CuriosOne

12 minutes ago, Sensei said:

...did not you just posted thread with title "what is the 3rd dimension?"..... ?
 

Yes, but I use 3d to create 3d video games, 3d models, basically 3d "toys" and when I render the images it looks very similar to images I see in the real world..

The same is true for animations that uses the evolution of time to move.

For instance, how do numbers "know" left, right, up or down??

If you say computer software programs them, then that's more than obvious, but say we do math on plain "paper" and pencil, how then can the numbers do what they do in 3d space x,y,z??

I guess I'm asking what links "instructions to numbers in the phyiscal world without computers??

The universe has been here for billions of years before the thought of the computer processor..

On 10/18/2020 at 12:17 AM, Sensei said:

Triangles (in 2D and 3D space) are exclusively convex and planar. Quads can be convex or concave. in 3D they are also often non-planar.

https://en.wikipedia.org/wiki/Convex_polygon

https://en.wikipedia.org/wiki/Concave_polygon

There is many ways quads (or n-gons, typical name for polygons with more than four vertexes) can be triangulated.

The most simple triangulation algorithms are triangle strip and triangle fan.

https://en.wikipedia.org/wiki/Triangle_strip

https://en.wikipedia.org/wiki/Triangle_fan

They work reliably only with quads and n-gons which are planar and convex. With non-planar and concave polygons they can result in many kinds of issues.

They are used just because they are extremely fast for CPU/GPU.

 

3D non real-time renderer is typically using ray-tracing. It is much faster to find intersection between ray and triangle (or ray and plane), than ray and quad/n-gon.

Example ray-triangle algorithm:

https://en.wikipedia.org/wiki/Möller–Trumbore_intersection_algorithm

 

Do you think that " triangulation" has something to do with the "ELECTRON CONFIGURATION" and color charge??

Afterall, pixels are made of 3 colors RGB and the "outer regions" of "objects" either obsorb or emit electromagnetic radiation.

But then, how do triangles, pythagoreon therom and electron configuration relate??

After all, we use the same visual perception using the Pythagorean Theorem.

Seems like everything uses 2 points in space shared by an origin in Cartessian Space..

On 10/19/2020 at 3:15 AM, Sensei said:

Download, install and play with some 3D software.

e.g. Blender, Lightwave, Maya etc (but majority of 3d apps have 30 day trials)..

 

I've been using Maya for the past 10 years, I love 3d...

On 10/15/2020 at 12:56 AM, studiot said:

Gosh you are a hard person to keep up with on ideas. I see you have started another thread this morning.

I seriously recommend you at least get to a sensible pause point with each one before moving on, we still have a long way to go in your calculus one.

Anyway swansont has answered your question but here is more on my comment.

Here is a brilliant experiment you can easily perform to gain insight.

You will need a cardboard box with all six sides intact.

Here is a quick blackboard sketch.

Rotations can be represented by complete circles.

Consider first one single space dimension.

There is nowhere for rotations to occur. You have to leave the dimension (employ another one) to even turn around. This is Fig 0.

Move up to two dimensions  _  I have modelled this as a plane in two dimensions in Fig 1
You can have a rotation about any point in the plane.

Draw this as a circle on one face of your box, as in Fig 2.
But any rotation is about an axis which has to be a line in a third dimension.
So if you extend a line through your point through the opposite side you have the z axis.
You can draw a circle round it though any plane parallel to the first side like the opposite side.

Now move up to 3 dimensions.
You have to pairs of sides you can draw rotation circles on to generate two more axes, making 3 in all.
As in Figs 3 and 4.
I have shown the conventional right handed rectangular xyz coordinate system.

Now comes the clever part  - your experiment.
Use the box to convince yourself that rotation on any plane at any angle has an axis within the 3D system.
You do not need to leave 3D and have a rotation axis pointing into a fourth or higher dimension.

This is what I mean when I say that 3D is complete for rotations.

Let us know how you get on with your box.

Still sort of confusing becuase all I see are 6 sides of 1 dimension ( 1 flat hyper plane of 6) and rotations look reflected on all 6 dimensions of the box. Its mind boggling...Atleast its how I perceive it to be.

