Modified sine function help

Pete2112

Pete2112

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I've been using Desmos unsuccessfully to try and produce a particular graph, but I'm not having much luck, so I'm asking here for help please.
Basically I'm looking for a sine (or cosine) type wave, that increases in amplitude whilst decreasing in wavelength. I need a section of about 6 cycles where the amplitude doubles whilst the wavelength halfs. So far, I can only find functions that increase both the amplitude and wavelength.
Any help would be appreciated
 
If [imath]T[/imath] is the starting period, a.k.a. wavelength, then this function goes from 1 at 0 to 2 at [imath]6T[/imath]:
[math]M(x) = 1 + \frac{x}{6T}[/math]and [imath]\frac{1}{M(x)}[/imath] does the opposite, i.e. goes from 1 to 1/2. As one of the solutions you can then use
[math]f(x) = M(x) \sin \frac{2\pi x M(x)}{T}[/math]
 
Pete2112 said:
I've been using Desmos unsuccessfully to try and produce a particular graph, but I'm not having much luck, so I'm asking here for help please.
Basically I'm looking for a sine (or cosine) type wave, that increases in amplitude whilst decreasing in wavelength. I need a section of about 6 cycles where the amplitude doubles whilst the wavelength halfs. So far, I can only find functions that increase both the amplitude and wavelength.
Any help would be appreciated
Click to expand...
Can you sketch by hand what you want it to look like, so we can be sure?

In particular, do you want a piecewise function that changes from high amplitude/short wavelength to low amplitude/long wavelength (changing suddenly), or smoothly varying between the two?
 
blamocur said:
If [imath]T[/imath] is the starting period, a.k.a. wavelength, then this function goes from 1 at 0 to 2 at [imath]6T[/imath]:
[math]M(x) = 1 + \frac{x}{6T}[/math]and [imath]\frac{1}{M(x)}[/imath] does the opposite, i.e. goes from 1 to 1/2. As one of the solutions you can then use
[math]f(x) = M(x) \sin \frac{2\pi x M(x)}{T}[/math]
Click to expand...

Note: I am assuming the total time is 6 starting periods here:
Figure_1.png
 
Sorry, but my requirements were a little difficult to describe without a sketch as @Dr.Peterson suggested. Thanks to @blamocur for your suggestions. As a maths forum, you may all have a giggle at this, but this curve is simply a template to help me cut out an unusual geometric woodworking project ?. So it doesn't matter which way the graph goes as I can simply turn the template around. I like woodwork with interesting mathematical elements, but whereas some people are happy working with rough sketches, which usually still creates a superb effect, It makes it much more interesting for me to understand the maths and be able to reproduce things accurately in wood to within a millimetre.
This mesmerising YouTube video shows what I want to do, and the first one and a half minutes explains the superb geometry of creating the curve in 3d.
youtube]bcHezjT2f54
 
Last edited by a moderator:
Pete2112 said:
Sorry, but my requirements were a little difficult to describe without a sketch as @Dr.Peterson suggested. Thanks to @blamocur for your suggestions. As a maths forum, you may all have a giggle at this, but this curve is simply a template to help me cut out an unusual geometric woodworking project ?. So it doesn't matter which way the graph goes as I can simply turn the template around. I like woodwork with interesting mathematical elements, but whereas some people are happy working with rough sketches, which usually still creates a superb effect, It makes it much more interesting for me to understand the maths and be able to reproduce things accurately in wood to within a millimetre.
This mesmerising YouTube video shows what I want to do, and the first one and a half minutes explains the superb geometry of creating the curve in 3d.
youtube]bcHezjT2f54
Click to expand...
I don't see anything to giggle about. I use various curves for my 3d printing projects, and, occasionally, for woodworking projects as well. Because of a complete lack of any art talents I rely on math to produce nice shapes.
 
blamocur said:
Because of a complete lack of any art talents I rely on math to produce nice shapes.
Click to expand...
I always thought of math as an art. You can dazzle people with your mathematical ability but you can't do that with ones engineering ability (there's one for you Subhotosh)
 
blamocur said:
I don't see anything to giggle about. I use various curves for my 3d printing projects, and, occasionally, for woodworking projects as well. Because of a complete lack of any art talents I rely on math to p
Click to expand...
Now that I mentioned it :

whole2.jpg
 
Steven G said:
I always thought of math as an art. You can dazzle people with your mathematical ability but you can't do that with ones engineering ability (there's one for you Subhotosh)
Click to expand...
Yep

People see pyramids and are dazzled by mathematics

People see Taj Mahal, and are dazzled by mathematics

People see Golden Gate Bridge and are dazzled by mathematics

I wish I met such people.........
 
