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B A single or multiple equations of 'y'?

  1. Jul 30, 2017 #1
    Hello.
    I have an object that I want to talk about.
    trieste-side-chair-o_zpsdom72n3a.jpg

    The object above does not seem to have numbers in a definite pattern, so, I think that this object will not have a single equation that describes the object, but instead there will be multiple number of equations of 'y' in 'x'. Correct?
     
  2. jcsd
  3. Jul 30, 2017 #2

    mfb

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    What is x, what is y, in which way is an equation supposed to describe the object?

    Nchairs=1 is an equation and it gives you a pretty good idea what the object is, I think, but that is probably not what you had in mind.
     
  4. Jul 30, 2017 #3
    I have the following equation in my mind and I want to do something similar with the object in my first post:
    x2 + y2 = 1
     
  5. Jul 30, 2017 #4

    mfb

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    All points satisfying this equation lie on a circle in the x-y-plane.

    You can make nearly arbitrary shapes if you are fine with very ugly expressions. The Batman Curve is an example.
     
  6. Jul 30, 2017 #5
    Is it possible to write only a single equation to describe any object?
    Are there any reasons that will compel me to write multiple equations of 'y', to describe an object?
     
  7. Jul 30, 2017 #6

    mfb

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    No. This is not even possible for real numbers. There are real numbers you cannot describe with an equation (the opposite of computable numbers.

    In practice these theoretical limitations rarely matter: For every possible black/white image on a computer screen, you can find an equation where the solutions will look exactly like this image.
     
  8. Jul 30, 2017 #7
    A question:
    The Batman Curve, given as an example, is the only way how the object could be described or is there any other way?
    Is the equation of Batman Curve built for a graph on a computer screen is same as that exists in reality?
    Is it already extended to three dimensions?

    Another question:
    So, by using multiple equations of 'y' in 'x', I can describe anything, or is there anything else that I should know?
    Can this be extended to three dimensions?
    I don't want to look at objects from software programmer's point of view but instead I want to look at objects from mathematician's point of view.
    If I am looking at an equation of an object that is on a graph on a computer screen then does this equation of the object on the graph on the computer screen represents the same object that exists in reality?
    Does the equation formed for the graph really represent the object that exist in reality?

    1.
    I thought, I need to find the "core" of the object and move inside-out mathematically (by using parenthesis) to describe the object.

    For example:
    y = ((a + b ) + (q + d(x(h+q))))
    The Inner-most parenthesis: h + q represents "core". As the evaluation goes inside-out, the object gets constructed by other elements: a, b, q, d.

    Does the approach of going inside-out mathematically make sense?
     
    Last edited: Jul 30, 2017
  9. Jul 30, 2017 #8

    mfb

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    There is an infinite set of ways to do that.
    What do you mean by "exists in reality"?
    It is an equation for this specific line in two dimensions, but you can create similar equations for higher-dimensional objects.
    Not literally everything, but you can make a lot with it, and the exotic cases that don't work are probably not interesting here anyway.
    This is independent of the number of dimensions considered.
    That is a philosophy question, I guess.
    I don't understand what you are asking here.
     
  10. Jul 30, 2017 #9

    Nidum

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    upload_2017-7-30_17-5-7.png

    Any 3D object can be described as an assembly of primitive elements . The simpler the elements are the easier it is to describe them mathematically using nodal coordinates and edge equations .

    (Apologies for the illustration quality - that's the only one I could find quickly on the web)
     
  11. Jul 30, 2017 #10

    No, this isn't how you would define a shape. You would have to go about entirely different means of deriving the equation.

    What mfb is trying to tell you is that you could define a chair in 3D, as a product of several rectangular prisms, arranged into a chair-like object. Actually, for a chair, you can use as few as 12 rectangular prisms. For 3D shapes, we use equations in 3 variables : f(x,y,z)

    An equation for a unit cube of edge length 2 :

    $$\big||x-y| + |x+y| -2z\big| + \big||x-y| + |x+y| +2z\big| = 1$$

    In order to stretch and scale a cube into a rectangle prism, we use additional coefficients like this :

    $$\big||bx-cy| + |bx+cy| -dz\big| + \big||bx-cy| + |bx+cy| +dz\big| = 1$$

    Changing the values of b , c , and d from 0 to any non-negative value can make any proportion you want. You likely won't use anything smaller than 1/10, or larger than 10, for this application.

    Next step is to input additional coefficients to allow the ability to translate the rectangle prism to anywhere in a 3D coordinate grid:

    $$\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| = 1$$

    You can now place the center the shape to the coordinates (f, g, h).

    To build a chair-like shape, you would want to position and scale each of the 12 rectangular prisms to their respective places (one for each wooden bar). You can combine all 12 surfaces into one expression by taking the product of all of them, using a single, repeated equation in general form like this:

    $$\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)$$

    where each individual 'factor equation' (for each wooden bar) has unique values for b, c, d, f, g, h .

    The end result will be a horrid monstrosity :

    $$\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)\cdot\left(\big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| -d(z-h)\big| + \big||b(x-f)-c(y-g)| + |b(x-f)+c(y-g)| +d(z-h)\big| - 1\right)$$


    And, this is how you can plot a chair graph in 3D. It is however possible to simplify it quite a bit, by combining pairs of surfaces in one equation, and using more absolute value expressions. You could condense this into a product of 4 equations, instead of 12.
     
  12. Aug 8, 2017 #11

    Nidum

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    @pairofstrings : Here's a stool for you that doesn't exist in reality . The image is created entirely from numerical data .
    Wooden chair v2.png
     
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