# Transformations for isotropy in terms of math

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• gionole
gionole
Probably, my last question about isotropy. This is the last thing that I want to double check.

We know that mathematically, passive and active transformation are both the same. In passive, coordinate frame is moved and nothing else, while in active frame, objects are moved and coordinate frame stays the same.

I'm trying to show that I can get the same mathematics by using active and passive for isotropy detection.

Experiment we can do consists of the system that includes earth and ball where ball is dropping to earth from some height.

In passive, we get the following math:

##\ddot y = -\frac{d}{dy}(gy)##
##\ddot y' = -\frac{d}{dy}(gycos\theta)## (in rotated frame)

In active(look at the attached image), I rotate earth and ball counter-clockwise by some ##\theta## angle. Note that calculations of old and new coordinate of the points(after rotation and before rotation) are given by the same mathematics as in passive, so I'm on the right track. ##y' = -xsin\theta + ycos\theta## and ##x' = xcos\theta + ysin\theta##(there's no prime concept here as we only got one fixed frame, but I call the new coordinates after rotation to be primed)

Now, what I do is: in 1st experiment, we got the same thing ##\ddot y = -\frac{d}{dy}(gy)##. In 2nd experiment, after earth and ball rotated, I write the following thing: ##\ddot y = -\frac{d}{dy}g(xsin\theta - ycos\theta)## (I replace ##y## by ##xsin\theta - ycos\theta##). The reason why I replace by such thing is that after we did rotating of earth, it got moved as well, so potential energy can't be represented by ##mgy## anymore. So I find by how much earth got moved in ##y## direction, which is given by ##-xsin\theta + ycos\theta - (-y) = -xsin\theta + ycos\theta +y## and then any point where the ball is can be given by ##y - (-xsin\theta + ycos\theta +y) = xsin\theta - ycos\theta## and then plugging in ##\ddot y = -\frac{d}{dy}(gy)##. Also note that on the left side, I don't replace it, because it would be incorrect, as we're still trying to find acceleration in the same ##y## frame.

So I end up with ##\ddot y = -\frac{d}{dy}g(xsin\theta - ycos\theta)## which results in ##\ddot y = \frac{d}{dy}(gycos\theta)## . Problem is this result doesn't have "-" in front of it, while from passive transformation calculations, it does. where did I make a mistake and how do I end up with the exact same math ?

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gionole said:
Problem is this result doesn't have "-" in front of it, while from passive transformation calculations, it does. where did I make a mistake and how do I end up with the exact same math ?
Consider a 2D object in relation to a set of 2D cartesian axes. If you rotate the object (active transformation), say counterclockwise, it's equivalent to rotating the axes (passive transformation) clockwise. In other words:
Active rotation by angle ##\theta## ##\Longleftrightarrow## Passive rotation by angle ##-\theta##.
So you need to flip the sign of the rotation-angle between your two calculations.

gionole

## What is isotropy in mathematical terms?

Isotropy in mathematical terms refers to the property of being directionally invariant. This means that a function, object, or space exhibits the same behavior or properties in all directions. In the context of transformations, it often implies that the transformation does not favor any specific direction.

## Why are transformations for isotropy important?

Transformations for isotropy are important because they ensure that models, equations, or systems behave uniformly regardless of direction. This is crucial in various fields such as physics, engineering, and computer graphics, where uniformity and consistency in all directions are required for accurate simulations and analyses.

## How can you achieve isotropy through transformations?

Isotropy can be achieved through transformations by applying rotationally invariant operations. For instance, using spherical coordinates instead of Cartesian coordinates can help in achieving isotropy in three-dimensional problems. Additionally, averaging over all possible orientations or using isotropic kernels in convolution operations are common techniques to ensure isotropy.

## What are some common mathematical transformations used to ensure isotropy?

Common mathematical transformations used to ensure isotropy include:- Rotational transformations, which involve rotating the coordinate system to eliminate directional bias.- Spherical harmonics, which are functions defined on the surface of a sphere and are inherently isotropic.- Fourier transforms, particularly in the context of isotropic kernels in image processing.- Scaling transformations that adjust the size of objects uniformly in all directions.

## Can isotropy be preserved in discrete systems?

Preserving isotropy in discrete systems is challenging but possible. Techniques such as using isotropic finite difference schemes, isotropic interpolation methods, and ensuring uniform sampling in all directions can help maintain isotropy in discrete systems. However, the discrete nature of these systems often introduces some level of anisotropy, which needs to be minimized through careful design and analysis.

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