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Area of Origin: Calculation and Interpretation

Given a set of trajectories from an impact pattern, we can examine these trajectories to determine the area where they converge — the so-called Area of Origin (AOO). The AOO helps in understanding the dynamics of an impact, and is represented as a 3D ellipsoid.

Area of Origin Example

Calculation of the Area of Origin

The AOO is represented by an ellipsoid that has two parameters: a center (x, y, z) and a size (sigma_x, sigma_y, sigma_z).

Ellipsoid Center

The ellipsoid's center is calculated by minimizing the following function:

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This function minimizes the sum of all squared angles between the stain trajectories and the lines connecting the AOO center with the stain coordinates. In other words, this function seeks the 3D location that is most likely to be the origin of the input trajectories.

Ellipsoid Size

Once we have the center of the ellipsoid, we can calculate its size using the following steps:

  • We calculate the closest point along each stain trajectory to the ellipsoid center, resulting in a set of the 3D points around the center.
  • The horizontal spread of the ellipsoid is determined by calculating the standard deviation of these closest points in the horizontal plane (xz-plane).
  • The vertical spread is obtained by calculating the standard deviation in the vertical direction (y-axis).

Both spreads are then used to draw the ellipsoid (of course, the spread for the x- and z-dimensions will always be the same).

For a more detailed explanation and validation of this method, please see the full publication

Interpretation of the Area of Origin

The 3D ellipsoid representing the AOO has two key properties: a center and size (or sigma), both of which are defined in three dimensions.

Ellipsoid center: The center of the ellipsoid indicates the exact center of the Area of Origin. This is expressed in world coordinates, as per the units specified in the project settings. In the example below, the center is positioned at [583mm, 129mm, 1049mm] with respect to the origin.

Ellipsoid sigma: The spread represents the uncertainty of the AOO, and provides insight into the variation of the calculated AOO. It is derived from the standard deviation of the closest points along each stain trajectory to the ellipsoid center. The uncertainty is quantified separately for horizontal (xz-plane) and vertical (y-axis) dimensions. Hence, the ellipsoid will always be circular when viewed from above.

Note that the ellipsoid in the 3D view is drawn using 3 times the standard deviation (sigma). In the example below, the horizontal sigma is 6.4cm, meaning that the total width of the ellipsoid in the 3D view is 38.4cm (6 times the sigma, three times in positive and negative direction).

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