INTERACTIVE ANIMATION : INERTIA
OF A CLOUD OF POINTS
This animation illustrates the concept of
(moment of) inertia of a cloud of points.
of Animations" on your computer
The animation has two operating modes :
- Projected inertia.
It opens in the "Basic" mode.
Illustrates the definition
and the fundamental property of the inertia.
The frame shows :
* A "cloud" of
five red points. All points have the same arbitrarily chosen initial weight of
* 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
* 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.
- 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
- 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
for multidimensional samples.
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
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.
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).
"PROJECTED INERTIA" MODE
Illustrates the concept of
"Direction of Maximum Projected Inertia".
Click on the "Projected Inertia"
radio button to enter this mode.
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
- 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
- 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.".
The display may now be made less cluttered by clicking
on the "Mask Max. Spread" radio button.
- 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).
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
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
- 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
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.
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.