Thread: curves

Hey guys,

I'm looking into curves at the moment, and to be quite honest I can't understand Lagrange polynomials or Hermite curves, I'm just not understanding them!

I do understand the line equation, L(x) = (1 - t)P0 + t(P1 - P0), where 0 <= t <= 1, which is easy enough, but smooth curves, finding that hard!

I've had a look at the Lagrange polynomial, and not quite understanding why it works... It seems to be a linear combination of many polynomial functions (n points give n * (n-1) polynomial combinations?)

Hope someone can help out here!

Thanks!

Originally Posted by rgba

Hey guys,

I'm looking into curves at the moment, and to be quite honest I can't understand Lagrange polynomials or Hermite curves, I'm just not understanding them!

I do understand the line equation, L(x) = (1 - t)P0 + t(P1 - P0), where 0 <= t <= 1, which is easy enough, but smooth curves, finding that hard!

I've had a look at the Lagrange polynomial, and not quite understanding why it works... It seems to be a linear combination of many polynomial functions (n points give n * (n-1) polynomial combinations?)

Hope someone can help out here!

Thanks!

You need to state a precise question.

To paraphrase Dirac, "I don't understand" is not a question.

Originally Posted by rgba

Hey guys,

I'm looking into curves at the moment, and to be quite honest I can't understand Lagrange polynomials or Hermite curves, I'm just not understanding them!

I do understand the line equation, L(x) = (1 - t)P0 + t(P1 - P0), where 0 <= t <= 1, which is easy enough, but smooth curves, finding that hard!

I've had a look at the Lagrange polynomial, and not quite understanding why it works... It seems to be a linear combination of many polynomial functions (n points give n * (n-1) polynomial combinations?)

Hope someone can help out here!

Thanks!

Hey don't feel bad, I do not even know what a polynomial is, but I don't want to know.

Sincerely,


William McCormick

Originally Posted by William McCormick

Hey don't feel bad, I do not even know what a polynomial is, but I don't want to know.

Sincerely,


William McCormick

I cannot imagine a more revealing statement in a mathematics forum.

Hey,

Sorry for late reply, been busy! Working on a hectic VFX job at the moment :P William, I find that a little odd, being interested in math, and you don't even want to know how a polynomial works? I'm interested in any kind of math, even if I don't use it, but in my case I'll be using the math behind curves alot (actually do work with curves quite a bit for VFX, for instance, making particles move along a curve, oscilliating around the curve by linear combinations of the normal and cross product of normal and tangent), but without understanding it totally ). I'm really trying to learn the math and understand why it works, I feel that I will gain alot of benefit from that kind of understanding!

DrRocket, yeah, I should have been a bit clearer about what I don't understand! Well, here goes:

The equation for a linear line is very straightforward, being a point plus a scaled vector, i.e. or . Parameter is in the range to force a convex combination of the two points. That's simple and straightforward.

However, for higher order curves, I'm simply not understanding the general concept behind them? That's my main problem, I'm looking at the Lagrange combination as it seems the simplest of them all?

From what I've read here: http://en.wikipedia.org/wiki/Lagrange_polynomial, it seems that any higher order curve is a combination of many different polynomials, one for every point over than the one in question? In that case I'm confused as to why a polynomial, but I'm probably missing the concept of them entirely. I do understand that you can generate curves with a polynomial function by plotting them on a 2D graph, but the use of polynomials is still slightly fuzzy for me I've never used one on a practical problem...

If someone could explain the general concept behind generating smooth curves, then that would help me move onto the math!

Btw: I'm not going to start learning calculus right now, I'm going to build up with algebra, then trigonometry, linear algebra and then calculus, I'd prefer to have a good foundation to work on, so going to start small and just ignore the more advanced stuff, and just be happy with how it works till I get to the point where I can start learning that stuff... Does that seem like a good idea?

Looking forward to a response! Thanks!

Edit: Looking at the image below:



I can see that the black curve is defined by the 4 different polynomials, one of each which passes through a point on the curve... Why!?!?!

