F(x, y) describing a distorted 3D Gaussian bell

In summary, the conversation discusses the creation of a function that would produce an egg-shaped 3D shape with a 2D cross-section. The suggested solution involves using a bivariate normal distribution, also known as a Gaussian distribution, and modifying it with a Crystal Ball function or a modified cos curve. The final result would be used for creating procedurally generated islands.
  • #1
corpsinhere
6
3
I am new to these forums - if I have posted in the wrong place please let me know.
Standard 3D Gaussian bell: z = e^-(x^2) * e^-(y^2)
From along the z-axis this looks "round".
I would like a generalized f(x, y) which would look egg-shaped from above - possibly quite distorted..
I thought at first that this would be easy - hours of experimentation later I have a new respect for this problem. I hope someone finds a simple solution - I would love to have missed something simple!
Thanks,
Broos
 
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  • #2
Basically you need a function of the form ##f(x,y) = e^{-( Ax^2 + Bxy + Cy^2)}##. Look up "bivariate normal distribution" , "bivariate gaussian distribution", or "multivariate normal distribution".
 
  • #3
That is super-cool, and will allow me to squish and rotate the bell; however, all the resultant shapes are still bilaterally symmetric. Not so much egg-shaped as oval. I would very much like to find a way of producing a 3D shape with a 2D cross-section similar to this:
https://www.screencast.com/t/ztmJfYS2Gen0

Looks like e^-(1*ln(x)2+ 1*ln(y)2)
get closer - from the side it looks like this: https://www.screencast.com/t/ZfDcDhGiTj
but from the top it looks sqare-ish: https://www.screencast.com/t/wV5Oz1G7
 
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  • #4
It's not clear from the picture what you want. In general you can easily distort a 2-d Gaussian so that the cross section looks like an ellipse. In this case, set up the ellipse and determine its center and the two axes. These then can be used for the 2-d Gaussian exponent.
 
  • #5
Something like the Crystal Ball function?

In general you can rotate your function by an angle ##\phi## by setting ##x' = \cos(\phi)x + \sin(\phi) y## and ##y' = \cos(\phi)y - \sin(\phi)x##.
 
  • #6
mathman said:
It's not clear from the picture what you want. In general you can easily distort a 2-d Gaussian so that the cross section looks like an ellipse. In this case, set up the ellipse and determine its center and the two axes. These then can be used for the 2-d Gaussian exponent.
@mathman - actually I am specifically *not* looking for a z-ward cross-section that looks like an ellipse - the shape I am looking for is distorted so that the cross-section looks like an exaggerated egg: one end narrow, the other wide. More like you took a regular Gaussian bell and smudged it in one direction.
 
  • #7
mfb said:
Something like the Crystal Ball function?

In general you can rotate your function by an angle ##\phi## by setting ##x' = \cos(\phi)x + \sin(\phi) y## and ##y' = \cos(\phi)y - \sin(\phi)x##.
@mfb - Oh that is excellent! I am not sure how to generalize it to a 3D shape; if we simply rotate around the maximum then our shape again becomes symmetrical. As in my reply to @mathman, the shape should look "like you took a regular Gaussian bell and smudged it in one direction."

FYI the way in which this will be used:
I am making procedurally generated islands. The base is Perlin noise. I am looking for a family of shapes to distort, rotate, and overlap to create an envelope over the Perlin noise. The egg, or teardrop z-ward outline is meant to roughly simulate arms of a central mountain system; the off-central maximum with exponential falloff should leave a rough, pleasing edge.
Just for fun here is an earlier attempt using Perlin noise and many overlapping Sinwaves: https://www.screencast.com/t/2w6unMYt9i7
Ty!
 
  • #8
A Crystal Ball in one dimension, a Gaussian shape in the other one will give some sort of egg-shaped peak.
 

1. What is the significance of F(x, y) describing a distorted 3D Gaussian bell?

F(x, y) describing a distorted 3D Gaussian bell is a mathematical function used to model the probability distribution of a continuous random variable. It is commonly used in fields such as statistics, physics, and engineering to describe various phenomena.

2. How is F(x, y) related to the normal distribution?

F(x, y) is a variant of the normal distribution, also known as the Gaussian distribution. In a 2D or 3D plot, the shape of the bell curve will be distorted, but the fundamental properties of the normal distribution still hold, such as the mean, median, and mode being equal and the majority of data falling within a certain number of standard deviations from the mean.

3. How is the 3D Gaussian bell distorted?

The distortion of the 3D Gaussian bell is determined by the values of the parameters in the function, specifically the standard deviations in the x, y, and z directions. These values control the spread and shape of the bell curve, with larger standard deviations resulting in a wider and flatter curve, and smaller standard deviations resulting in a taller and narrower curve.

4. What are some real-life applications of F(x, y) describing a distorted 3D Gaussian bell?

F(x, y) is used in a wide range of fields and applications, including image processing, signal processing, financial modeling, and weather forecasting. It is also commonly used in machine learning and artificial intelligence algorithms to model and analyze data.

5. How is F(x, y) calculated and plotted in a 3D space?

F(x, y) is calculated using a combination of the x and y values and the parameters of the function. The resulting values are then plotted on a 3D graph, with the height of the curve representing the probability of a given combination of x and y values occurring. This allows for a visual representation of the distribution and can provide insights into the behavior of the data.

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