How to find values of variables when a value is known (after using curve fitter)?

I have used curve fitter app for my xyz data. The custom equation I used is :

a0*exp(-(((x-x0)*cos(phi)+(y-y0)*sin(phi))^2/0.5*ax^2) - (((y-y0)*cos(phi)-(x-x0)*sin(phi))^2/0.5*ay^2))

The equation has six unknown variables : a0, ax, ay, phi, x0, and y0. After fitting the equation on the data, the values obtained were as follows

a0 = 2.7451, ax = 8.1356, ay = 10.3918, phi = 4.1032, x0 = 88.3697, and y0 = 22.5765

I wanted to find the values of ax and ay when the value of a0 is 61% of its value (i.e., 1.6745). How can I do so?

Thank you

Regards

[Stephen23]

An equation must be equal to something... but you have not told us what that formula is equal to.

Should we assume zero?

[Deepshikha Deo]

The z values are treated as equal to the equation. Thus, in curve fitter toolbox you only need to type equation, i.e., z= f(x,y)= equation

And the x,y,z values are used as inputs

Answer 1

[Drew]

Assume this is a surface fit z = f(x,y) as in Curve Fitter app. If it is given that a0 = 1.6745, then substitute that into your given custom equation

a0*exp(-(((x-x0)*cos(phi)+(y-y0)*sin(phi))^2/0.5*ax^2) - (((y-y0)*cos(phi)-(x-x0)*sin(phi))^2/0.5*ay^2))

which results in the new custom equation

1.6745*exp(-(((x-x0)*cos(phi)+(y-y0)*sin(phi))^2/0.5*ax^2) - (((y-y0)*cos(phi)-(x-x0)*sin(phi))^2/0.5*ay^2))

Finally, use the new custom equation in the curve fitter app to complete a new surface fit on your data. The new surface fit will solve for values of ax, ay, phi, x0, and y0.

If this answer helps you, please remember to accept the answer.

[Deepshikha Deo]

Thank you for the answer Drew. After some brainstorming, I also thought the same but after some time I realised I missed an important point. What I want is that the surface should be drawn from the maximum value to its 61 percent fall and the values of ax and ay needs to be found at that point. In this first image, values of ax and ay are found at the dotted line which is above the zero. The image is taken from "https://doi.org/10.1016/j.jclepro.2022.133720".

Here is another example in 2D where the golden region is above zero and the blue line used to represent ax (above 0). The image is taken from "doi:10.3390/a8020082".

So, I want to find the values of ax and ay at 61 percent of its maximum value but it should not be from zero. Any suggestion?