3 hours ago, Sensei said:

No. It is used to calculate distance between two points. Length of line/segment. Pythagorean Theorem is used at the lowest-level. I mentioned example usages that you might know from your daily life.

Zooming-in might not work if programmer wanted/predicted it. Curve can be dynamically sub-divided to single pixel precision on the display..

In typical 3D games everything is made of triangles. If you would be able to zoom-in infinitely, you will reach triangle, with flat surface. Even though object looked like sphere at full scale, at the end it is made of triangles, in enough quantity to pretend sphere.

But if programmer knew users will be zooming in, could use adaptive sub-division techniques.

https://www.google.com/search?q=adaptive+subdivision

Seriously? Do you know C/C++?

Calculate point on Catmull-Rom curve in C/C++ might look like:

CVector GetSplinePointAt(
 const CVector &position0,
 const CVector &position1,
 const CVector &position2,
 const CVector &position3,
 double time )
{
 ENTER( "GetSplinePointAt()" );

double time2 = time * time;
 double time3 = time2 * time;
 double h2 = 3.0f * time2 - time3 - time3;
 double h1 = 1.0f - h2;
 double h4 = time3 - time2;
 double h3 = h4 - time2 + time;
 CVector v1 = position1 - position0;
 CVector v2 = position2 - position1;
 CVector v3 = position3 - position2;
 double v1length = v1.Length();
 double v2length = v2.Length();
 double v3length = v3.Length();
 CVector r1 = ( v1 * v2length + v2 * v1length ) / ( v1length + v2length );
 CVector r2 = ( v2 * v3length + v3 * v2length ) / ( v2length + v3length );
 CVector position = h1 * position1 + h2 * position2 + h3 * r1 + h4 * r2;

 LEAVE( "GetSplinePointAt()" );
 return( position );
}

where CVector class method Length() implementation is:

double CVector::Length( void ) const
{
 double x = m_Vector[ 0 ];
 double y = m_Vector[ 1 ];
 double z = m_Vector[ 2 ];
 double temp = ( x * x ) + ( y * y ) + ( z * z );
 if( temp > 0.0 )
 {
  return( sqrt( temp ) );
 }
 return( 0.0 );
}

Which is equation from Pythagorean Theorem.

If you never heard of Catmull-Rom spline:

https://en.wikipedia.org/wiki/Catmull-Rom_spline

eventually

https://en.wikipedia.org/wiki/Centripetal_Catmull–Rom_spline

In programming it can/will be brute-force summation of the all small segments with required precision (i.e. if something is smaller than 1 pixel on user screen, it is enough, in typical usage).

 

 

BTW, at the first sight, the above code has error: division by zero. But when we will look at overloaded division operator code, it is handled:

CVector operator / ( const CVector &vector1, double value )
{
 CVector vector = vector1;
 if( value != 0.0 )
 {
  vector.m_Vector[ 0 ] /= value;
  vector.m_Vector[ 1 ] /= value;
  vector.m_Vector[ 2 ] /= value;
 }
 return( vector );
}

I created my 1st ever computer program 20 years ago in C++ for video games until i found an easier and faster method with Python, but C++ is very time consuming..It is however industry standard and professionally efficient I must say..

About our distances and 2 points, is zooming in a subdivision process of triangulation? 

I keep thinking about the rendering engine itself, the smoother a mesh the more render time...((IE more triangles = more processing power..))

Quads to Triangles final process. I've always wondered why that was...

Just want to add a point is dimensionless right?

I'm a CGI artist as well, very knowledgable in this field..

"One Very Important Question"

What units do theses points of the extremely small use??? 

The points that The Pythagorean Theorem predict between a triangle to its hypotenuse??

And are theses in 3 since a triangle has 3 points..

3 hours ago, Sensei said:

Nope.

It is just useful mathematical tool.

