New math, plus script to download latex images of equations (locusrobotics#44)

* New math, plus script to download latex images of equations

* General Taylor Series derivation

Author: DLu

dlux_global_planner/Derivation.md

@@ -0,0 +1,78 @@
+# Full Derivation
+We start with our basic equation definitions.
+ * ![P(X) = \frac{-\beta + \sqrt{\beta^2 - 4 \gamma}}{2}](doc/PX_frac_beta_sqrtbeta_2_4gamma2.gif)
+ * ![\beta = -\Big(P(C) + P(A)\Big)](doc/beta__BigPC_PABig.gif)
+ * ![\gamma = \frac{P(A)^2 + P(C)^2 - h^2}{2}](doc/gamma_fracPA_2_PC_2_h_22.gif)
+
+## Reformulate
+First we want to reformulate in terms of our new ![\delta](doc/delta.gif) variable.
+ * ![\delta = \frac{P(C) - P(A)}{h}](doc/delta_fracPC_PAh.gif)
+
+We can rewrite this so we have a new definition of ![P(C)](doc/PC.gif)
+ * ![P(C) = h\delta  + P(A)](doc/PC_hdelta_PA.gif)
+
+This allows us to derive new values for the other equations.
+ * ![\beta = -\Big(P(C) + P(A)\Big)](doc/beta__BigPC_PABig.gif)
+ * ![\beta = -\Big(h\delta + P(A) + P(A)\Big)](doc/beta__Bighdelta_PA_PABig.gif)
+ * ![\beta = -h\delta - 2 P(A)](doc/beta__hdelta_2PA.gif)
+ * ![\gamma = \frac{P(A)^2 + P(C)^2 - h^2}{2}](doc/gamma_fracPA_2_PC_2_h_22.gif)
+ * ![\gamma = \frac{P(A)^2 + \Big(P(A)+h\delta\Big)^2 - h^2}{2}](doc/gamma_fracPA_2_BigPA_hdeltaBig_2_h_22.gif)
+ * ![\gamma = P(A)^2 + h\delta P(A) + \frac{h^2\delta^2 - h^2}{2}](doc/gamma_PA_2_hdeltaPA_frach_2delta_2_h_22.gif)
+ * ![P(X) = \frac{-\beta + \sqrt{\beta^2 - 4 \gamma}}{2}](doc/PX_frac_beta_sqrtbeta_2_4gamma2.gif)
+ * ![P(X) = \frac{-(-h\delta - 2 P(A)) + \sqrt{\Big(-h\delta - 2 P(A)\Big)^2 - 4 \Big(P(A)^2 + h\delta P(A) + \frac{h^2\delta^2 - h^2}{2}\Big)}}{2}](doc/PX_frac__hdelta_2PA_sqrtBig_hdelta_2PABig_2_4BigPA_2_hdeltaPA_frach_2delta_2_h_22Big2.gif)
+ * ![P(X) = \frac{h\delta + 2 P(A) + \sqrt{\Big(-h\delta - 2 P(A)\Big)^2 - 4 P(A)^2 -4 h\delta P(A) - 2h^2\delta^2 + 2h^2}}{2}](doc/PX_frachdelta_2PA_sqrtBig_hdelta_2PABig_2_4PA_2_4hdeltaPA_2h_2delta_2_2h_22.gif)
+ * ![P(X) = \frac{h\delta + 2 P(A) + \sqrt{4P(A)^2 + 4h\delta P(A) + h^2\delta^2 - 4 P(A)^2 -4 h\delta P(A) - 2h^2\delta^2 + 2h^2}}{2}](doc/PX_frachdelta_2PA_sqrt4PA_2_4hdeltaPA_h_2delta_2_4PA_2_4hdeltaPA_2h_2delta_2_2h_22.gif)
+ * ![P(X) = \frac{h\delta + 2 P(A) + \sqrt{-h^2\delta^2 + 2h^2}}{2}](doc/PX_frachdelta_2PA_sqrt_h_2delta_2_2h_22.gif)
+ * ![P(X) = P(A) + \frac{h}{2} \Big(\delta + \sqrt{2-\delta^2}\Big)](doc/PX_PA_frach2Bigdelta_sqrt2_delta_2Big.gif)
+
+We're going to call this a new function ![f(\delta)](doc/fdelta.gif), so now ![P(X)](doc/PX.gif) is in terms of ![f(\delta)](doc/fdelta.gif)
+ * ![f(\delta) = P(A) + \frac{h}{2} \Big(\delta + \sqrt{2-\delta^2}\Big)](doc/fdelta_PA_frach2Bigdelta_sqrt2_delta_2Big.gif)
+
+## Taylor Series Expansion
+For computational efficiency, we are going to compute the Taylor series expansion. So first, we'll need the derivatives of ![f(\delta)](doc/fdelta.gif).
