145
Each fillet span π − α	2.53073
Developed length ℓ	7.37252
Max join gap	0
Max unit-tangent gap	0
Center residual vs R+r	0
Fillet-center distance 2d	1.35364
Sampled non-adjacent clearance	0.343338

The crown span is greater than π, so the drawn crown is the major arc. The fillet-center distance exceeds 2r, so the two fillet disks are disjoint. The clearance row samples non-adjacent pieces. It is not a proof that the curve or a wire tube is free of intersections. The crown length is R(2π − 2α). It is not the archive measurement C = aπ.

Three fixed cases. Each row recomputes coincident joins, center separation R+r, and crown span in binary64.
Case	R	r	α	L	Center residual	Span / π
illustrative	1.00000	0.180000	35°	0.700000	0	1.61111
case-2	2.00000	0.500000	30°	1.00000	0.00000	1.66667
case-3	1.50000	0.200000	20°	0.400000	0.00000	1.77778

Five pieces, left to right

Each piece uses its own u in [0, 1]. Those speeds are not one global C1 parameterization. Where the join residuals above vanish, position and unit tangent agree, so arc length is a regular tangent-continuous parameter. That observation is binary64, not an exact certificate. There is no sharp corner at the crown/fillet joins. The two ends stay apart: the curve is open.

\gamma_1(u)=(-d-L+Lu,\ 0,\ 0).

\theta=-\pi/2+u(\pi-\alpha),\qquad \gamma_2(u)=(-d+r\cos\theta,\ r+r\sin\theta,\ 0).

\theta=-\pi/2-\alpha-u(2\pi-2\alpha),\qquad \gamma_3(u)=(R\cos\theta,\ h+R\sin\theta,\ 0).

\theta=\pi/2+\alpha+u(\pi-\alpha),\qquad \gamma_4(u)=(d+r\cos\theta,\ r+r\sin\theta,\ 0).

\gamma_5(u)=(d+Lu,\ 0,\ 0).

In this left-to-right order the signed curvatures are 0, +1/r, −1/R, +1/r, 0. Numerically: 0.00000, 5.55556, -1.00000, 5.55556, 0.00000. Reversing the traversal reverses every sign. A jump in curvature is not, by itself, infinite stress or an infinite field.

Three constructions

Let sigma be arc length, T the unit tangent, and n = (−T_y, T_x, 0) the in-plane unit normal. Sigma is not the archive half-chord s = sin(ε/2). The profile alone does not determine torsion, chirality, or a magnetic moment.

Extruded midsurface, depth b > 0. A thin wall of thickness t uses the normal offsets. Ends are open.

X(\sigma,v)=\gamma(\sigma)+v e_z,\qquad -b/2\le v\le b/2.

\gamma(\sigma)\pm(t/2)\,n(\sigma).

A bonded baseline skin would be more material and another load path. Only with that closure would the section be a cell. Attachment and boundary conditions are not supplied here.

Circular wire. Local regularity uses 0 < a_w < min(R, r). Global clearance is a separate sampled check, not a certificate. Ends need their own geometry, which is not drawn.

X(\sigma,\theta)=\gamma(\sigma)+a_w[\cos\theta\, n(\sigma)+\sin\theta\, e_z].

Shell of revolution about an axis offset by B_rev. The profile’s x becomes the axial coordinate. This is not a view of the extrusion. The condition used here is B_rev > 0, which keeps B_rev + y > 0 because this profile has y ≥ 0.

X(\sigma,\theta)=\big((B_{\mathrm{rev}}+y(\sigma))\cos\theta,\ (B_{\mathrm{rev}}+y(\sigma))\sin\theta,\ x(\sigma)\big).

Not the six-mark construction

The archive’s marked construction uses centered coordinates a(−A, −B, −s, s, B, A), with A = s + φ², B = s + 1, and s = sin(ε/2), gap ratios (φ, 1, 2s, 1, φ), and selected measurements G = aφ, C = aπ. Those constraints are additional structure. A general R, r, α, L profile does not have them. An identification would need an explicit map that keeps label order, measurements, frame, and scale. No such map is stated. In particular the major crown is not silently replaced by C = aπ. The marked kernel’s cameras are drawn on the canonical marked view. Spine measurements and the y-axis revolution are stated on the geometry page. Front, side, and oblique on this page name a camera for this proposed profile only. An offset here is not a spine length. Revolution uses the axis you set, and a point-sized section does not by itself keep features apart.

R(2\pi-2\alpha)\ne \pi R\quad\text{for the angles used on this page.}

Conditional statements, not results

For an ideal sharp wedge of angle β > π and suitable homogeneous Dirichlet data, a leading harmonic corner term can behave as below. Its gradient is singular when the coefficient is nonzero. That mechanism needs the sharp geometry and the stated problem. The joins drawn here are tangent. Fillets, material response, forcing, and boundary conditions matter, and none of them is solved on this page.

u\sim c\,\rho^{\pi/\beta}\sin(\pi\theta/\beta).

For a thin isotropic closed cell, under the assumptions of Bredt–Batho torsion, torsional rigidity can be modeled as follows. A_m is the enclosed median-line area. The formula is not evaluated. It does not prove optimality of an Omega section, remove warping, or set a universal stiffness. Composite laminates need their own constitutive model. The open profile is not that closed cell.

K_t=4A_m^2\Big/\int_{\mathrm{closed}}\frac{ds}{G(s)\,t(s)}.

Reciprocal omega-type magnetoelectric response is a real engineering class. It is not implied by this silhouette. For compatible dipole and field normalizations, the cross-polarizabilities satisfy the transpose relation below. Reciprocal matched single-mode scattering ports obey S21 = S12, while S11 and S22 can differ. Polarization-converting or multimode comparisons have to name the corresponding modes. Double-negative parameters and application performance do not follow from the drawing. The wire-view arrows are possible dipole axes, not a computed moment.

\alpha_{em}=-\alpha_{me}^{\mathsf T}.

Five pieces, left to right

Three constructions

Not the six-mark construction

Conditional statements, not results

Sources checked for this interpretation