Invariant under rotation: Banal, obvious, or noteworthy?

[Trysse]

TL;DR Summary

You can find the coordinates of a given point P in a Cartesian coordinate system, by imagining a sphere on which the point P and the origin are antipodal. The points where the axes intersect this sphere mark the coordinates of the point. Is this relationship noteworthy or banal?

Given a cartesian coordinate system with a fixed point of origin and three axes, it is a fact, that the coordinates of a point P change, when the coordinate system is rotated around its point of origin. The distance between the origin and point P is of course unaffected by such a rotation. What is consequently unaffected is the square over the distance $OP$.

Going back to the theorem of Pythagoras, the squares of each of the coordinates added together is equal to the square over the distance $OP$. This is common knowledge.

However, there is a different way to think about the separation (respectively the relation) between the point of origin $O$ and the point $P$. Imagine the points $O$ and $P$ to be antipodal points of a sphere. The surface area of this sphere is related to the distance between the two points by the ratio

$$A=(PO)^2*\pi$$

This is not too remarkable. However, what I found remarkable is the fact, how each of the three cartesian axes intersects with the sphere in two points. One of the points does not offer any additional information: This point is $O$, the origin of the coordinate system. However, the other point in which the axis intersects the sphere is the coordinate of the point on this axis.

For visualization, you can look here: https://www.geogebra.org/classic/gwas9fwd

Once this relation between the sphere and the coordinates is pointed out, it becomes obvious. In the “classic” introduction of the cartesian coordinate system, the three coordinates can be seen as describing a rectangular cuboid.

Point $O$ is one of the apices of the cuboid and point $P$ is the apex that lies diagonally across the cuboid. The sphere on which the points O and P are antipodal is the sphere that circumscribes this cuboid.

So why did I find it noteworthy?

• The surface area of the cuboid is not invariant under the rotation of the coordinate axes. However, the sphere (and its surface area) does not change as I rotate the coordinate system. So every point in space can be described by a sphere on which point O is antipodal to the described point.
  Every sphere can be described by four points. If I use the points where the Cartesian axes intersect with the sphere, these four points are: The origin $O= (0,0,0)$ and the three points $(x,0,0)$, $(0,y,0)$, and $(0,0,z)$.
• To know the coordinates of a given point, I would normally first drop a perpendicular line from the point to the plane described by two axes and then I draw a second perpendicular line on that plane to the axis of which I want to know the coordinate.
  If I consider the sphere, I must only look, at where the axes intersect with the sphere.
  If I think of the sphere as the naturals shape in space, then this way of thinking about the coordinates might be considered more natural than the cuboid. It is a “natural” way to determine each of the coordinates (i.e. the distance along the x-, y- and z-axis not the numerical values).

In a plane, this relationship can be shown with a circle instead of a sphere.

I have noticed this relation between the cartesian coordinates, the two antipodal points and the sphere while considering the cartesian coordinate system and the Theorem of Pythagoras in three dimensions. After I noticed this relationship, I have tried to find any mention of it in the literature. However, I was unable to find it anywhere. So, the question to myself was: Is this relation so obvious and banal that it is not worth mentioning, or has it been mentioned anywhere and I just did not find it?

Consequently, my questions to you are: Can you point me to a book in which this relation is mentioned? If you don’t know of any book, were you aware of this relationship before you read this post? If you were not aware, do you find it noteworthy or banal? If you were aware, did you ever wonder, why it is not commonly mentioned?

 

[BvU]

I think it's pretty trivial. Works in 2d as well

high school books on math should treat this adequately ...

$\$

 

[Trysse]

I don't know if it is trivial, because I am hopeless when it comes to proofs. However, I am not so sure if it is obvious. Many geometric relations are obvious once you know them. So maybe I should rephrase my question: "Is it common knowledge?"

What do I mean by this: If I ask random people, what they can say about a right-angled triangle I guess many will immediately say something related to the theorem of Pythagoras. They will do so because Pythagoras's Theorem is common knowledge However, what will they say, when you ask them to say something about a circle in a coordinate system where the origin and some point $P=(x,y)$ are antipodal? I am not sure how many people will answer that such a circle intersects the axes in the points $(x,0)$ and $(0,y)$.

high school books on math should treat this adequately ...

But do you know of any math book that has a statement like: "If you consider the points $O=(0,0,0)$ and $P=((x,y,z)$ in a Cartesian coordinates system to be the antipodal points on a sphere than this sphere will intersect the x-, y-, and z-axis in the points $(x,0,0)$, $(0,y,0)$ and ((0,0,z)## regardless of the orientation of the axes"? Or something in that direction that makes this relationship obvious for somebody who is not aware of this fact.
I can of course say, that this fact can be derived from other facts (e.g. considering the equation of the sphere). However, the question is, how many people will consider these things beyond what is written in a book? As long as it is not explicitly stated, that knowledge is contingent.

What I have seen about how coordinate systems are treated in textbooks it is mainly about how to use them and not how they "work".

I guess the intention of this question is to find a book in which this relationship is mentioned to give me a feeling, that somebody else found these thoughts relevant enough to consider and write down.

And as I think about it, I have come to a "practical" application: Imagine I have a rotating coordinate system and I want to keep track of the coordinates of points that are not rotating. This can be nicely visualized:

https://www.geogebra.org/classic/aepr7aqn