+ * ![f'(\delta) = \frac{h}{2} (1 - \frac{\delta}{\sqrt{2-\delta^2}})](doc/f_delta_frach21_fracdeltasqrt2_delta_2.gif)
+ * ![f''(\delta) = \frac{-h}{(2-\delta^2)^\frac{3}{2}}](doc/f__delta_frac_h2_delta_2_frac32.gif)
+
+### 0th Order Taylor
+ * ![f_0(\delta, a) = f(a)](doc/f_0delta_a_fa.gif)
+ * ![f_0(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2}\Big)](doc/f_0delta_a_PA_frach2Biga_sqrt2_a_2Big.gif)
+
+### 1st Order Taylor
+ * ![f_1(\delta, a) = f(a) + f'(a) (\delta - a)](doc/f_1delta_a_fa_f_adelta_a.gif)
+ * ![f_1(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2}\Big) + \frac{h}{2} (1 - \frac{a}{\sqrt{2-a^2}}) (\delta - a)](doc/f_1delta_a_PA_frach2Biga_sqrt2_a_2Big_frach21_fracasqrt2_a_2delta_a.gif)
+ * ![f_1(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2} + (1 - \frac{a}{\sqrt{2-a^2}}) (\delta - a)\Big)](doc/f_1delta_a_PA_frach2Biga_sqrt2_a_2_1_fracasqrt2_a_2delta_aBig.gif)
+ * ![f_1(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2} + \delta - a - \frac{a\delta}{\sqrt{2-a^2}} + \frac{a^2}{\sqrt{2-a^2}}\Big)](doc/f_1delta_a_PA_frach2Biga_sqrt2_a_2_delta_a_fracadeltasqrt2_a_2_fraca_2sqrt2_a_2Big.gif)
+ * ![f_1(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \sqrt{2-a^2} + \frac{a^2}{\sqrt{2-a^2}}\Big)](doc/f_1delta_a_PA_frach2Bigdelta_fracadeltasqrt2_a_2_sqrt2_a_2_fraca_2sqrt2_a_2Big.gif)
+ * ![f_1(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \frac{2}{\sqrt{2-a^2}}\Big)](doc/f_1delta_a_PA_frach2Bigdelta_fracadeltasqrt2_a_2_frac2sqrt2_a_2Big.gif)
+
+### 2nd Order Taylor
+ * ![f_2(\delta, a) = f(a) + f'(a) (\delta - a) + \frac{f''(a)}{2!}(\delta - a)^2](doc/f_2delta_a_fa_f_adelta_a_fracf__a2_delta_a_2.gif)
+ * ![f_2(\delta, a) = f_1(\delta, a) + \frac{1}{2} \frac{-h}{(2-a^2)^\frac{3}{2}} (\delta - a)^2](doc/f_2delta_a_f_1delta_a_frac12frac_h2_a_2_frac32delta_a_2.gif)
+ * ![f_2(\delta, a) = f_1(\delta, a) + \frac{h}{2} \frac{-1}{(2-a^2)^\frac{3}{2}} (\delta^2 - 2\delta a + a^2)](doc/f_2delta_a_f_1delta_a_frach2frac_12_a_2_frac32delta_2_2deltaa_a_2.gif)
+ * ![f_2(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \frac{2}{\sqrt{2-a^2}}\Big) + \frac{h}{2} \frac{-1}{(2-a^2)^\frac{3}{2}} (\delta^2 - 2\delta a + a^2)](doc/f_2delta_a_PA_frach2Bigdelta_fracadeltasqrt2_a_2_frac2sqrt2_a_2Big_frach2frac_12_a_2_frac32delta_2_2deltaa_a_2.gif)
+ * ![f_2(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \frac{2}{\sqrt{2-a^2}} + \frac{-1}{(2-a^2)^\frac{3}{2}} (\delta^2 - 2\delta a + a^2)\Big)](doc/f_2delta_a_PA_frach2Bigdelta_fracadeltasqrt2_a_2_frac2sqrt2_a_2_frac_12_a_2_frac32delta_2_2deltaa_a_2Big.gif)
+ * ![f_2(\delta, a) = P(A) + h\Big(c_0(a) + c_1(a)\delta + c_2(a) \delta^2\Big)](doc/f_2delta_a_PA_hBigc_0a_c_1adelta_c_2adelta_2Big.gif)
