## The problem

Most of the time just drawing a simple Bézier curve is not enough. You need to join together a few curves to create a complex shape.

I encountered this problem while coding for Diagramo and tried to solve it. (Took me ages to de-rust my math skills)

An option might be to draw n grade Bézier curve for complex shapes but most of the time we are not so lucky to squeeze our shapes even into a single complex curve.

Here are a few example of bad joins. I just added the tangents at joining points so later I can better explain the problem. ## Solution

There are two levels of smoothness. First level just makes the transition from one curve to another to look decent while second level make the transition look natural.

### Level 1 smoothness

First level of smoothness is reached when the tangent on joining points of the curves are equals. As you can see the tangent of those 2 curves coincide and the transition is smoother.

Mathematically this means that the first derivative of first curve must be equal to the first derivate of the second one (at the glue point) : In the Formulas section you will find the formulas for Cubic, Quadratic curves and their first derivatives.

So all you have to do is to solve those equations (or keep them valid – in case you do something dynamic)

Even if now the curves join together nicely the join still looks "unnatural". That is because the curvature of those 2 curves do not coincide in joining point.

### Level 2 smoothness

In order to make the the join more natural you have to have first derivate (tangent) and second derivative (curvature) equal.  These conditions ensure that the transition from a curve to the next is smooth and natural.

## Formulas

The equation for Quadratic (Q(t)) is with start point ( ), one control point ( ) and an end point ( ) .

Derivative formulas are: ### Cubic curve

The equation for Cubic (C(t)) curves is with one start point ( ), two control points ( and ) and an end point ( ).

Derivative formulas are 