You use Pythagorean Theorem  when you're calculating distance between two points, in e.g. Euclidean space:

a2+b2=c2

transform it to solve c :

c=a2+b2−−−−−−√

Then in physics we often see it as:

ds=dx2+dy2−−−−−−−−√

and in three dimensions:

ds=dx2+dy2+dz2−−−−−−−−−−−−−√

Programmers can use it to e.g. calculate length of curve. Divide curve to sequence of (very short) straight lines and then sum their lengths together.

App which tells how how much you walked per day used by sportsmen or people on diet, taxi/Uber/TIR driver, airplanes etc. , to measure distance of travel, will use it too with coordinates (latitude,longitude) returned by GPS, then transformed to 3D coordinate on sphere.

Never knew The Pythagorean Theorem had multi puposes use, especially the sequencing and summing up the very short straight lines part, or zooming into to them..

Can I see an example of this please?

Reason I ask is becuase It clarifies my knowledge on calculus since it uses it too..

1 hour ago, studiot said:

To continue where I left of and pick up and develop  joigus points about limits

 

 

The f(x) and y =y(x) notation comes in useful if you want to get it all onto one line so the derived functions become f'(x) and y'(x).

This is not directly related to any of the d notation variants and also becomes very clumsy when many variables are involved.

So let us look at the d notation, which is, after all, what you are asking about.

So we have 

Greek capital delta   Δ

Greek lower case delta   δ

Roman Capital D

Roman lower case d

Eighteenth century European stylised script  d    ∂

We have all of them because each perform a different operation (remember them) on the symbol that follows them.

The first one is not important to us because we use it for the difference between two specific values of the variables represented by the symbol that follows.

So the difference in height, Δh ,  between the ground and the gutterline of my house is 18 feet.   Δh may be large, but must be exact. That is it is a number. As it is a number it can legitimately be zero.

The second one is important to us because it is not only about small differences in the values of the variable that follows it, it is about arbitrarily small differences.
This means that  δx is not exact but may be made smaller and smaller indefinitely. So it is not a number it is a sort of function. Also since it is not a number it can never be zero, since zero is a number.

This brings us neatly to the idea of limits.

 

 

This is a sequence

1,4,9,4,16,...

If we put in addition signs it becomes a series

1+4+9+16...

And if we work out the differences between each term or the changes from term to term

357

The three dots at the end is the convention for continuing indefinitely.

It can immediately be seen that all the values for both the sequence and series and for the differences increase with each additional term added so each get larger and larger.

This is called divergence and the sequence and series are called divergent.
Such series do not have limits.

 

 

If however we take the reciprocal of these sequences they get smaller and smaller, whilst the partial sums for the series gets closer and clsoer to a specific number (1.64)

11+14+19+116+...

1+0.250+0.111+0.063...

0.750.1390.048

This behavious is called convergence.
The specific number is called the limit.

Because the process can continue indefinitely we write

limn→∞∑1n2=1.64

Next time, this will lead directly into the limits 

limδx→0

and

limδx→0,δy→0

 

So we have numbers, functions, variables, operators, limits sequences and series all mentioned before.

An easy way to look at operators are that simple functions work on numbers to output other numbers and operators work on simple functions to output other functions.

You will need to do some work  on these to use all the to get a handle on what calculus is, what it can do for you and what it can't do for you.

So please let us know any ideas in this development you didn't get a hold of properly so we can help correct that.

 

 

 

 

Very insightful thXxxx...

The definition of zero being a number now answers most of the point of calculus, "But" in regards to arbitrary small differences not being numbers, what are they??

functions??  I read??

My focus then would be on the operators on simple functions that output other functions not numbers as converged "things" whatever they connect to at this point.

From what I'm understanding it sounds like a terrasac hypercube with infinite limits of functions within functions of operators...Is this right?

If "things" are converged I can see why this would be the case in regards to a refference atlas like cartessian space and the "points" on it in 3d, hieght, width and depth..

Is this the way to think about it?

The Greek lower case delta   δ

has my attention now.

This Makes Sense ThnXxxx

The second one is important to us because it is not only about small differences in the values of the variable that follows it, it is about arbitrarily small differences.
This means that  δx is not exact but may be made smaller and smaller indefinitely. So it is not a number it is a sort of function. Also since it is not a number it can never be zero, since zero is a number.

