Functions: instead of plotting points, can you move planes?

[grisman]

This is a question that has been burning for some time, I have been wondering, instead of plotting the different points of a function onto a steady x and y axis, is it possible to have a single point (at the origin) and have the planes move instead. The space moving around the point.

When I think about it, there are a lot of problems. Let's say y=sinX. Our plane I imagine as y=0 and x=0. Our point begins at the origin and itself does not move. Towards the interval x=pi/2 I imagine the plane has now shifted to the x-axis being defined as y=-1 (to account for the upward movement of the stationary point. I imagine a pen held steady while I move a piece a piece of paper. I pull down to draw the ink upward) and our y-axis as now x=-pi/2.

And I also wonder with directions, say for the same function, if I wish to picture both directions occurring at the same time, my plane has now split into two planes which began at y=0, x=0 as my axis' and (0,0) as my only point.

Instead of equations describing relationships between the dimensions and the points, if instead there were ways to desribe moving planes with a stationary point defined by the origin.

And I wonder if this is a redundant way to think of it, as I still am relating my new planes, to the original plane, and/or if the current system can already be thought of it as this way.

I'm sorry to ask such a question as I know I haven't worded it all that well, its just been on my mind a LOT lately. I just wish to know if this is a valid idea somewhere in mathematics, and to keep it in mind; or if its just not applicable. I don't require a complex answer, I just want to know to know if this has any worth to it.

thank you if you have read this far

 

[Mark44]

Sure, you could think of things this way - as a fixed point with the plane moving under it. There is some precedence for this in geology, in relation to the volcanoes in the Hawaiian Islands. If you think of the seafloor as being fixed, it appears that volcanoes have erupted along a roughly NE to SW line, with older volcanoes at the NE end and younger ones in the SW. The current thinking is that there is a fixed hot spot in the Earth's mantle, and the crust itself (the Pacific Plate) has been moving over the hot spot.

As far as mathematics is concerned, it's probably more complicated to describe the motion of the coordinate plane than to describe the motion of a point on a curve, but the two ways of looking at things would be equivalent. Your idea is used, to some extent, when we describe complicated equations as translations of simpler equations.

For example, the graph of y = (x - 2)² + 3 is nothing more than a translation to the right by 2 units, and a translation up by 3 units of the graph of y = x².