+ * ![c_0(a) = \frac{1}{2}\Big(\frac{2}{\sqrt{2-a^2}} - \frac{a^2}{(2-a^2)^\frac{3}{2}}\Big)](doc/c_0a_frac12Bigfrac2sqrt2_a_2_fraca_22_a_2_frac32Big.gif)
+ * ![c_0(a) = \frac{4 - 3a^2}{2(2-a^2)^\frac{3}{2}}](doc/c_0a_frac4_3a_222_a_2_frac32.gif)
+
+ * ![c_1(a) = \frac{1}{2}\Big(1 - \frac{a}{\sqrt{2-a^2}} + \frac{2a}{(2-a^2)^\frac{3}{2}} \Big)](doc/c_1a_frac12Big1_fracasqrt2_a_2_frac2a2_a_2_frac32Big.gif)
+ * ![c_2(a) = \frac{-1}{2(2-a^2)^\frac{3}{2}}](doc/c_2a_frac_122_a_2_frac32.gif)
+
+## Exact coefficients
+Now that we have the general equations for the Taylor series, we can evaluate it at different values of a in the range `[0, 1]`.
+
+| ![a](doc/a.gif) | ![c_0(a)](doc/c_0a.gif) | ![c_1(a)](doc/c_1a.gif) | ![c_2(a)](doc/c_2a.gif) |
+| ------ | ----------- | ----------- | ----------- |
+|  0.0   |    0.7071   |    0.5000   |   -0.1768   |
+|  0.5   |    0.7019   |    0.5270   |   -0.2160   |
+|  1.0   |    0.5000   |    1.0000   |   -0.5000   |
+
+Historically, the values used by [`navfn`](https://github.com/ros-planning/navigation/blob/1f335323a605b49b4108a845c55a7c1ba93a6f2e/navfn/src/navfn.cpp#L509) are
+
+|  ![c_0](doc/c_0.gif)   |  ![c_1](doc/c_1.gif)   |  ![c_2](doc/c_2.gif)   |
+| ----------- | ----------- | ----------- |
+|    0.7040   |   0.5307    |   -0.2301   |
+
+You can see these values plotted [here](https://www.desmos.com/calculator/vbpkey1mt6).
+
+The historical values are pretty close to the values for ![\delta=0.5](doc/delta_0_5.gif), although the exact reason for the difference is unknown, but its close enough to not be overly concerning.

dlux_global_planner/README.md

@@ -111,23 +111,24 @@ Below, we provide a "brief" mathematical derivation of the calculation.
  `D` are on the other axis.
  * The cost of moving to a cell is the cost from the costmap, which we'll call `h` (to match the paper's notation)
  * We assume, without loss of generality that
-  * `P(A) <= P(B)`
-  * `P(C) <= P(D)`
-  * `P(A) <= P(C)`
- * If `P(C)` is infinite, that is, not initialized yet, the new potential calculation is straightforwardly  ![P(X) = P(A) + h](https://latex.codecogs.com/gif.latex?P%28X%29%20%3D%20P%28A%29%20&plus;%20h).
- * Otherwise, we want to find a value of `P(X)` that satisfies the equation ![(P(X) - P(A))^2 + (P(X) - P(C))^2 = h^2](https://latex.codecogs.com/gif.latex?%5CBig%28P%28X%29-P%28A%29%5CBig%29%5E2&plus;%5CBig%28P%28X%29-P%28C%29%5CBig%29%5E2%3Dh%5E2)
- * It's possible there are no real values that satisfy the equation if ![P(C) - P(A) >= h](https://latex.codecogs.com/gif.latex?P%28C%29%20-%20P%28A%29%20%5Cgeq%20h) in which case the straightforward update is used.