Subhotosh Khan said:
Yep

People see pyramids and are dazzled by mathematics

People see Taj Mahal, and are dazzled by mathematics

People see Golden Gate Bridge and are dazzled by mathematics

I wish I met such people.........
Click to expand...
I'm a Physicist. I get dazzled by intense self-reinforcing electromagnetic waves with wavelengths in the range of 400 nm to 700 nm.

-Dan
 
Thank you @blamocur your method does the trick (first screenshot), just needed the curve stretching out a bit to look exactly how I want it ?
Second image shows pretty much the same curve done a slightly different way by my son. He's will soon be starting his 4th year of Natural Science at uni.

@blamocur what is that object? I thought at first it was a type of oloid, which has very interesting mathematical and physical properties.

I see a few of you, just like me, are fascinated by mathematical art and geometry and see maths in all manner of things ??.

The third pic is the graph of x^x where the third dimension is in the complex plane. There are an infinity of complex values making up an infinity of spirals to actual produce a 'solid' 3d surface, but only a few spirals are shown in this graph. I found this pic on the internet but somewhere on my computer (lost it temporarily) my son has programmed a beautiful 'solid' rotatable 3d rendition of x^x using Julia/Python with Blender. When I get access to, or buy my own lathe, I intend to reproduce this shape in some nice grained wood and make it into a table lamp or bed knobs or both. Could be a bit of a talking point, though probably above some people's comprehensive. I think it looks great anyway ?.
 

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Here's a simple bit of mathematical garden woodwork I made. It is a wind spinner. People usually make these with no maths involved. Really you can just cut any old shape, even a rectangle out of the slats of wood laid side by side, then twist them around to make the spinner. That method didn't mean anything to me, so I found myself a nice looking 4th order quadratic in Desmos (as you do), planned it all out on a paper template first. It even lent itself nicely to a bit of calculus to work out exactly the total length of slats the shape required. Nice to find a practical use for calculus. I think it looks good in the garden and is 750mm tall. I can drastically change the shape of it by fanning out the slats with just slightly different overlaps if I want to change how it looks, so I have several different wind spinners in one. I'm trying to get my son to code up a program for me that will render these to any shape or curve and show the different spiral shapes when wound up with various slat overlaps ?.
 

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Pete2112 said:
Here's a simple bit of mathematical garden woodwork I made. It is a wind spinner. People usually make these with no maths involved. Really you can just cut any old shape, even a rectangle out of the slats of wood laid side by side, then twist them around to make the spinner. That method didn't mean anything to me, so I found myself a nice looking 4th order quadratic in Desmos (as you do), planned it all out on a paper template first. It even lent itself nicely to a bit of calculus to work out exactly the total length of slats the shape required. Nice to find a practical use for calculus. I think it looks good in the garden and is 750mm tall. I can drastically change the shape of it by fanning out the slats with just slightly different overlaps if I want to change how it looks, so I have several different wind spinners in one. I'm trying to get my son to code up a program for me that will render these to any shape or curve and show the different spiral shapes when wound up with various slat overlaps ?.
Click to expand...
Lovely. I see you've even changed the angle of every slat's end-cuts to make the shape even smoother!

--

The initial video in post#5 was very interesting. The end result is very nice, but I would prefer it if the leg shape had curve from all angles (from some viewpoints the legs look straight). On the other hand, this sets it apart from a lathed/ turned or spiral leg appearance.
 
@blamocur what is that object? I thought at first it was a type of oloid, which has very interesting mathematical and physical properties.
Click to expand...

This is a hook for a car charging cable, and I used many different curves, starting with ellipses, to get the shape I wanted.
 
Subhotosh Khan said:
Yep

People see pyramids and are dazzled by mathematics

People see Taj Mahal, and are dazzled by mathematics

People see Golden Gate Bridge and are dazzled by mathematics

I wish I met such people.........
Click to expand...
We need to meet as I am one of those people. These beautiful constructions you speak of could not be made without mathematics! There would be no engineers, computers, television, etc etc etc, if not for mathematics (and physics).
 
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