Growing Mountains from Math

This week, I did two things. A) Got bored and made a wordpress blog, and B) Wondered how to model a mountain.

I started out researching various base points to start from. I imagined a mathematical model for a mountain being vaguely like a normal distribution, peaked at (0, 0), and the limit at infinity would be zero. I tried various curves, until I mistakenly found the Lorenz Hat equation (Which I now can’t find the source).

f(x,y) = \frac{1}{x^2 + y^2 + 1}

Now, this gave me a nice, tidy hat type deal.


Nice, but not nearly noisy enough for a mountain. I then added some “noise” functions, like cos(x*y), etc.


This was nice, but I also wanted a much higher render quality. At this point, I switched over to Blender to do the heavy lifting (Rendering-wise).

First, to export it, I created a gray scale image using mathematica:


After getting the map, I used Gimp to edit it. It took a lot of Gaussian Blurring, but I managed to “smooth” it out mathematically.


Next, I created a Mesh in Blender, then used a displace modifier. I received the displacement data from the texture above, this resulted in a much nicer looking deal.


Here’s a view from the top down. I love how you can still partially see the cos(x*y) in the mapping.Image

Lastly, here’s a beauty shot with a mildly realistic texture.


Thanks for reading!

– Tsoccer93