+  * ![P(A) <= P(B)](doc/PA__PB.gif)
+  * ![P(C) <= P(D)](doc/PC__PD.gif)
+  * ![P(A) <= P(C)](doc/PA__PC.gif)
+ * If ![P(C)](doc/PC.gif) is infinite, that is, not initialized yet, the new potential calculation is straightforwardly  ![P(X) = P(A) + h](doc/PX_PA_h.gif).
+ * Otherwise, we want to find a value of `P(X)` that satisfies the equation ![\Big(P(X) - P(A)\Big)^2 + \Big(P(X) - P(C)\Big)^2 = h^2](doc/BigPX_PABig_2_BigPX_PCBig_2_h_2.gif)
+ * It's possible there are no real values that satisfy the equation if ![P(C) - P(A) \geq h](doc/PC_PAgeqh.gif) in which case the straightforward update is used.
  * Otherwise, through clever manipulation of the quadratic formula, we can solve the equation with the following:
-   * ![P(X) = 0.5 * (-beta + sqrt(beta^2 - 4 * gamma))](https://latex.codecogs.com/gif.latex?P%28X%29%20%3D%20%5Cfrac%7B-%5Cbeta%20&plus;%20%5Csqrt%7B%5Cbeta%5E2%20-%204%20%5Cgamma%7D%7D%7B2%7D)
-   * ![beta = -(P(A) + P(C))](https://latex.codecogs.com/gif.latex?%5Cbeta%20%3D%20-%5CBig%28P%28A%29&plus;P%28C%29%5CBig%29)
-   * ![gamma = 0.5 * (P(A)^2 + P(C)^2 - h^2)](https://latex.codecogs.com/gif.latex?%5Cgamma%20%3D%20%5Cfrac%7BP%28A%29%5E2%20&plus;%20P%28C%29%5E2%20-%20h%5E2%7D%7B2%7D)
- * That all looks complicated, and computationally inefficient due to the square root operation. Hence, we calculate a second-degree Taylor series approximation at ![x=P(C)+h/2](https://latex.codecogs.com/gif.latex?x=P(C)&plus;h/2) as
-  * ![delta = (P(C) - P(A)) / h](https://latex.codecogs.com/gif.latex?%5Cdelta%3D%5Cfrac%7BP%28C%29-P%28A%29%7D%7Bh%7D)
-  * ![P(X) = P(A) + h (c_2 \delta^2 + c_1\delta + c_0)](https://latex.codecogs.com/gif.latex?P%28X%29%20%5Capprox%20P%28A%29%20&plus;%20h%20%28c_2%20%5Cdelta%5E2%20&plus;%20c_1%5Cdelta%20&plus;%20c_0%29)
-  * ![c_2 = -0.2301, c_1 = 0.5307, c_0 = 0.7040](https://latex.codecogs.com/gif.latex?c_2%20%3D%20-0.2301%2C%20c_1%20%3D%200.5307%2C%20c_0%20%3D%200.7040)
- * You can compare the two curves on [this plot](https://www.desmos.com/calculator/axz378nfqg)
-
-It is this final approximation that we use in our potential calculation. Although these constants have been used
+   * ![P(X) = \frac{-\beta + \sqrt{\beta^2 - 4 \gamma}}{2}](doc/PX_frac_beta_sqrtbeta_2_4gamma2.gif)
+   * ![\beta = -\Big(P(A)+P(C)\Big)](doc/beta__BigPA_PCBig.gif)
+   * ![\gamma = \frac{P(A)^2 + P(C)^2 - h^2}{2}](doc/gamma_fracPA_2_PC_2_h_22.gif)
+ * That all looks complicated, and computationally inefficient due to the square root operation. Hence, we reformulate the equation in terms of a new variable ![\delta](doc/delta.gif) and calculate a second-degree Taylor series approximation as
+  * ![\delta = \frac{P(C) - P(A)}{h}](doc/delta_fracPC_PAh.gif)
+  * ![P(X) = P(A) + \frac{h}{2} (\delta + \sqrt{2-\delta^2})](doc/PX_PA_frach2delta_sqrt2_delta_2.gif)
+  * ![P(X) \approx P(A) + h (c_2 \delta^2 + c_1\delta + c_0)](doc/PXapproxPA_hc_2delta_2_c_1delta_c_0.gif)
+ * If you're really interested, you can look into the [full derivation](Derivation.md)
+ * You can compare these equations on [this plot](https://www.desmos.com/calculator/p7x6d0kg6t)
+
+It is this final approximation that we use in our potential calculation. Although this method has been used
 [since the origins of the nav stack](https://github.com/ros-planning/navigation/blob/1f335323a605b49b4108a845c55a7c1ba93a6f2e/navfn/src/navfn.cpp#L509),
 this is likely the first time the meaning of the constants has ever been documented.
