Coordinate System is just a 'social construct'

It is quite a well known fact that the Coordinate System is just a way to visualize or rather a labeling scheme where each point in space has a unique set of numbers denoting its ‘identity’. We are also familiar with the fact that each person can have a different perspective to look at the same space (basically a different set of basis vectors). But there are many other properties that are rather inherently related to the specific labeling than the axes we choose. If you are looking for a much more concrete sense of where we are going with this, by the end of this article, you can confidently answer which parabola that passes through (1,2),(3,7) and whose axis makes 30° of angle with the x-axis.

Part 1: Starting off Slow

Some of you must be familiar with the concept of family of curves. Often times the style of generating such families is quite algebraic rather than intuitive. Let us take an example

We can see the above construction of equations very clearly.

$$x^2 + y^2 = 4 \\$$

This is a circle with radius 2 and center at origin.

$$y = x$$

This is a line with slope 1 that passes through the origin.

$$x^2 + y^2 - 4 + k(y - x) = 0$$

If we can notice something very unique that can explain the nature of this curve without any effort, It would of huge profit. Particularly the line and the circle intersect at 2 points and thus we can notice quite easily that for every k belonging to R, those 2 points always satisfy the equation. If we are keen enough to see that the simplification of the expression also results in an expression which is a circle independent of k. We can firmly state that the above expression is a circle passing through the 2 points. Also notice that how i attached the 2 equations with a sum instead of a multiplication. As it is widely known that additions corresponds to an AND gate and multiplications correspond to an OR gate. Using an addition works because both curves cover both the points. But we aren’t here to talk about generating such curves , are we ?

You could use this logic of OR and AND gates to combine different curves and lines to form many different families of curves. As far as I have seen, we can generate many families of curves (Ellipses, Hyperbolas, Circles, Parabolas, etc) which contain upto 4 points of our liking !! 4 POINTS. Such a generation of family of curves can be very useful for any person to find a specific curve passing through a number of known points.

Part 2: Involvement of Axes

Up until this point, I am sure all of you must be quite familiar with all this information. But lets take a very acute turn in our interpretation of family of curves.

$$y = (x-1)(x-2)$$

It is a very standard upwards facing parabola that we all know about where we can distinctly identify that 1 and 2 are roots of the expression. But what I propose is to not see those numbers as root but rather lines L1:(x-1) and L2:(x-2) and L3:y=0. What i mean is to view it likewise.

$$(y-0)-(x-1)(x-2)=0$$

Looking at it like this , we can easily identify that the point (1,0),(2,0) are present on both sides of the AND gate , namely (y-0=0 and (x-1)(x-2)=0) and because only one of the points are covered each by (x-1) and (x-2) , we link them using a multiplication (OR gate).

There is a key insight that (x-1=0) and (x-2=0) are parallel. And soon , we are going to enter the good phase itself.

Part 3: Replacing of Axes with Lines

I say that the axes themselves are just lines that we consider very special but although aren’t that special at all. It is almost as if that axes are just a reference that we choose to actually label all the points in space, isn’t it ? And all our equations when taking x or y into consideration are also just lines that we measured. For example,

$$(y-0)-((x-0)-1)((x-0)-2)=0$$

How about we replace the reference of these lines , namely L1:y=0 and L2:x=0 with some other lines ? let us replace it L1 with y-x and L2 with x+y.

It is a parabola but rotated by 45 degrees !! We essentially rotated a parabola but using nothing but change of axis. Further investigating onto its fixed points.

we can see once again see that the lines (x+y-1=0) and (x+y-2=0) are parallel. Essentially , whenever 2 parallel lines and a line intersecting them both are combined in this special way, we can assure that it will always be a parabola whose axis has the same slope as the slope of the parallel lines and it will always pass through the 2 points.

Lets take all of it a bit more slowly, we have 2 main points for this discovery,

• We can assure that such a parabola will always pass through 2 points because of the gates argument.

• We can also assure that the slope of the axis will be what we want because we just replaced the axes lines (eg. y=0 and x=0) with some other set of axes.

Therefore, here we can firmly understand that just taking into consideration that the axes are nothing but a social construct and that we are free to change it whenever we like, we can construct a huge amount of very simple equations that depict rotated parabolas , hyperbolas and even ellipses in many cases.

For Example, Below I show that how simple it could be to just write a family of parabola that passes through (1,2) and (3,5) whose axis makes 30 degrees of angle with positive x-axis.

In wrapping up, this article delves into the intriguing notion of coordinate systems as adaptable constructs rather than fixed entities. By viewing axes as mere reference lines, we unlock the potential to reinterpret and manipulate mathematical expressions creatively. This shift in perspective not only deepens our understanding of geometric transformations but also simplifies the construction and analysis of complex curves. As we explore the broader implications of this idea, we may discover applications across diverse fields, from physics to computer graphics, where the flexibility of coordinate systems can lead to innovative solutions. Embracing this adaptability fosters a more dynamic approach to mathematical problem-solving, enriching our appreciation of the mathematical world. I hope you gained a good perspective on the coordinate systems and can also expand over to hyperbolas and ellipses alike just like we have done for parabolas. Have a Great Journey Ahead !