Find intersection in a 4-dimensional space

[alx]

Hello everyone!
While working on a piece of code, I got stuck with a problem but my lack of math background makes it hard to even phrase the question to google, as whatever I find doesn’t seem to fit. Could you please push me in the right direction?

I have 4 points defining a 4 dimensional space. To be clear with my definitions, the first point in the table below is: x = 0, y = 900, z = 205, w = 526.
How to find x and y (is it a range?) for any given pair of z and w? In the table, z = 200, w = 1100, x = ?, y = ?


I tried starting with 2D intersections but that seems to lead nowhere.
Thank you!

 

[blamocur]

I cannot understand the problem yet:

I have 4 points defining a 4 dimensional space.

What does this mean? Do you mean you have 4 points in a 4-dimensional space? Please clarify
What is known about dependencies between x,y,z and w?
What are other numbers in the table?
Maybe you can provide a fuller description of your problem, not necessarily in mathematical terms.

 

[alx]

Yes, 4 points in a 4-d space. I was trying to visualize the space with the table, but perhaps just this would be easier:
A (0, 900, 205, 526)
B (0, 100, 80, 350)
C (100, 900, 410, 1119)
D (100, 100, 100, 1007)

In x-y dimensions, the 4 points always form a rectangle. The other two coordinates are variable.
Having the above, can I find x and y for another point E intersecting that plane, given two other coordinates?
E (x, y, 200, 1100)


Am I’m overcomplicating it by putting it in 4D? It can also be seen as 4 points forming a rectangle in a 2D space; each of them has two independent parameters, say z and w:
A (0, 900), z=205, w=526
B (0, 100), z=80, w=350
C (100, 900), z=410, w=1119
D (100, 100), z=100, w=1007

The z and w can be linearly interpolated between the points. With that, I’d need to find the x-y location of a point E, given its z and w:
E (x, y), z=200, w=1100

Hope that makes it more clear?

 

[blamocur]

Yes, 4 points in a 4-d space. I was trying to visualize the space with the table, but perhaps just this would be easier:
A (0, 900, 205, 526)
B (0, 100, 80, 350)
C (100, 900, 410, 1119)
D (100, 100, 100, 1007)

In x-y dimensions, the 4 points always form a rectangle. The other two coordinates are variable.
Having the above, can I find x and y for another point E intersecting that plane, given two other coordinates?
E (x, y, 200, 1100)


Am I’m overcomplicating it by putting it in 4D? It can also be seen as 4 points forming a rectangle in a 2D space; each of them has two independent parameters, say z and w:
A (0, 900), z=205, w=526
B (0, 100), z=80, w=350
C (100, 900), z=410, w=1119
D (100, 100), z=100, w=1007

The z and w can be linearly interpolated between the points. With that, I’d need to find the x-y location of a point E, given its z and w:
E (x, y), z=200, w=1100

Hope that makes it more clear?

This is much, much clearer -- thank you.

You have a function ("map") from 2D to 2D, but the nature of this function is difficult to figure out from just 4 points. It is clear that the map is not linear or affine because parallel lines aren't mapped to parallel ones, as can be seen in the attached graph. We can try fitting a projective transformation to you data, but if you supplied more points it might be easier to guess the nature of the function.
Do you have more data than you posted?

 

[blamocur]

Below is what I got for projective transforms between [imath](x,y)[/imath] and [imath](z,w)[/imath] -- let me know whether it looks right, and what you get for point [imath]E[/imath].

[math]z_k = \frac{a_{1,1} x_k + a_{1,2} y_k + a_{1,3}}{a_{3,1} x_k + a_{3,2} y_k + 1}[/math][math]w_k = \frac{a_{2,1} x_k + a_{2,2} y_k + a_{2,3}}{a_{3,1} x_k + a_{3,2} y_k + 1}[/math]
[math]x_k = \frac{b_{1,1} x_k + b_{1,2} y_k + b_{1,3}}{b_{3,1} x_k + b_{3,2} y_k + 1}[/math][math]y_k = \frac{b_{2,1} x_k + b_{2,2} y_k + b_{2,3}}{b_{3,1} x_k + b_{3,2} y_k + 1}[/math]where:

[imath]a_{1,1} = -0.581842[/imath]

[imath]a_{1,2} = 0.228045[/imath]

[imath]a_{1,3} = 59.798821[/imath]

[imath]a_{2,1} = -1.154890[/imath]

[imath]a_{2,2} = 0.398329[/imath]

[imath]a_{2,3} = 321.556739[/imath]

[imath]a_{3,1} = -0.007884[/imath]

[imath]a_{3,2} = 0.000325[/imath]

[imath]b_{1,1} = 9.293146[/imath]

[imath]b_{1,2} = -6.600246[/imath]

[imath]b_{1,3} = 1566.634358[/imath]

[imath]b_{2,1} = -43.670445[/imath]

[imath]b_{2,2} = -3.493936[/imath]

[imath]b_{2,3} = 3734.939933[/imath]

[imath]b_{3,1} = 0.087474[/imath]

[imath]b_{3,2} = -0.050896[/imath]

 

[blamocur]

The green dot on this graph shows point E with [imath]z=200, w=1100[/imath] and the red dot shows which [imath]x,y[/imath] would it correspond to:

 

[alx]

Thank you! Can I ask where do values for [imath]a[/imath] and [imath]b[/imath] come from?

From manual guessing, the expected values are x = 109.3, y = 340 (with some rounding), so the expected point is
E(109.3, 340, 200, 1100)