add approx description

Signed-off-by: Mamoru Sobue <mamoru.sobue@tier4.jp>

Author: soblin

common/autoware_trajectory/README.md

@@ -47,7 +47,7 @@ This section introduces strict definition of several words used in this package
 | `curve`         | `curve` is an oriented bounded curve denoted as `(x(s), y(s), z(s))` with additional properties, parameterized by `s` (`s = 0` at the start).                                                                                                                                                                                                                                    | ![curve](./images/nomenclature/curve.drawio.svg)<br>[View in Drawio]({{ drawio("/common/autoware_trajectory/images/nomenclature/curve.drawio.svg") }})<br>There are 5 `underlying` points<br>$\mathrm{P0} = (0, 0, 0)$<br>$\mathrm{P1} = (1/ \sqrt{2}, 1/ \sqrt{2}, 0)$<br>$\mathrm{P2} = (1/ \sqrt{2}, 1+1/ \sqrt{2}, 0)$<br>$\mathrm{P3} = (2/ \sqrt{2}, 1+2/ \sqrt{2}, 0)$<br>$\mathrm{P4} = (2/ \sqrt{2} + 1/ \sqrt{6}, 1+2/ \sqrt{2} + 1 / \sqrt{3}, 1 / \sqrt{2})$<br>and the `arc length` between each interval is $1, 2, 1, 1$ respectively, so $\mathrm{start} = 0$ and $\mathrm{end} = 5$. |
 | `underlying`    | `underlying` points of a curve refers to the list of 3D points from which the curve was interpolated.                                                                                                                                                                                                                                                                            |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
 | `arc length`[m] | `arc length` denotes the approximate **3D** length of of a curve and is computed based on the discrete `underlying` points.                                                                                                                                                                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
-| `s`[m]          | `s` denotes the **3D** `arc length` coordinate starting from the base point (mostly the start point) of the curve and a point is identified by `trajectory(s)`.<br>Due to this definition, the actual _curve length_ and `arc length` have subtle difference as illustrated.                                                                                                     | ![approximation](./images/nomenclature/approximation.drawio.svg)<br>[View in Drawio]({{ drawio("/common/autoware_trajectory/images/nomenclature/approximation.drawio.svg") }})<br>The point for $s = 0.5$ is the purple dot, but the _curve length_ from $\mathrm{P0}$ to this point does not equal to $0.5$.                                                                                                                                                                                                                                                                                        |
+| `s`[m]          | `s` denotes the **3D** `arc length` coordinate starting from the base point (mostly the start point) of the curve and a point is identified by `trajectory(s)`.<br>Due to this definition, the actual _curve length_ and `arc length` have subtle difference as illustrated.                                                                                                     | ![approximation](./images/nomenclature/approximation.drawio.svg)<br>[View in Drawio]({{ drawio("/common/autoware_trajectory/images/nomenclature/approximation.drawio.svg") }})<br>The point for $s = 0.5$ is the purple dot, but the _curve length_ from $\mathrm{P0}$ to this point does not equal to $0.5$.<br>The exact _curve length_ is $\int \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2} dt$, which cannot be obtained in an analytical closed form.                                                                                                                      |
 | `curvature`     | `curvature` is computed **using only X-Y 2D coordinate**. This is based on the normal and natural assumption that _roads are flat_. Mathematically, it asserts that [Gaussian curvature](https://en.wikipedia.org/wiki/Gaussian_curvature) of road is uniformly 0.<br>The sign of curvature is positive if the center of turning circle is on the left side, otherwise negative. | ![curvature](./images/nomenclature/curvature.drawio.svg)<br>[View in Drawio]({{ drawio("/common/autoware_trajectory/images/nomenclature/curvature.drawio.svg") }})                                                                                                                                                                                                                                                                                                                                                                                                                                   |
 
 ## API