A little more of what I understand:

Use a set of points to define a function. Can then sample points along that function to find new points. Two possibilities:

Interpolation: passes through points
Approximation: passes near points (this would be a higher order curve right? except where the curve passes through one of the points defined in the set?)

Function maps paramter to point on line . Commonly broken into . Called a space curve, of which derivatives are the tangents, .

Linear interpolation: Two points define a line, can generate new points between them. We do this using the parametrization of line, i.e. . Parameter controls position on line, and only defined where in range , i.e. convex combination of points. We can linearly interpolate between points in a set of points defining the line by piecewise linear interpolation, but we get discontinuity at each point on the line as the tangents are flat (?).

Continuity defines a function that has no 'gaps'. continuity is where the tangent is the first derivative (not too sure what this means, but from the definition of a derivative, it's the rate of change of line (acceleration?) at that point. Tangents that join give a continuous function, i.e. no sudden changes in direction as in linear curve.

The above pretty much makes sense to me, but this:

Hermite curve, one solution. Curve is defined by cubic curve between each point and the previous. Tangents at each point specify 'shape' of curve.

Piecewise hermite spline:

For points and tangents function is:



Now that's just plain confusing to me! Hopefully someone can elaborate a little more! Thanks

Originally Posted by rgba

A little more of what I understand:

Use a set of points to define a function. Can then sample points along that function to find new points. Two possibilities:

Interpolation: passes through points
Approximation: passes near points (this would be a higher order curve right? except where the curve passes through one of the points defined in the set?)

Function maps paramter to point on line . Commonly broken into . Called a space curve, of which derivatives are the tangents, .

Linear interpolation: Two points define a line, can generate new points between them. We do this using the parametrization of line, i.e. . Parameter controls position on line, and only defined where in range , i.e. convex combination of points. We can linearly interpolate between points in a set of points defining the line by piecewise linear interpolation, but we get discontinuity at each point on the line as the tangents are flat (?).

Continuity defines a function that has no 'gaps'. continuity is where the tangent is the first derivative (not too sure what this means, but from the definition of a derivative, it's the rate of change of line (acceleration?) at that point. Tangents that join give a continuous function, i.e. no sudden changes in direction as in linear curve.

The above pretty much makes sense to me, but this:

Hermite curve, one solution. Curve is defined by cubic curve between each point and the previous. Tangents at each point specify 'shape' of curve.

Piecewise hermite spline:

For points and tangents function is:



Now that's just plain confusing to me! Hopefully someone can elaborate a little more! Thanks

I think part of the problem is that you are trying to get to advanced concepts a bit too quickly. To explain very much of this you really do need calculus.

Here is some simple stuff.

A general curve is nothing more than a function defined on some interval or the entire real line. That is so general that it is not particularly useful. Most curves are assumed to be continuous. To define a continuous curve rigorously you really need calculus, but basically a continuous curve is one that you can draw without lifting your pencil (there are some pathalogical examples that are exceptions but this is accurate most of the time). A differentiable curve, a , curve is one that is continuous and also has no kinks, or corners -- again you need calculus to define this properly. You can also look at curves that are differentiable as many times as you please -- infinitely differentiable curves. Many curves are in fact infinitely differentiable and even some fairly non-intuitive curves can be that smooth. There are so-called "bump functions" that are infinitely differentiable, but are also zero except on a fixed interval.

You know what a line is, and the equation for a line is y = ax + b. Lines are pretty useful and in fact a good deal of calculus is learning how to study more general curves by approximating them with straight lines, near a fixed point on the curve. That is the guts of differential calculus.

A slightly more general class of curves are those with equations of the form . Curves like this are called polynomials and the highest power of x that occurs, n in this example, is called the degree of the polynomial. It is a general result that any n+1 points with no 2 of them on a vertical line can be connected with a polynomial of degree n -- a generalization of the fact that 2 points determine a line. Polynomials are a simple class of curves with great utility. So one studies polynomials and means of approximating more general functions with polynomials.

