Bezier Curves

Learning Bezier curves and illustrating in Rust.

What is Bezier curve?

It's a curve that was invented in 1960s by french engineer Pierre Bézier to design extraordinary shapes.
It's not easy to represent Bézier curve as a set of simple mathematical functions. This curve is parameterized with some parameter and set of points in space and it can turn into any smooth shape you want.

How does this work?

Let's define some concepts:

$t$ — is a curve parameter, assume that $t \in \left[ 0;1 \right]$
$P_i$ — is a i-th point that defines shape of curve, i.e. control point
$B_n\left( t \right)$ — is a point on Bézier curve that uses n + 1 control points

Bézier curve properties that we need for definition:

Curve begins at first control point and ends at last control point
The start and end of the curve is tangent to lines between first two and last two segments
Curve can be split at any point into two subcurves which is also a Bézier curve

Simple case of bezier curve is just a point:

$$ B_0\left( t \right) = P_0 $$

A bit more complicated case is a two-point curve, which is just linear interpolation of these control points:

$$ B_1\left( t \right) = \left( 1 - t \right)P_0 + t P_1 $$

Let's assume that we have n + 1 points so using third property $B_n\left( t \right)$ can be defined recursevily:

$$ B_0\left( t \right) = P_0 $$

$$ B_n\left( t \right) = \left( 1 - t \right) B_{n-1}\left( t \right) + t B_{n-1}\left( t \right) $$

However all coefficient before control points can br calculated and curve can be expressed also as:

$$ B_n\left( t \right) = \sum\limits_{i=0}^n {n \choose i} \left( 1 - t \right)^{n - i} t^i P_i $$

In general case to compute point on a curve with given parameter $t$ we use matrix form:

$$ B_n\left( t \right) = \left( {\begin{array}{cccc} P_{0x} & P_{1x} & \dots & P_{nx} \\ P_{0y} & P_{1_y} & \dots & P_{ny} \\ \end{array} } \right) \left( {\begin{array}{cccc} b_{0,n,n} & \cdots & \dots & b_{0,n,0} \\ \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots \\ b_{n,n,n} & \dots & \dots & b_{n,n,0} \\ \end{array} } \right) \left( {\begin{array}{c} t^n \\ t^{n-1} \\ \dots \\ 1 \\ \end{array} } \right) $$

Build/Run

To build and run the application you need to install Rust and type following:

git clone https://github.com/ivan0sokin/bezier-curves.git
cd bezier-curves
cargo run --release