# ABACUS R8 harmonic tripod geometry authority

## 1. Cross-sectional field

```text
k(q,u) = 2/W(u) · [1 + ε(u) cos(6πq)]
ψ(q,u) = 2πq + ε(u)/3 · sin(6πq)
γ(q,u) = πW(u) ∫₀q exp(iψ(t,u))dt − c(u)
S(q,u) = (Re γ(q,u), Im γ(q,u), Hu)
```

## 2. Axial fields

```text
W(u) = Wb + m u + (Wt − Wb − m)(3u² − 2u³)
m = −2pH
ε(u) = ε₀(1 − 6u² + 4u³)
```

## 3. Tripod lower boundary

Let the bottom curvature poles occur at normalized arclength:

```text
q = 0, 1/3, 2/3
```

The unique lowest-frequency, reflection-symmetric, triaxial lift field with zero lift at those poles is:

```text
a(q) = [1 − cos(6πq)]/2 = sin²(3πq)
```

The lower edge is:

```text
z₀(q) = hₐ a(q)
u₀(q) = z₀(q)/H
B(q)  = S(q,u₀(q))
```

This is not a second modeled surface. It is the analytic boundary of the original surface domain.

## 4. Least-action character

Among periodic, threefold, reflection-symmetric fields, `cos(6πq)` is the first admissible nonconstant harmonic. Holding amplitude and phase constraints fixed, it minimizes every derivative-energy functional:

```text
Jm = ∫ |dᵐa/dqᵐ|² dq
```

relative to any addition of higher harmonics. The arch profile therefore adds the minimum possible spectral complexity.

## 5. Phase lock to curvature

At the bottom:

```text
k(q,0) = k₀[1 + ε cos(6πq)]
```

For `ε > 0`:

```text
a(q) = [kmax − k(q,0)] / [kmax − kmin]
```

The feet are the maximum-curvature poles. The arch apices are the minimum-curvature side midpoints.

## 6. Intake-area forcing

The app does not expose `hₐ` as a shape slider.

Target:

```text
Aeffective = Afan
Afan = πD²/4
Ageometric,target = Afan / φ
```

with fixed baseline:

```text
D = 140 mm
φ = 0.80
```

For the zero-taper estimate, every constant-width section obeys Barbier's theorem:

```text
L = πWb
```

and the harmonic's mean lift is `hₐ/2`, giving:

```text
Ageometric = Lhₐ/2
hₐ = D²/(2φWb)
```

The implementation then solves `hₐ` by bisection against the actual triangulated intake curtain of the tapered analytic surface.

## 7. Physical support

The exact shell boundary reaches the desk at three mathematical poles. Real load is carried by three concealed replaceable elastomer pads centered just inside those poles.

This preserves:

- exact visible surface;
- three-point non-rocking support;
- furniture protection;
- vibration isolation;
- casting-tolerance accommodation.