You can go a bit beyond simple polynomials and look at functions defined by a power series, which is basically a polynomial of infinite degree. To make sense of such a thing you again need calculus. But functions that can be described by power series are particularly nice, and the study of functions of a complex variable is very much involved with such functions. Power series are also used to approximate functions with polynomials, by simply stopping the series at some finite point. This technique is widely used by physicists and engineers -- sometimes even used correctly.

You can also take a set of points and solve a set of equations to fit those points to a polynomial using a measure of goodness of fit called "least squares", which again requires some calculus to explain properly. This technique is widely used in applied science and in statistics. Because polynomials are easy to work with, polynomial approximations provide great utility in applied science.

There are also some special polynomial classes, and you seem to have discovered at least two of them. But the study of special functions like those really requires that you have a sound understanding of some more advanced mathematics.

Originally Posted by DrRocket

Originally Posted by William McCormick

Hey don't feel bad, I do not even know what a polynomial is, but I don't want to know.

Sincerely,


William McCormick

I cannot imagine a more revealing statement in a mathematics forum.

I can make or build almost anything on earth, and almost have. Ha-ha.

But I have yet to hear of these polynomials as such.

I have found that whenever I hear of something that is not easily understood. Or lack of its knowledge is "so revealing".
That when I do get to the bottom of it. It either has no use in the real world. Or is in use already under other much simpler terms. Or its actual term used in industry is something comically simple.

And you know in a real math forum, if there is something great. Someone is instantly happy to share it. Not try to puff it up or make it bigger then what it is.

I had hit on this subject already, and I was directed to Wikapedia. Not to put down the intended purpose of Wikapedia. However they are often far to deep and do not properly put the worth of the information in proper perspective. And usually omit the simple information needed to understand it.

Like I said there is nothing I cannot build, so why would I want to know what it is? If it is all this complicated?

Math is not about math. Math is about applying math to real things in real life, to make them better. Usually in a math forum the members cannot do that. Math is not supposed to be a distraction from real life. That often kills people. Sudoku is an honest distraction from life using math.

If you said to me "Bill tomorrow I am going to be rolling an offset in a none transition elbow, and someone was saying that I had to roll it like a frustum".

And I said that is very revealing. I would go and hang up.

I would just say it is not a big deal, it just gets rolled along its center. That you line up perpendicular to the roller. And that would be it.

Or you would ask another question. That would probably be my fault, that you had to ask it. Because I had not relayed simple information to you, the beginner or novice. That I should be able to relay with brevity and simplicity, if I am any good.

There are some that would give you miles of confusing information. Or direct you to information that I cannot even understand. And I successfully make them all the time.

So it is revealing actually.

Sincerely,


William McCormick

I'm quite sure there are many more things you haven't built than there are things you have. Have you built a Burr puzzle? Have you built a house? Have you built a computer? Have you built a transistor? Have you built a nuclear reactor? Have you built a perpetual motion machine?

I could continue, but as for that last one, I thought you said you'd post plans for one.

Originally Posted by William McCormick

Originally Posted by DrRocket

Originally Posted by William McCormick

Hey don't feel bad, I do not even know what a polynomial is, but I don't want to know.

Sincerely,


William McCormick

I cannot imagine a more revealing statement in a mathematics forum.

I can make or build almost anything on earth, and almost have. Ha-ha.

But I have yet to hear of these polynomials as such.

I have found that whenever I hear of something that is not easily understood. Or lack of its knowledge is "so revealing".
That when I do get to the bottom of it. It either has no use in the real world. Or is in use already under other much simpler terms. Or its actual term used in industry is something comically simple.

And you know in a real math forum, if there is something great. Someone is instantly happy to share it. Not try to puff it up or make it bigger then what it is.

I had hit on this subject already, and I was directed to Wikapedia. Not to put down the intended purpose of Wikapedia. However they are often far to deep and do not properly put the worth of the information in proper perspective. And usually omit the simple information needed to understand it.

Like I said there is nothing I cannot build, so why would I want to know what it is? If it is all this complicated?

Math is not about math. Math is about applying math to real things in real life, to make them better. Usually in a math forum the members cannot do that. Math is not supposed to be a distraction from real life. That often kills people. Sudoku is an honest distraction from life using math.

