# Algorithm to generate a hill

Setup
I recently started to work with Unity. I want to generate a custom terrain at runtime. To do this i take a grid with a variable amount of squares. For each of the squares i calculate the height with a exchangeable height function. The height function takes the X and Z coordinate as arguments. X and Z range between 0 and 1.
I wrote some height function using perlin noise and random numbers.

Problem
I want to generate a hill. The hilltop (height 1.0) should be in the center (coordinate 0.5, 0.5). And the edges mark the bottom of the hill (height 0.0).

My best bet was a gaussian like function. The result is quite what i am looking for. A smooth top and leveled at the edges. The problem is that the edges are always a bit raised. You can reduce that by scaling X and Z but this results in a smaller hill.

I also tried other functions. But they all have shortcomings like sharp edges or raised edges.

Can anyone help me pick an appropriate function? Or maybe a combination of functions?

Here is a simple possibility:

Let $p(t) = (1-t)^2(1+t)^2$ and $f(x,y) = p(\min(1,\sqrt{x^2+y^2}))$.

This gives a hill of height $1$ at the origin and zero outside $[-1,1]^2$.

Looks like: To get a hill that peaks at $({1\over 2},{1 \over 2})$ and is zero outside $[0,1]^2$, use $g(x,y) = f(2x-{1 \over 2}, 2y-{1 \over 2})$.

• By the way: By converting the Cartesian representation of the graph to the Polar coordinates representation of the graph, you can get better looking results for the mesh that will be used to render the hill on the computer. (Since the hill has "circular" structute that can be represented efficiently using Polar coordinates) Aug 24, 2015 at 8:30

You are going to have problems with a Gaussian because it vanishes only at infinity. You can however construct a "modified Gaussian", that vanishes (with all its derivatives) at the endpoints of an interval $[a, b]$ and equals $1$ at the midpoint: \begin{equation} G_{a, b}(x)=\exp\left( \frac{4}{b-a} + \frac{-1}{x-a} + \frac{-1}{b-x}\right). \end{equation} To construct a hill in the rectangle $[a_x, b_x]\times [a_y, b_y]$ you just need to multiply together two modified Gaussians:

\begin{equation} G_{a_x, b_x}(x)\cdot G_{a_y, b_y}(y). \end{equation}

Here's the result for the square $[0,1]\times[0,1]$: You can use a product of the classical "bump" function which is extremely smooth ( has infinitely many smooth derivatives everywhere ).

http://en.wikipedia.org/wiki/Bump_function In one dimension defined as:

$$b(x) = \begin{cases}e^{-\frac{1}{1-x^2}} & |x|<1\\0 & |x|>1\end{cases}$$

Then $b(x)^{c_x}b(y)^{c_y}$ would be a hill for any positive $c_x$ and $c_y$.

Gaussian sounds good for a hill. I suggest you try dropping it everywhere by a little, but not allowing it to go below zero.

So first calculate the heights the way you already have.

Let $drop$ be the height at $(0,0.5)$

For every point where you have calculated the height, let $height_{new} = max(height_{old} - drop,0)$