 

dlux_global_planner/doc/BigPX_PABig_2_BigPX_PCBig_2_h_2.gif

@@ -0,0 +1,78 @@
+# Full Derivation
+We start with our basic equation definitions.
+ * !eq[P(X) = \frac{-\beta + \sqrt{\beta^2 - 4 \gamma}}{2}]
+ * !eq[\beta = -\Big(P(C) + P(A)\Big)]
+ * !eq[\gamma = \frac{P(A)^2 + P(C)^2 - h^2}{2}]
+
+## Reformulate
+First we want to reformulate in terms of our new !eq[\delta] variable.
+ * !eq[\delta = \frac{P(C) - P(A)}{h}]
+
+We can rewrite this so we have a new definition of !eq[P(C)]
+ * !eq[P(C) = h\delta  + P(A)]
+
+This allows us to derive new values for the other equations.
+ * !eq[\beta = -\Big(P(C) + P(A)\Big)]
+ * !eq[\beta = -\Big(h\delta + P(A) + P(A)\Big)]
+ * !eq[\beta = -h\delta - 2 P(A)]
+ * !eq[\gamma = \frac{P(A)^2 + P(C)^2 - h^2}{2}]
+ * !eq[\gamma = \frac{P(A)^2 + \Big(P(A)+h\delta\Big)^2 - h^2}{2}]
+ * !eq[\gamma = P(A)^2 + h\delta P(A) + \frac{h^2\delta^2 - h^2}{2}]
+ * !eq[P(X) = \frac{-\beta + \sqrt{\beta^2 - 4 \gamma}}{2}]
+ * !eq[P(X) = \frac{-(-h\delta - 2 P(A)) + \sqrt{\Big(-h\delta - 2 P(A)\Big)^2 - 4 \Big(P(A)^2 + h\delta P(A) + \frac{h^2\delta^2 - h^2}{2}\Big)}}{2}]
+ * !eq[P(X) = \frac{h\delta + 2 P(A) + \sqrt{\Big(-h\delta - 2 P(A)\Big)^2 - 4 P(A)^2 -4 h\delta P(A) - 2h^2\delta^2 + 2h^2}}{2}]
+ * !eq[P(X) = \frac{h\delta + 2 P(A) + \sqrt{4P(A)^2 + 4h\delta P(A) + h^2\delta^2 - 4 P(A)^2 -4 h\delta P(A) - 2h^2\delta^2 + 2h^2}}{2}]
+ * !eq[P(X) = \frac{h\delta + 2 P(A) + \sqrt{-h^2\delta^2 + 2h^2}}{2}]
+ * !eq[P(X) = P(A) + \frac{h}{2} \Big(\delta + \sqrt{2-\delta^2}\Big)]
+
+We're going to call this a new function !eq[f(\delta)], so now !eq[P(X)] is in terms of !eq[f(\delta)]
+ * !eq[f(\delta) = P(A) + \frac{h}{2} \Big(\delta + \sqrt{2-\delta^2}\Big)]
+
+## Taylor Series Expansion
+For computational efficiency, we are going to compute the Taylor series expansion. So first, we'll need the derivatives of !eq[f(\delta)].
