Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Residual cones and true smoothing displacements

Tensor conventions and separated characteristic geometry › Residual cones and true smoothing displacements · lines 611–663

Em01-F@3#eq:residualcones, Em01-F@3#eq:residualconstraint, Em01-F@3#eq:shift1, Em01-F@3#eq:shift2

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Write

r = A_{r} ℓ_{n} + B_{r} k_{n} + Z_{r}^{A} E_{A} .

Then the additional internal cone equations are exactly

(b - z)^{2} = 4 (β - A_{r}) (u - B_{r}) - ∣ Z_{r} ∣^{2}, r^{2} = 4 A_{r} B_{r} - ∣ Z_{r} ∣^{2} .

The $K_{5, +}$ tip line has argument $- r$ , and the $K_{5, -}$ tip line has argument $r$ ; their retarded time orientations must not be interchanged.

If $β - A_{r} > 0$ , the equation $(b - z)^{2} = 0$ gives

B_{r} = u - \frac{∣ Z _{r} ∣ ^{2}}{4 ( β - A _{r} )} .

This is a geometric constraint, not a scaling rule for $A_{r}$ .

For actual smoothing displacements

H_{i} = A_{i} ℓ_{n} + B_{i} k_{n} + Z_{i}^{A} E_{A}, d = β - α,

one has

(y - a - δ H_{1})^{2} (y - b - δ H_{2})^{2} = 4 (α - δ A_{1}) (v - δ B_{1}) - ∣ ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (u - v + δ B_{2}) - ∣ ξ - δ Z_{2} ∣^{2} .

Thus

(y - b - δ H_{2})^{2} = (b - y)^{2} + 2 δ (b - y) \cdot H_{2} + δ^{2} H_{2}^{2} .

These equations retain the residual variable and its cone-tip transitions. The geometry does not assign a common rescaling weight to $r$ .

Displayed formulas in this passage

Em01-F@3#eq:residualcones · line 618
$(b - z)^{2} = 4 (β - A_{r}) (u - B_{r}) - ∣ Z_{r} ∣^{2}, r^{2} = 4 A_{r} B_{r} - ∣ Z_{r} ∣^{2} .$
Em01-F@3#eq:residualconstraint · line 631
$B_{r} = u - \frac{∣ Z _{r} ∣ ^{2}}{4 ( β - A _{r} )} .$
Em01-F@3#eq:shift1 · line 643
$(y - a - δ H_{1})^{2} (y - b - δ H_{2})^{2} = 4 (α - δ A_{1}) (v - δ B_{1}) - ∣ ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (u - v + δ B_{2}) - ∣ ξ - δ Z_{2} ∣^{2} .$
Em01-F@3#eq:shift2 · line 643
$(y - a - δ H_{1})^{2} (y - b - δ H_{2})^{2} = 4 (α - δ A_{1}) (v - δ B_{1}) - ∣ ξ - δ Z_{1} ∣^{2}, = 4 (d + δ A_{2}) (u - v + δ B_{2}) - ∣ ξ - δ Z_{2} ∣^{2} .$
unlabeled · line 614
$r = A_{r} ℓ_{n} + B_{r} k_{n} + Z_{r}^{A} E_{A} .$
unlabeled · line 638
$H_{i} = A_{i} ℓ_{n} + B_{i} k_{n} + Z_{i}^{A} E_{A}, d = β - α,$
unlabeled · line 654
$(y - b - δ H_{2})^{2} = (b - y)^{2} + 2 δ (b - y) \cdot H_{2} + δ^{2} H_{2}^{2} .$