INTERACTIVE ANIMATION : INERTIA OF A CLOUD OF POINTS

This animation illustrates the concept of (moment of) inertia of a cloud of points.

 The "Book of Animations" on your computer

The animation has two operating modes :

1. Basic.
2. Projected inertia.

It opens in the "Basic" mode.

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"BASIC" MODE

Illustrates the definition and the fundamental property of the inertia.

Frame

The frame shows :

* A "cloud" of five red points. All points have the same arbitrarily chosen initial weight of 5.

* The barycenter of the cloud, denoted by a blue point.

* A reference point denoted by a black cross. Inertia is measured with respect to this reference point.

* Green lines connecting the reference point to every point in the cloud.

At the bottom right of the animation is displayed the value of the total inertia of the cloud (with respect to the reference point). This value is :

Inertia = Σi mi.d²i

where :

* mi  is the weight of point i.

* d²i is the square of the distance between point i and the reference point (black cross), that is, the length of the corresponding green line.

No reference frame is shown to emphasize the fact that inertia is defined with respect to a point, not with respect to axes.

Animation

Drags

• Drag red points about with your mouse, and observe the corresponding changes in the inertia. In particular, observe that as a point is dragged away from the reference point, the inertia increases. So, in a loose way, for a given set of weights, inertia is a measure of the spatial extension of the cloud.
• Do the same with the reference point (black cross). Observe that the inertia is indeed minimal when the reference point coincides with the barycenter. If all weights have the same value w (as is the case at the opening of the animation), the definition of inertia is now formally identical to that of n.w times the sample variance for a 1-dimensional set of points. Inertia may then be perceived as an extension of the concept of variance for multidimensional samples.

Weight changes

Double-click on any red point. The current weight of the point appears in a pop-up display. Use the control buttons to change the weight of the point. Close the display by clicking anywhere in the scene.

Observe that as the weight of a point increases all things being equal, the inertia increases.

Number of Points

Use the "Nb Points" controls to change the number of points. As new points are added, the inertia increases even if the spatial repartition of the points in the cloud remains roughly unchanged.

Reduce the number of points down to 1. Drag the point away from the reference point and observe that the inertia increases more rapidly than the distance between the point and the reference point : the inertia is quadratic in distances (but linear in weights).

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"PROJECTED INERTIA" MODE

Illustrates the concept of "Direction of Maximum Projected Inertia".

Click on the "Projected Inertia" radio button to enter this mode.

Frame

If the weights are not all equal, two lines are displayed :

• A blue line, which is the direction of Maximum Spread of the cloud.
• A black line, which is the direction of Maximum Projected Inertia of the cloud.

• The blue line is such that, if weights are ignored (or all equal), then the projection of the cloud on this line has the largest possible variance. The blue dot is the projection of the center of gravity of the positions of the points .  For more details, please see Covariance Matrix.
• The cloud is projected on the black line. The projected points (green) are assigned weights equal to that of the points themselves. The inertia of the set of green points with respect to their barycenter is called the Projected Inertia of the cloud on the line.
The black line is the direction of Maximum Projected Inertia (MPI) of the cloud. No other line will yield a projected cloud with a larger inertia (with respect to the barycenter). The value of the Maximum Projected Inertia is displayed at the bottm right of the animation. It is of course less than the value of the total inertia.

The blue and black lines are made to run through the barycenters for convenience only. Their actual positions are irrelevant, only their directions matter.

If all the points have approximately the same weight, the blue and black lines have approximately the same direction.

Animation

• The Direction of Maximum Spread and the Direction of MPI can be quite different.
* Create a 3-point cloud such that a heavy point sits between two nearby light points. The blue and black line are almost identical.
* Now drag the heavy point perpendicularly to the segment defined by the two light points. At first, the (direction of the) blue and black lines are unchanged, but the the black line ends up being unstable and finally becomes orthogonal to blue line : the directions of Maximum Spread and of MPI are then "as different as can be". This extreme situation is pathological : both directions are then very unstable, and all directions exhibit about the same spread and the same projected inertia.

* Keep dragging the heavy point until the blue line abruptly rejoins the black line :  the directions of Maximum Spread and of MPI are again almost identical (but orthogonal to their original directions).

• Increase the weight of a point : it "attracts" the black line (the blue line remains, of course, unchanged). In the vocabulary of Principal Components Analysis or Correspondence Analysis, "Heavy points have a greater influence on determining the direction of Maximum Projected Inertia than light points.".

• Drag a (preferably heavy) point so that its projection on the black line is close to the barycenter (the point itself doesn't have to be close to the barycenter). Move it about perpendicularly to the black line, and observe that the direction of the black line remains unchanged (remember that the position of the line is immaterial).
Then drag the point so that its projection on the black line is now far away from the barycenter, and move it about with the same amplitude as before. The black line now "follows" the point by sort of "pivoting" around the barycenter.
In the vocabulary of Principal Components Analysis or Correspondence Analysis "For a given weight, points that project far away from the barycenter have a greater influence on determining the direction of Maximum Projected Inertia than points that project close to the barycenter.".

• You may now check that the black line is indeed the direction of Maximum Projected Inertia. Left-click anywhere in the empty space between points, and drag your mouse with the button down. The black line turns purple and starts spinning around the barycenter. It is not the direction of MPI anymore, just an arbitrary direction of projection. The value of the inertia of the projected cloud on this line is displayed at the bottom right of the animation ("Projected Inertia").
Drag points about and change their weights so that it becomes hard to guess the direction of MPI. As you do so, the purple line translates parallel to itself so as to always run through the barycenter, but remember that this is just for convenience : only the direction of the line matters.
Then spin the purple line until you get the largest possible value of the projected inertia. You have indentified "manually" the direction of MPI.
To confirm this identification, double-click in an empty region of the scene : the purple line becomes black again and snaps back to the true direction of MPI.