Geometric Algebra

[agor95]

In searching for a mathematical model that is suitable for our needs.

I have looked at

Revision of 101 mathematics
Matrix Algebra in relation to transformations
Partial Differentials
Complex Numbers for rotation on 2d plans
Quaternions Rotation in a 4d plans
Lagrangian Mechanics

And now this Algebra

https://www.youtube.com/watch?v=2hBWCCAiCzQ
https://www.youtube.com/watch?v=tX4H_ctggYo
https://www.youtube.com/watch?v=60z_hpEAtD8

[agor95]

https://enkimute.github.io/ganja.js/exa ... _and_lines

[agor95]

'You get out more from a subject than you put in.'

https://geometricalgebratutorial.com/

Regards

[agor95]

I admit this is an example from the Ganja Coffee Shop site.

I wanted to build from the ground up. So there is a HTML file zipped up attached.

Your choice by placing this on your Desktop and viewing it in your internet browser - Terrible things will happen - yes you will learn.

You should see a moving scene and dragging with left mouse down will rotate the whole scene.

I am working on several aspects of the code.

What does it mean?
How do I get it to work for me?
How do I mash-up this scripting language with the others I have learned?

Anyway the first question: What are you looking at?

Before you is a volume of space, a box, with two points on the back wall P1 and P2.
A pendulum R and another point moving from P1 to P2. A little pointless even though it's mainly points.
There are two plains p1 and p2 with another plain moving from one plain position to the other.

Also there are two lines l1 and l2 with an other line doing the same thing on the front left wall.

What has this got to do with Bessler?

What I think is possible is the creation of Rigid Body Dynamics models in our browsers.

So all the lovely things we talk about can be presented https://en.wikipedia.org/wiki/Rigid_body_dynamics.

Now were are my headache pills.

P.S. That is Projected Geometric Algebra 3D not Professional Golfers' Association.

[agor95]

I have completed my first read of the material on 'Rigid Body Dynamics'.

The example zipped below is something for you too play with after it stops moving.

You can drag the two top dots around.

[Senax]

I enjoyed that. (-:

[agor95]

I trust those who used the last demo tried dragging the background as well as the dots?

[Senax]

I trust those who used the last demo tried dragging the background as well as the dots?

I hadn't tried that but I have now.
Fascinating.

[agor95]

Hi All

For years I have been trying to find a maths that represents dynamic behaviour of movement
with something better than the current methods. This maths is just that 'something better'.

https://en.wikipedia.org/wiki/Conformal ... ic_algebra

We are involved in the rotation of weights and their movement restricted to radial distances.
Any pivoted mass is restricted to this reality.

That is why I prefer polar coordinates over cartesian.

This algebra describes lines as large loops.

So a straight lines are paths of infinite loops.
As we see a straight horizon from our perspective and from space we see a sphere like object.
Therefore the horizon is a curved loop like the equator.

This algebra increases my chances at presenting the movement with more easy than matrix algebra
,Lagrangian or Hamiltonian mechanics [formula].

The target is a mathematics that does not use conservation of energy as a short cut to describing motion.

Regards

[agor95]

Hi All

Well I am still staring at this subject and glimmers of understanding appear from time to time.
It is still weird and dives into physics beyond by pay grade.

So what is weird about it?

Well you may know about Complex Numbers where you have Real Number added to a weird Imaginary Number.
An example 10 + 5i and the i is the square root of minus 1; It gets worse.

There are Quaternions Number 10 + 5i + 2j + 3k all these are imaginary except the first Real Number.

These get replaced with a new thing which gets even worse.

Regards

[agor95]

Hi All

These get replaced with a new thing which gets even worse.

Those who have read up on this subject already know this 'even worse'.

The imaginary number squares to -1 and a variable that does this is indicated as e_1.
The worse is there are two others.

The first squares to +1 as e+1. The other that squares a variable to 0 which is indicated with e_0.

So what does that do for use.

Well if you have a formula then you can find it's normal anywhere along it's length.

That is like a radial line on a circle. Turning this normal by 90 degrees gets you to the tangent.

That is in effect differentiation.

If I understand this correctly then you can have as many variables [dimensions] as you need.
Also vertical tangents are catered for so there are no exceptions.

[agor95]

Hi TTIF

I have been looking at this part of Geometric Algebra for a month.

Still waiting for it to click in place.

Ultimately it should give use solid state dynamics with the ability to link components together.

So for the first time we could simulate double pendulums with dynamic anchor points and more with a stream lined approach.

This removes matrix rotation and complex partial derivative formulations.

Summary

A solid beam with a square cross section. That has a mass i.e. inertial. This could have s force anywhere along it's surface from any direction in 3 dimensions applied to the object.

Then it would accelerate [translate and rotates] without doing the complex maths.

Also two such objects could be linked by a spring and interact as both would experience the spring's force.

There is no reason why these could not be anchored with one of the components to a disc [cylinder].

The cylinder would then rotate under it's inertia by the force from that component.

Regards

[agor95]

Hi TTIF

To unpack the image the best I can at this time.

The cryptic parts are the top row and the bottom box that hold two formulas.

The B is the rate a mass moving both translational & rotational.
The reason for the 'B' is the concept of vectors!

The 'B' combines both in a Bi-vector.
Analogy of this is a complex number where the Real is velocity and
the Imaginary is rotation.

If you Integrate B between Time zero to 10 seconds and the Real part was 2 km/s.
Then the unit distance moved would be 20km.

If a force were applied then a mass would accelerate.
You will see an odd thing 'forque'. This is not a spelling mistake!

If we push an object it will translate when force in alignment with it's centre of mass.
But we would get rotation if the force was a glancing blow!

The mix of Force to Torque is combined and are not separate. Therefore we have FORce torQUE.

With a 3 dimensional object this happens simultaneously in all of them!

When the forque has a fixed value then the 'B' changed over time and that is why we have a B dot.

Think of pushing a pencil in line with it's axis it translates.
However if slightly off then the rotation increases over time.

So we have two Integration steps 1. to add up the 'B' change and 2. adding up the resulting 'B' to integrate into the total movement.

No wonder simulation are so error prone.

The key is the inertia the forque pushes against.

The reveres of that is an inertial mass and it's momentum.
A forque [force] is generated when it's translation/rotation changes.

In the image you see a change in momentum over time [differentiation] results is a force.

So when two objects collide forque is created as one slows down.
That forque speeds up the other.

And at the same time rotation many result.

...

[agor95]

Hi TIFF

There is a starting base example of this Algebra in 2 dimensions and one projective dimension.


In English that means two parallel lines up a sheet of paper looks like they meet at infinity when you look along the sheet from the bottom.
The old railway rails going off into the distance.

Regards

[agor95]

Hi TIFF

It would appear if you ask very tight questions to ChatGPT. Then it outputs text that can be used too assist in developing mathematical Models.

For those in the know Cl(4,1,0) is the Clifford Algebra signature being studied.

Note.

It appears the LLM model used by ChatGPT was given most of the current literature
on this topic. Therefore asking a tight question gives you the text relating to that question.

P.S. No stone will be left unturned.

Regards