This brings us neatly to the idea of limits.

4 minutes ago, joigus said:

It could be a value not accessible by elementary operations. It could be pi, or e. In that sense, it's at the core of space, rather than at the edge.

The "camera lens" is made of deltas and epsilons that I defined for you before. Calculus allows you to forget about the lens and operate with the points.

This is getting very intresting now...

I'm glad I posted this OP bold question, this information is inspiring thnXxxxx.

22 minutes ago, studiot said:

Gosh you are a hard person to keep up with on ideas. I see you have started another thread this morning.

I seriously recommend you at least get to a sensible pause point with each one before moving on, we still have a long way to go in your calculus one.

Anyway swansont has answered your question but here is more on my comment.

Here is a brilliant experiment you can easily perform to gain insight.

You will need a cardboard box with all six sides intact.

Here is a quick blackboard sketch.

Rotations can be represented by complete circles.

Consider first one single space dimension.

There is nowhere for rotations to occur. You have to leave the dimension (employ another one) to even turn around. This is Fig 0.

Move up to two dimensions  _  I have modelled this as a plane in two dimensions in Fig 1
You can have a rotation about any point in the plane.

Draw this as a circle on one face of your box, as in Fig 2.
But any rotation is about an axis which has to be a line in a third dimension.
So if you extend a line through your point through the opposite side you have the z axis.
You can draw a circle round it though any plane parallel to the first side like the opposite side.

Now move up to 3 dimensions.
You have to pairs of sides you can draw rotation circles on to generate two more axes, making 3 in all.
As in Figs 3 and 4.
I have shown the conventional right handed rectangular xyz coordinate system.

Now comes the clever part  - your experiment.
Use the box to convince yourself that rotation on any plane at any angle has an axis within the 3D system.
You do not need to leave 3D and have a rotation axis pointing into a fourth or higher dimension.

This is what I mean when I say that 3D is complete for rotations.

Let us know how you get on with your box.

Will definetly try this out, looks fun thnXxxxx...I will try too slow down too, I agree with you...

1 minute ago, joigus said:

Exactly, you're getting closer. Literally.

Calculus is about formalising the operation of "getting closer."

You can use it to calculate the gentlest slope on a mountain, where the summit is, etc. It's not necessarily about "orbits."

If getting closer

"Where is x at then?"

At the edge of space??? 

And how do we know how far to zoom in and out, x is not a camera lense..😎

7 minutes ago, joigus said:

The ancient Egyptians used a primitive version of calculus to make the pyramids, rather. (Small incremental sums.)

Maybe you mean "similar to KE=(1/2)mv²

Or perhaps F=ma?

F=mv² is no standard physics.

 KE = (1/2) mv^2 looks correct...

4 hours ago, MigL said:

All of Newton's equations of motion were derived using calculus.
As a matter of fact, he invented the calculus for just that purpose.
( also, independently, by G Leibniz )
Newtonian mechanics, while not as 'sophisticated as Lagrangian or Hamiltonian mechanics, can still calculate the 'fall 'of an artillery shell at several miles distance, or even enable putting a rocket in lunar orbit.

So, I would say, quite useful.
Your criticism of calculus, on the other hand, totally useless.

"I do get what your saying"

But its said Newton used the pyramids to create calculus...Is this true??

The extent of pi and the pythagorean theorem used in calculus "right angle" triangles leads one to think the pyramid contribution may be true...

Even E=mc^2 looks very familiar to F=mv^2....

1 hour ago, joigus said:

It's hard for me to understand what your point about the point of calculus is. But as I can't sleep very well tonight, I've thought I might as well tell you a little bit about what your problem may be.

Your problem may be that you don't understand real numbers. Real numbers go beyond what intuitive numbers (numbers you may be used to) are. They're not like the number of people in a room, nor like the reading of a ruler, nor like the money in an account or exchange rates between currencies.

These numbers can be classed into a list:

Counting numbers: 1, 2, 3,... (natural numbers)

Whole numbers: ..., -3, -2, -1, 0, 1, 2, 3,...

