[Insight-developers] itkPoint from a math perspective

[Luis Ibanez]

On Tue, 17 Jul 2001, Paul Hughett wrote:
>
> Thanks for the explanation!  Could you put this into the user's manual
> somewhere?  It's too good an explanation to left to languish in our mail
> archives.
>

Sure, I'll make a Doxygen page for it.
There is now a "Geometry" group in the Modules. It contains
itkPoint,itkVector,itkCovariantVector and will also contain
the Transforms. Seems like a good place for describing how
they work together.

>
> A couple of follow-on questions:
>
> 1. I seem to recall that covariant vectors are somehow also associated
> with differentiation.  Is this true and what is the connection?
>

You're right, in fact, what we use to call "derivative" is strictly
a "covariant derivative", the book of Dodson & Poston gives a pretty
clear explanation on the subject (it is not the case for other tensor
calculus books). The reason is that a derivative is oriented othogonal
to an isovalue curve of the original function. If a transformation is
applied to the space in which a function is defined, the transformation
should map gradient-vectors in such a way the they are always orthogonal
to the function iso-values in the new space.


>
> 2. Tensor analysis, as I recall it, makes a distinction between covariant
> and contravariant tensors (or was it indices?).  Is there also such a thing
> as a contravariant vector?
>

Right again,
Strictly speaking, our itkVector should be called itkContravariantVector,
but for simplicity we keept itkVector. In the mathematical notation
super-indices are used to indicate contravariant and sub-indices are used
to indicate covariant (if i remember correctly). Vectors are a particular
case of Tensors. Tensors are usually represented by Matrices, but the
matrix notation in itself doesn't contain the information about how a
tensor will behave.


Luis