[Insight-users] difference between vector and covariantvector

[JSW]

Hi,

I must say that, apparently, ITK has a rather curious definition of 
covariant vectors. First of all, a vector is either
covariant or contravariant. Which one it is, depends on its 
tranformation properties under coordinate changes.
See: http://mathworld.wolfram.com/CovariantTensor.html

Formally the definition: Vector=Point1-Point2, is not even a vector, 
that statement only holds in flat Euclidean
space (and not on a sphere for example). The definition 
CovariantVector=Vector1xVector2, also isn't a vector formally,
but an axial vector (axial vectors do not change sign under inversion, 
while vectors do).

A gradient is indeed a covariant vector (in fact, in differential 
geometry it is the basis of all vector spaces). Note that if the space 
is endowed with a metric, you can transform between contravariant and 
covariant vectors easily.

I'm not objecting to the name "vector" for a difference of two points, 
as long as we're in good old flat space, nothing
bad will happen. However, the definition of a CovariantVector given 
below, is misleading in my opinion. Here's why:
Under rigid transformations (rotations for example), the difference 
between two points transform covariantly. The
cross-product of two vectors does not transform covariantly (at least 
not as a vector, it does transform covariantly
as a 2-form or anti-symmetric rank two tensor).

If the goal of the documentation is to express that there are certain 
vectors that are transformed by ITK, and certain
vectors that are not, then I would recommend using different names for 
it (for example FixedVector and VariantVector).

best,
Jeroen Wijnhout

Luis Ibanez wrote:
>
> Hi Yannick,
>
> A Vector describes the relative position between two points in space.
>
>             Vector = Point1 - Point2
>
>
> A CovariantVector describes the direction orthogonal to a surface.
>
>          CovariantVector = Vector1 x Vector2
>
> where "x" is a cross product of the two vectors.
>
>
> CovariantVectors and Vectors behave differently under
> Affine transformation. That is one of the reasons why
> it is important to make a distinction between them in ITK.
>
>
> For example:
>
>  Gradients of functions are CovariantVectors (not Vectors).
>
>
>
>    Regards,
>
>
>
>        Luis
>
>
>
> ----------------------
> yannick pannier wrote:
>> Hi everybody,
>>
>> I'm learning how to use ITK's library and I don't understand very 
>> well the difference between itk::CovariantVector and itk::Vector 
>> classes.
>>
>> The ITK's software guide say :
>>
>> //  covariant vector differs from a vector in the way they behave
>> //  under affine transforms, in particular under anisotropic
>> //  scaling. If a covariant vector represents the gradient of a
>> //  function, the transformed covariant vector will still be the valid
>> //  gradient of the transformed function, a property which would not
>> //  hold with a regular vector.
>>
>> Does anyone could give me more explanations ?
>>
>> Thanks,
>>
>> Yannick
>>
>>
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