If you said to me "Bill tomorrow I am going to be rolling an offset in a none transition elbow, and someone was saying that I had to roll it like a frustum".

And I said that is very revealing. I would go and hang up.

I would just say it is not a big deal, it just gets rolled along its center. That you line up perpendicular to the roller. And that would be it.

Or you would ask another question. That would probably be my fault, that you had to ask it. Because I had not relayed simple information to you, the beginner or novice. That I should be able to relay with brevity and simplicity, if I am any good.

There are some that would give you miles of confusing information. Or direct you to information that I cannot even understand. And I successfully make them all the time.

So it is revealing actually.

Sincerely,


William McCormick

Build this.

http://upload.wikimedia.org/wikipedi..._Waterfall.jpg

Originally Posted by DrRocket

Build this.

http://upload.wikimedia.org/wikipedi..._Waterfall.jpg

That is just a mis-drawn picture. You cannot create 2D drawings to build it for that reason.

It is just a poor attempt at 3D art.


There are mechanical ways to move water like that though, using special impellers that require little energy, but the initial energy to turn them. The use of tank tread water wheels would also make for a more efficient system instead of the water wheel.

Basically large open water tanks connected, mounted to battle tank like treads. So that many could be filled, and a longer straighter distance would see maximum use from the raised water.

We have had perpetual motion since the early 1900's some say since the beginning of time.

Sincerely,


William McCormick





















Sincerely,


William McCormick

Originally Posted by William McCormick

Originally Posted by DrRocket

Build this.

http://upload.wikimedia.org/wikipedi..._Waterfall.jpg

That is just a mis-drawn picture. You cannot create 2D drawings to build it for that reason.

It is just a poor attempt at 3D art.


There are mechanical ways to move water like that though, using special impellers that require little energy, but the initial energy to turn them. The use of tank tread water wheels would also make for a more efficient system instead of the water wheel.

Basically large open water tanks connected, mounted to battle tank like treads. So that many could be filled, and a longer straighter distance would see maximum use from the raised water.

We have had perpetual motion since the early 1900's some say since the beginning of time.

Sincerely,


William McCormick





















Sincerely,


William McCormick

It is not a poor attempt at art at all. It is in fact a rather well known and respected piece of art by Escher. It is an astoundingly successful example of real artistry.

Your lack of appreciation of art matches you lack of appreciation of mathematics.

Neither mathematics nor fine art are confined to the mechanical drawings and capabilities of the machinists art. You need to learn to distinguish among science, mathematics, art, engineering and manufacturing. They are rather different. Each has its place.

Originally Posted by DrRocket

It is not a poor attempt at art at all. It is in fact a rather well known and respected piece of art by Escher. It is an astoundingly successful example of real artistry.

Your lack of appreciation of art matches you lack of appreciation of mathematics.

Neither mathematics nor fine art are confined to the mechanical drawings and capabilities of the machinists art. You need to learn to distinguish among science, mathematics, art, engineering and manufacturing. They are rather different. Each has its place.

It is an illusion not depicting reality, could it be art. I suppose it could be to someone, but I don't consider it quality art at all.

I see it more like someone that either gave up on perpetual motion, or is attacking it. Or making fun of one of their own ideas for it. In any case probably not art. That is the kind of art that would need an explanation of why it is art.

Kind of like a piece of dung on the Mona Lisa painting. One day this painting might be a well respected piece of art to.

Mathematics is only to help in real things in real life. Can you use it for enjoyment as a distraction sure. But then it is just that.

There is nothing that complicated about, math, art or anything good really.


Sincerely,


William McCormick

William, could you show use any pictures or something of your fantastic machines like your perpetum mobile?

Originally Posted by William McCormick

Originally Posted by DrRocket

It is not a poor attempt at art at all. It is in fact a rather well known and respected piece of art by Escher. It is an astoundingly successful example of real artistry.

Your lack of appreciation of art matches you lack of appreciation of mathematics.

Neither mathematics nor fine art are confined to the mechanical drawings and capabilities of the machinists art. You need to learn to distinguish among science, mathematics, art, engineering and manufacturing. They are rather different. Each has its place.