+ * !eq[f'(\delta) = \frac{h}{2} (1 - \frac{\delta}{\sqrt{2-\delta^2}})]
+ * !eq[f''(\delta) = \frac{-h}{(2-\delta^2)^\frac{3}{2}}]
+
+### 0th Order Taylor
+ * !eq[f_0(\delta, a) = f(a)]
+ * !eq[f_0(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2}\Big)]
+
+### 1st Order Taylor
+ * !eq[f_1(\delta, a) = f(a) + f'(a) (\delta - a)]
+ * !eq[f_1(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2}\Big) + \frac{h}{2} (1 - \frac{a}{\sqrt{2-a^2}}) (\delta - a)]
+ * !eq[f_1(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2} + (1 - \frac{a}{\sqrt{2-a^2}}) (\delta - a)\Big)]
+ * !eq[f_1(\delta, a) = P(A) + \frac{h}{2} \Big(a + \sqrt{2-a^2} + \delta - a - \frac{a\delta}{\sqrt{2-a^2}} + \frac{a^2}{\sqrt{2-a^2}}\Big)]
+ * !eq[f_1(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \sqrt{2-a^2} + \frac{a^2}{\sqrt{2-a^2}}\Big)]
+ * !eq[f_1(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \frac{2}{\sqrt{2-a^2}}\Big)]
+
+### 2nd Order Taylor
+ * !eq[f_2(\delta, a) = f(a) + f'(a) (\delta - a) + \frac{f''(a)}{2!}(\delta - a)^2]
+ * !eq[f_2(\delta, a) = f_1(\delta, a) + \frac{1}{2} \frac{-h}{(2-a^2)^\frac{3}{2}} (\delta - a)^2]
+ * !eq[f_2(\delta, a) = f_1(\delta, a) + \frac{h}{2} \frac{-1}{(2-a^2)^\frac{3}{2}} (\delta^2 - 2\delta a + a^2)]
+ * !eq[f_2(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \frac{2}{\sqrt{2-a^2}}\Big) + \frac{h}{2} \frac{-1}{(2-a^2)^\frac{3}{2}} (\delta^2 - 2\delta a + a^2)]
+ * !eq[f_2(\delta, a) = P(A) + \frac{h}{2} \Big(\delta - \frac{a\delta}{\sqrt{2-a^2}} + \frac{2}{\sqrt{2-a^2}} + \frac{-1}{(2-a^2)^\frac{3}{2}} (\delta^2 - 2\delta a + a^2)\Big)]
+ * !eq[f_2(\delta, a) = P(A) + h\Big(c_0(a) + c_1(a)\delta + c_2(a) \delta^2\Big)]
+ * !eq[c_0(a) = \frac{1}{2}\Big(\frac{2}{\sqrt{2-a^2}} - \frac{a^2}{(2-a^2)^\frac{3}{2}}\Big)]
+ * !eq[c_0(a) = \frac{4 - 3a^2}{2(2-a^2)^\frac{3}{2}}]
+
+ * !eq[c_1(a) = \frac{1}{2}\Big(1 - \frac{a}{\sqrt{2-a^2}} + \frac{2a}{(2-a^2)^\frac{3}{2}} \Big)]
+ * !eq[c_2(a) = \frac{-1}{2(2-a^2)^\frac{3}{2}}]
+
+## Exact coefficients
+Now that we have the general equations for the Taylor series, we can evaluate it at different values of a in the range `[0, 1]`.
+
+| !eq[a] | !eq[c_0(a)] | !eq[c_1(a)] | !eq[c_2(a)] |
+| ------ | ----------- | ----------- | ----------- |
+|  0.0   |    0.7071   |    0.5000   |   -0.1768   |
+|  0.5   |    0.7019   |    0.5270   |   -0.2160   |
+|  1.0   |    0.5000   |    1.0000   |   -0.5000   |
+
+Historically, the values used by [`navfn`](https://github.com/ros-planning/navigation/blob/1f335323a605b49b4108a845c55a7c1ba93a6f2e/navfn/src/navfn.cpp#L509) are
+
+|  !eq[c_0]   |  !eq[c_1]   |  !eq[c_2]   |
+| ----------- | ----------- | ----------- |
+|    0.7040   |   0.5307    |   -0.2301   |
+
+You can see these values plotted [here](https://www.desmos.com/calculator/vbpkey1mt6).
+
+The historical values are pretty close to the values for !eq[\delta=0.5], although the exact reason for the difference is unknown, but its close enough to not be overly concerning.