pairofstrings said:
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?
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.
