The inertia tensor
Contents
- A proof of the formula
- Covariance of a canonical triangle
- Covariance of the triangle with a vertex in the origin
- Covariance of the triangle in question
- References
where covariance is defined as area integral over the triangle:
Covariance for a triangle in three-dimensional space, assuming that mass is equally distributed over the surface with unit density, is
where
Substitution of triangle covariance in definition of inertia tensor gives eventually
A proof of the formula
The proof given here follows the steps from the article.
Covariance of a canonical triangle
Let's compute covariance of the right triangle with the vertices (0,0,0), (1,0,0), (0,1,0).
Following the definition of covariance we receive
The rest components of
As a result,
Covariance of the triangle with a vertex in the origin
Consider a linear operator
that maps the canonical triangle in the triangle
Covariance of the triangle in question
The last thing remaining to be done is to conceive how covariance is changed with the translation of all points on vector
where
is the centroid of dashed triangle.
It's easy to check now that all coefficients in