It is an illusion not depicting reality, could it be art. I suppose it could be to someone, but I don't consider it quality art at all.

I see it more like someone that either gave up on perpetual motion, or is attacking it. Or making fun of one of their own ideas for it. In any case probably not art. That is the kind of art that would need an explanation of why it is art.

Kind of like a piece of dung on the Mona Lisa painting. One day this painting might be a well respected piece of art to.

Mathematics is only to help in real things in real life. Can you use it for enjoyment as a distraction sure. But then it is just that.

There is nothing that complicated about, math, art or anything good really.


Sincerely,


William McCormick

I don't expect you to understand this, but perhaps some others will. To mathematicians mathematics is art.

Yeah, and sometimes it's just a hard slog; but we love it anyway.

But I am more interested in an answer to this question: William: If your opinion of mathematics (and by implication, of its practitioners) is so low, why do you post in this sub-forum?

Or, to put it another way. Not everybody finds math interesting, rather fewer, I suspect, find it fun. This is absolutely OK (I assume) with all those here that do.

So, I repeat, if you don't find math interesting or fun, WTF are you doing posting your garbage in a Math subforum.

Originally Posted by thyristor

William, could you show use any pictures or something of your fantastic machines like your perpetum mobile?

I never built one. The only reason is that you do not need that much power from the original device to power a whole city.

So if I go off and design a more efficient water wheel. It would seem like I don't understand electricity.

A more efficient water wheel would be Sudoku at best. All you need is a spark.

Sincerely,


William McCormick

Originally Posted by Guitarist

Yeah, and sometimes it's just a hard slog; but we love it anyway.

But I am more interested in an answer to this question: William: If your opinion of mathematics (and by implication, of its practitioners) is so low, why do you post in this sub-forum?

Or, to put it another way. Not everybody finds math interesting, rather fewer, I suspect, find it fun. This is absolutely OK (I assume) with all those here that do.

So, I repeat, if you don't find math interesting or fun, WTF are you doing posting your garbage in a Math subforum.

I truly find math interesting and fun. I use it everyday. I have a blast with it. I program with it. I have made accounting software. I have made programs to calculate ohms law.

I do not like the way it is being portrayed. As something complicated or for intelligent people only. I know those that are more, or most intelligent, are usually a fairly sick lot.

Truly intelligent individuals would be doing a few kindergarten equations and coming to the conclusion, that they are in deep trouble. And not at all good mathematicians or individuals.

Sincerely,


William McCormick

Wow, flame war coming on

DrRocket, thanks for your helpful reply, yeah, I guess I'm trying to learn this stuff a little too quickly, I'm gonna take it a bit slower and have a look at calculus a little later, just wanna learn a bit more algebra and trig first, I guess that's the logical thing to do! But thanks for the heads up, I get the general gist of things, but not really the workings of :P

William, I get what you are saying, math doesn't need to be over complicated with fancy terminology and whatnot, but like it or not we're stuck with it...

And sorry guys for the late reply, been under the grindstone all week, lots of dreadful overtime work on a vfx job we've been working on! Finally getting the whole week off! (hopefully)

Originally Posted by MagiMaster

I'm quite sure there are many more things you haven't built than there are things you have. Have you built a Burr puzzle? Have you built a house? Have you built a computer? Have you built a transistor? Have you built a nuclear reactor? Have you built a perpetual motion machine?

I could continue, but as for that last one, I thought you said you'd post plans for one.

I don't even know what a burr puzzle is.

I have built tubes that conduct electricity when the diode is broken. Basically a triac.

Everything else I have done. The nuclear reactor got a bit out of hand. Fortunately I knew what could happen. Or else I could have really gotten hurt. What happened was much worse then what I had been told to expect.


Sincerely,


William McCormick

Ok... It's been a really long time since I posted that. Anyway, you should look up Burr puzzles. I think it'd be exactly your sort of thing.

Saying you've done something is easy. Showing it isn't.

"don't tell me, show me" - a quote i remember from someone i talked to a while ago