# Useful functions

The following mathematical functions are useful for smooth movements of two-dimensional objects.

## Smooth transition from origin to point

### Function

```math
f(x) = \frac{-2k}{n^3} * x^3 + \frac{3k}{n^2} * x^2
```

### Preview

*  [Preview](https://www.desmos.com/calculator/kxsbfgjnnz)

Example with $`k=3`$ and $`n=2`$:

<img src="https://gitlab.paulaumann.com/paul/functions/-/raw/master/imgs/func01.png"  width="300" height="300">

### Description

Graph starts at origin `(0,0)` and ends at `(n,k)` with a smooth transition between these points.

## Slight variation of the first function

### Function

```math
f(x) = \frac{2k}{n^3} * x^3 - \frac{3k}{n^2} * x^2 + k
```

### Preview

* [Preview](https://www.desmos.com/calculator/lsaoejeuwt)

Example with $`k=2`$ and $`n=3`$:

<img src="https://gitlab.paulaumann.com/paul/functions/-/raw/master/imgs/func02.png"  width="300" height="300">

### Description

Graph starts at `(0,k)` and ends at `(n,0)` with a smooth transition between these points.