# Bonus: Equations from de Casteljau's algorithm

Challenge question: can you work out the equations for n-degree curves generated by de Casteljau's algorithm?

## Parametric equation for a line

In the first step of de Casteljau's algorithm we define a point along a line in terms of $t$. For example, if we have a line between two points, $\blue{A}$ and $\blue{B}$, then we can define a point, $P(t)$ on that line.

The equation for the point is:

As $t$ goes from $0$ to $1$, $P(t)$ traces out the line from $\blue{A}$ and $\blue{B}$. The equation is linear, so the line can be considered a degree $1$ curve.

### Degree $2$ curves

When we create a degree $2$ curve (a parabola), we use three points, $\blue{A}$, $\blue{B}$, and $\blue{C}$

Now we get this equation for a point on the curve:

### Degree $3$ curves

### Degree $4$ curves

### Degree $n$ curves

Now let's see if we can spot any patterns in these equations that will allow us to find a general equation that uses $n + 1$ points, $\blue{A_0}, \blue{A_1}, ..., \blue{A_{n-1}}, \blue{A_n}$, to define an $n$ degree curve.

Now, the hardest part: look at the remaining terms in each of the above equations. Notice that each term includes:

- a constant
- $(1 - t)$ raised to a power
- $t$ raised to a power

For example, for a degree $2$ curve, the $\blue{A_1}$ term is $2(1 - t)t$, so the constant term is $2$, the exponent on $(1 - t)$ is $1$, and the exponent on $t$ is $1$.

In the coefficient for the $\blue{A_i}$ term in an equation for an $n$ degree curve:

### Extra Super Bonus Challenge

Can you find a formula for the constant term for $\blue{A_i}$? Once you have done that, can you combine all these parts into an equation for $P(t)$ for an $n$ degree curve?