Ratios: 2/3, -1/5, 10132/11, (plus all the whole numbers), etc. (rational numbers)

In order to define real numbers, besides all the usual algebraic assumptions, you need this axiom:

It is impossible to approach a number arbitrarily closely without that number being part of my system of numbers.

This is called "completeness." You can rephrase it as "limits of numbers must be numbers."

Calculus was being used very fruitfully by many mathematicians and natural philosophers for almost 200 years before mathematicians like Cauchy and Weierstrass defined it rigorously.

<ignore if you don't understand>

Another way to understand an idea could be to understand when it fails and how. This link may be interesting in that regard:

https://amsi.org.au/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html

There you can find a function (Weierstrass' function) which cannot be differentiated meaningfully:

 

f(x)=∑n=1∞12ncos(4nx)

 

And a graphic representation to an approximation by taking only 50 terms of the sum:

 

y=∑n=15012ncos(4nx)

 

</ignore if you don't understand>

Now let's go with your example. First, you've copied the formula wrongly. It should read:

 

limx→0x2−25x−5=5

 

What does that mean? (what you wrote has no meaning.) It means you can get as close as you want to number 5 by substituting in the expression for f(x),

 

f(x)=x2−25x−5

 

a number x as close to 0 as you want. The key that may be confusing you is that "as close as you want."

Now, when x is not 5, the expression,

 

x2−25x−5

 

of course simplifies to,

 

x+5

 

And it is obvious that you can get as close as you want to 5 by substituting x in x+5 for a number as close as you want to zero. That's the key to the "delta" that seems to ring a bell to you, but not the right bell. Here's the rigorous definition of limit: A function f(x) of one variable x as limit at x=a, and the limit is L if,

 

∀ε>0∃δ>0/|x−a|<δ⇒|f(x)−L|<ε

This explanation is very helpfull, thnXxx and your right about x=0 

If x=5 but you can't reach zero, is there a sequence of numbers going on, such as a list for x and y values?? I assume that's what they are..Maybe you can show that list, as I just place it in my calculator and press generate list this is what I got...

I'm seeing many calculus problems that use the ideas of right angle triangles, but seldom do I come across the use of our Pythagorean Theorem spot light to the rescue in them.

Infact I see much confusing or complex "hard to understand" functions and or equations, that never tell you, "You need the Pythagorean Theorem" to solve this problem. Maybe a technique to keep readers flipping pages whom knows..

In regards to rates again we have it here, a calculus problem...

Please note:

I'm seeing most of the time that there appears to be 2 things in motion at the same time in calculus problems, 2 changing lenths or distances but "one" time component "in the general sense."

ds/dt anyone??

2 hours ago, swansont said:

Whichever one you want it to. You can choose your own coordinate system. You want that to be the z direction, it can be z. If you want it to be y, it can be y. You can call it the first, second or third dimension. It won’t affect the physics.

(coordinate system choice will affect how hard it is to solve a physics problem, though)

Understood, and thnXxx.

Now if u may please, for the time dimension (t) is this the fourh dimension?

If so does it work the same as with the 3rd dimension ie x y z translations?

Do we even use the fourth dimension for any purpose other than time?

2 hours ago, studiot said:

I'm sorry to have to tell you that your video is a sorry mixture of fact and fiction that reaches some startling unsupportable conclusions.

You will only confuse yourself at best and start to 'believe'  junk Physics if you try to study from stuff like this.

 

You question is, however a valid one that is worth a considered answer.

The third spatial dimension is necessary in our physical world to achieve closure for a set of rotations.

This is also why a fourth spatial dimension is not needed and we do not observe one.

Which dimension goes up and down?

Say as we perceive falling objects from a building down to the ground with the force of gravity of coarse acting on it  bouncing up then repeating till it stops.

15 minutes ago, Phi for All said:

Everything is in motion. Imagine 100 people are in a big gymnasium with you (this is actually a two-dimensional representation, since you can't fly). You're walking around in a circle holding a throwing dart, and everybody else is also walking around in differing sized circles. It's complete chaos, but one of those people is also holding a dart board as they move (the asteroid). Can you imagine how you'd have to figure out when/how hard/what angle to throw the dart so it doesn't hit anyone or anything else but the dart board?

