Finding Equation to Describe Graph of Perceived vs Real Angles

[PatternSeeker]

I show a graph where errors in perceived angles of stimuli increase with the underestimation of lengths of stimuli. I want to find an equation that will describe when maximum error should occur. Specific questions are below. Please see the attached graph. I will be happy with any suggestions that will help me figure this out.

GRAPH DESCRIPTION
It shows a difference between perceived and real angles between edges of certain stimuli (Y-axis) as a function of real angles of stimuli (X-axis).
Separate lines are for different % errors in the perceived lengths of these stimuli. Negative values (-10 to -80 %) stand for % underestimation of lengths.

As you can see difference between perceived and real angles increases with % error in estimation of lengths.

MY QUESTIONS:
Do you have any suggestions about how to generate an equation that will describe this graph? How can I show, using an equation, where the maximum error in perceived angles should occur? That is at what angle (x-axis) would I observe the largest difference between perceived and real angles (y-axis)? Also, how can I show what is the maximum difference between perceived and real angles for specific % errors in perceived lengths?

I hope I am in the right forum :)

 

[LCKurtz]

I don't think there is an exact answer to your question, but there likely is an acceptable one. If I understand your graph correctly, what you have here is data from a function of two variables, the real angle and the % deviation. You would like to approximate this data with a smooth function that interpolates the data and "fits well". You could then use this function to answer both of your questions, at least approximately.

I don't know your mathematical background, but what I would suggest is you look at bivariate interpolation. Your data looks pretty smooth and could probably be approximated well with a either bivariate quadratic or cubic splines. I just checked my version of Maple and it only has built in procedures for one dimensional splines. Too bad because perhaps I could have run your data for you.

 

[jedishrfu]

your data looks a lot like a ball being shot from a cannon at varying angles of elevation with wind resistance factored in.

http://en.wikipedia.org/wiki/Wind_resistance

 

[PatternSeeker]

Hi LC Kurtz,

Thank you for your kind email. This is not exactly what I was looking for though. What I'm after exactly is finding a formula that will tell me where the maximum change between reported and physical angles should occur as a function of (1) % error in reported lengths and (2) physical angle of stimuli.

The graph is based on hypothetical data. Here, the change between reported (perceived/estimated) and physical (i.e., real) angles varies as a function of (1) physical angles, (2) % error in reported lengths of stimuli. For example, as % error increases, the change between reported and physical angles also increases. The greatest change may occur at a specific physical angle. In one of my other threads, I gave a specific example of stimuli involved. Let me know if that would be helpful.
Now, I generated the graph, but it would like to find a formula where I can quickly show if maximum change should occur at a specific physical angle (e.g., 20 deg) given specific % error (e.g., 80 %).

If you had a suggestion regarding relevant methods in math, that would be be useful for me too. I was thinking of using differential calculus to find maximum errors/changes. I took undergraduate calculus a few years ago, that's about it, but I'm a quick study. :)

 

[PatternSeeker]

Thanks jedishrfu.
That does look interesting. I will take a look at this & see if I could use it.