# How to Make a Cubic Bézier Spline

## The continuity of a cubic Bézier spline

The red curve below is made by gluing together two cubic Bézier curves. The first Bézier curve is defined by the anchor points $$A_0, A_1$$ and control points $$C_0, C_1$$. The second is defined by anchor points $$A_1, A_2$$ and control points $$C_2, C_3$$.

An object $$s$$ moving along the curve, has the velocity vector $$v$$ and acceleration vector $$a$$. These vectors can be placed anywhere. By placing them at a non-moving point, you can study the curves they trace out. The green curve is the curve of the velocity, and the blue curve is the curve of the acceleration.

A curve is continuous if you could draw it with a pen without lifting the pen from the paper. When measuring the smoothness of a curve, a curve is said to be $$C^0$$ continuous if it is continuous. It is $$C^1$$ continuous if both the curve and its velocity curve are continuous. It is $$C^2$$ continuous if the curve itself, and the velocity curve, and the acceleration curve, are all continuous.

### Calculus to explain the smoothness of A-shape

As was show on the previous page De Casteljau's Algorithm and Bézier Curves, the Bézier curves $$s_0(t)$$ and $$s_1(t)$$ are defined by:

\begin{align*} s_0(t) &= (1-t)^3A_0+3(1-t)^2tC_0+3(1-t)t^2C_1+t^3A_1, t\in [0,1] \\ s_1(t) &= (1-t)^3A_1+3(1-t)^2tC_2+3(1-t)t^2C_3+t^3A_2, t\in [0,1]. \end{align*}

Differentiating one time, we get the velocity curves $$v_0(t)$$ and $$v_1(t)$$:

\begin{align*} v_0(t) &= 3 \left( (1-t)^2(C_0-A_0)+2(1-t)t(C_1-C_0)+t^2(A_1-C_1) \right), t\in [0,1] \\ v_1(t) &= 3 \left( (1-t)^2(C_2-A_1)+2(1-t)t(C_3-C_2)+t^2(A_2-C_3) \right), t\in [0,1]. \end{align*}

Differentiating again, we get the acceleration curves $$a_0(t)$$ and $$a_1(t)$$:

\begin{align*} a_0(t) &= 6\left( (1-t)(C_1-2C_0+A_0)+t(A_1-2C_1+C_0) \right), t\in [0,1] \\ a_1(t) &= 6\left( (1-t)(C_3-2C_2+A_1)+t(A_2-2C_3+C_2) \right), t\in [0,1]. \end{align*}

In order for the velocity curve to be continuous, we must have that $$v_0(1)=v_1(0)$$. In order for the acceleration curve to be continuous, we must have that $$a_0(1)=a_1(0)$$. Hence we have two conditions

\begin{align} A_1-C_1 &=C_2-A_1 \label{eq:one} \\ C_0-2C_1 &=C_3-2C_2 \label{eq:two}. \end{align}

### Showing the conditions geometrically

The conditions can be written using vectors.

Let $$O$$ be the origin. A point $$A$$ can be represented by its position vector $$\vec{OA}$$, i.e. the vector having $$O$$ as the start point and $$A$$ as its terminal point. The first condition can now be written as

$\vec{OA_1}-\vec{OC_1} = \vec{OC_2}-\vec{OA_1}.$

By noting that $$\vec{OB}-\vec{OA} = \vec{AB}$$ (see Linear Algebra - Vectors), the condition can be written as

$\vec{C_1A_1}=\vec{A_1C_2}.$

As for the second condition, it can be rewritten as

$2C_1-C_0=2C_2-C_3 \Longleftrightarrow C_1+(C_1-C_0) = C_2+(C_2-C_3).$

Using vectors, we get that

$\vec{OC_1}+\vec{C_0C_1}=\vec{OC_2}+\vec{C_3C_2}.$ The velocity is continuous if $$C_2$$ is the mirror point of $$C_1$$ in $$A_1$$.

The acceleration is continuous if the mirror point $$C_0'$$ of $$C_0$$ in $$C_1$$, coincides with the mirror point $$C_3'$$ of $$C_3$$ in $$C_2$$.

One way to make the red curve $$C^2$$ continuous, is to let $$C_2$$ be the mirror point of $$C_1$$ in $$A_1$$, and let $$C_3$$ be the mirror point of $$C_0'$$ in $$C_2$$.

Another way to construct points on an A-shape is described next.

## Making a $$C^2$$ continuous cubic Bézier spline defined by points along a path

A spline is a piecewise defined function used for making a smooth curve defined by points along some path. In the example below, the yellow curve is a cubic Bézier spline defined by the red points.

In order to make a cubic Bézier spline, the control points and anchor points must be constructed. Let $$\left( P_i \right)_{i=1}^{n}$$ be a sequence of $$n$$ points (the red points in the example above).

When making an open spline, we need $$n$$ anchor points and $$2(n-1)$$ control points. In order to make points on A-shapes, make control points such that each pair of control points divide the corresponding line segment into three equal parts. As for the anchor points, let the first anchor point be $$P_1$$ and the last anchor point $$P_n$$. Let the other $$n-2$$ anchor points be midpoints between control points as shown above.

When making a closed spine, we need $$n+1$$ anchor points and $$2n$$ control points. The control points are made as in the case of the open curve, but add a pair of control points between $$P_n$$ and $$P_0$$. The anchor points are midpoints between control points, as shown above. The first and last anchor points coincide.

In order to make a cubic Bézier spline where the anchor point are the same as the points $$\left( P_i \right)_{i=1}^{n}$$, another method must be used. For more information see this pdf by Kirby A. Baker, Mathematics Department, UCLA.

In the demo below there is a delay to make the original strokes non-smooth.

## The code

The processing-code to the examples can be seen here: Continuity of Bézier splines, Making a Bézier spline, Drawing a Bézier spline.