What?! No, it works anywhere in the universe. It works somewhat differently when we're on the surface of the planet, but the concept of degrees of freedom holds true.

Sounds very intresting...."in regards to tossing a dart on a board bull's eye with both in circular motion"...

Where could I see a formula for this if you may?

7 minutes ago, Phi for All said:

Dimensions measure degrees of freedom within our universe. As a coordinate system, dimensions don't care about scaling, they're just used to measure what we observe. I can measure a building to get coordinates for the three spatial dimensions, then I can add extra floors to make it taller, or add extra rooms to make it longer and wider. 

Since Earth is moving along with the asteroid I want to land on, I have to plot where the asteroid will be in the future when my rocket arrives there (the x, y, z, and t coordinates for my arrival). The ideal is to aim it just right (NASA calls this the "window", the path where the rocket won't run into anything else on the way), figure out exactly how much fuel I need to get there, and at the right moment I fire the thrusters. I can't drive there like an automobile does on the surface of a planet, using a steering wheel. My rocket launches in a completely straight line, affected by spacetime curvatures caused by energy and mass (gravity), and the asteroid moves into my path because that's how I calculated it, and we land safely. 

Understood, but earth and the asteroid are in motion right? There is a reason I ask.

Sounds like this only works for close range proximity? I assume..

Sounds like calculus..😎

2 minutes ago, Phi for All said:

Charles3781's explanatory skills are an illusion. He was talking about the 2-dimensionality of images on your computer screen which simulates 3-dimensional objects. It was a confusing reply to someone like you asking a legitimate question.

We CAN perceive the third dimension, it's not an illusion. We can measure length, width, and height to determine the parameters of an object or phenomenon. How tall something is, how thick it is, how deep the water is, all these are visualizations of the third dimension.

This is more clearer and thanks..

I'm trying to tie this concept in with the pythagorean theorem since it does use some concept of geometry..

6 minutes ago, Charles 3781 said:

No,  it's real, but we can only visually perceive its reality because we  have two eyes.  If we were one-eyed creatures like a Cyclops, we'd see everything as optically flat and 2-dimensional.

We'd have to use our hands and possibly other appendages, to feel the three-dimensional physicality of bodies.

Much like music in mono and stereo, there is a big difference, but it still is what it is....Hymmm maybe being an advance human has its advantages who knows..

11 minutes ago, Phi for All said:

Length, width, and height are the three spatial dimensions we use in our coordinate system, or x, y, and z mathematically. You can visualize each by moving 90 degrees away from every point along the previous dimension (moving from a line to a plane, then from a plane to a cube). The third dimension is equally pictured as height, depth, and thickness, and lends a quality to objects we think of as solid.

Time (t) is a temporal dimension in a continuum with the spatial dimensions (spacetime). I can give you specific x, y, z, and t coordinates so we can meet for lunch on the 45th floor of the Empire State Building, or plot where you need to aim your rocket to reach a specific asteroid, and when to launch it so you can hit it without other maneuvering.

3rd dimension equivalent to height ,depth and thickness, what about 'scaling' making something bigger or smaller?

Can I see an  example of the rocket reaching an asteroid, and the time example to hit the asteroid  please..

9 minutes ago, Charles 3781 said:

I would say, that for humans, the 3rd dimension is "depth" . Like when we look at the colourful images which you kindly posted, in your OP. 

These are attractive, but exist only on our screens, in the form of flat 2-dimensional images.

If we could perceive them in 3 dimensions as solid objects, we would be able to truly visual their 3rd dimensional nature.

So the 3rd dimension is an illusion?

1 hour ago, Phi for All said:

!

Moderator Note

The way you've framed the OP, a reader is forced to watch that video, which is against our rules. Please summarize your question in context.

 

What is the 3rd dimension and how do we visualize it?

Why do we need to imagine it as I thought our concept of geometry was 3 dimensional..

