Can a three dimenstional equation be calculated from 1- and 2-d functions??

Factual Questions
1

Assuming that the two quadratic equations and one linear equation relate all possible combinations of the three variables?

I have three variables: n, t, and p

So if I know that these combine to form three equations (two quadratic; one linear) as follows:

T (p) = ap^2 +bp + c

N(p) = dp^2 + ep+f

and

T(n) = gn

Can I somehow combine all those to figure out P(t,n)?

As always, not homework. Just farting around with some data I derived.

I look forward to either the answer, or to being told the many ways I didn’t ask the question right, in which case I’m happy to clarify as needed! :wink:

2

If I understand what you’re saying, T is always g times as big as n. So telling me either one tells me both, so there’s never need to be told both. This also means that a b and c must be g times as big as d e and f.

So all you have to do to determine p from n or t is invert either one of the first two equations with the quadratic formula. There will either be two or zero possible p’s

3

What OldGuy said. n and T are both proportional to a quadratic in P. After you do all the substitutions, you end up with a quadratic a’ P^2 + b’ P + c’ = 0 (I’ll leave calculating a b & c prime for you). So P has two possible values, and then you can plug that in to find T and N. To get a more interesting answer, you probably need to make T(n) another quadratic or something else nonlinear.

Poking at the math I remember from college and generalizing, you have 3 equations and 3 variables, which IIRC is enough to define a surface in 3-dimensional space. To find out what happens in the volume inside the surface, you need to define some sort of boundary condition. Once you have all that, you can take a slice through z-axis=0 to get the values on the xy plane, which sounds kinda like what you’re looking for.

4

Gotcha ya! So P(t,n) would be a parabola, stretched out edgewise into the third dimension. Sort of as if you took a circle in two dimensions, and stretched it into a column in 3, right?

5

Not sure that “stretched” is the right word. The ensemble of points (n,t,p) that satisfy the equations is a line in 3D space.

Imagine that you make a 3D plot the usual way (i.e. plot this line in a cube) and then project it on the (n,t), (t,p) and (n,p) planes (i.e. look at the cube from both sides or from the top).

In the (t,p) plane, you see a parabola.
In the (n,p) plane, you see a parabola.
In the (n,t) plane, you see a straight line (t=g*n).