And I need to add this question:

Is Time One Dimensional?

8 hours ago, LaurieAG said:

While I studied calculus at high school we didn't cover infinitesimals until university.

https://www.math.wisc.edu/~keisler/calc.html

 

 

Thnks for the link...I do hope however I can have some answers to my question about limits earlier....in regards to:

Lim x->0 = x^2 -25/ x-5

BEFORE" we add numbers let's speak reason "please."

does x^2 -25 represent y the numerical value, the value that does not change?

and does x-5 represent the changing variable x?

On 10/13/2020 at 9:34 AM, HallsofIvy said:

Let me add this:  one of the things Sir Isaac Newton was concerned with was the force that kept the planets in motion about the sun.  He knew "mass times accelertion equals force" from Gallileo and knew, from Copernicus' measurements, that planets closer to the sun move faster than when they were farther from the sun.  From that he believed the force, and so the acceleration depended upon the distance from the sun.

But "acceleration" is "change in speed divided by change in time" just as "speed" is "change in position divided by change in time" while "distance" is given at a specific time, not over a change in time!  That was why Newton had to invent the Calculus- in order to be able to define "instantaneous speed", "instantaneous acceleration" or, generally, the rate of change at a specific time rather than over a span of time.

Where do these changes in position take place??? 

Do they take place on the orbital paths themselves, since "force" causes the planets to move??

Looks like orbital paths are "discountious" to me, hence the Title of my post.

So Calculus is a tool for orbits and nothing else??

6 hours ago, J.C.MacSwell said:

I could be wrong but I don’t think calculus will answer that. You might come up with a function that could model something they did or thought they did and calculus could give some answers based on that. 

Someone  better versed in math may add to (or correct) that. 

I've read number theory could be the more suitable approach as to use the number theory finds and apply them or plug them into our f(x) function limit issues.

After all, there has to be many variations that define the same value as there is many ways to play the drums to a melody....I guess it depends on reliability, trust and efficiency at this point if the world depended on such a system.

5 hours ago, MigL said:

I'm still not sure what you are asking for, since you keep throwing different scenarios out.

If you are asking about the limit formulation of a derivative ( with x cannot equal 0 ) that is introductory differential calculus, so I find it hard to believe you've been doing it for 20 years.

If we define f(x) to be a function of x, the derivative function of f at x is given by: df (x) = lim h→0 { f(x + h) − f(x) }/ h
If the limit exists, f is said to be differentiable at x, otherwise f is nondifferentiable at x.
If y = f(x) is a function of x, then we also use the notation dy/dx to represent the derivative of f.
You should also note that dy and dx are not numbers but differentials.   

Sorry about the lack of LaTex.   

Yes, differential calculus focuses on rates of changes and the ""Pythagorean Therom"" is extensively used with it.

The notation that expresses these functions also change which gets very frustrating, here is an example:

dy/dx or df/dx or df(x)/dx or D, xy or d/dx or f'(x) or y' and etc etc

"All Mean The Same Thing." 

Ive been observing this for years now of which had me confused for about 4 years until I caught onto it..

let me show a number example of these limits finally "I found one" and we can dissect this from here as a visual guide for direction, becuase the functions are confusing me and maybe others..

Lim x->0 = x^2 -25/ x-5

as "delta" x approaches 0

"BEFORE" we add numbers let's speak reason "please."

does x^2 -25 represent y the numerical value, the value that does not change?

does x-5 represent the value that does change?

24 minutes ago, J.C.MacSwell said:

X can equal 0. If you are interested in when X equals 0, or through a range of numbers where X equals zero you include that point with your function of X. 
if not you don’t include it. It’s not that x can or can’t equal 0, just that you have no interest in the function at that point. 

You get to set limits depending on what you have interest in solving with the function. 

Let's say the interest is 2 stationary vibrating "particles" in space (a,b )and we want to know how the "vibrations" of both interact within a certain "limit "before" change...Or even better, we want to know "which vibration is a or b, since they are all mixed in with each other...If that's even possible if this were a case involving trillions of stationary particles vibrating in space..