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## Pixar in a Box

### Course: Pixar in a Box>Unit 10

Lesson 2: Mathematics of animation curves

# 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, $A$ and $B$, then we can define a point, $P\left(t\right)$ on that line.
The equation for the point is:
$P\left(t\right)=\left(1-t\right)A+tB$
As $t$ goes from $0$ to $1$, $P\left(t\right)$ traces out the line from $A$ and $B$. 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, $A$, $B$, and $C$
Now we get this equation for a point on the curve:
$P\left(t\right)=\left(1-t{\right)}^{2}A+2\left(1-t\right)tB+{t}^{2}C$

### Degree $3$‍  curves

If we create a degree $3$ curve using four points, $A$, $B$, $C$, and $D$, is the equation for a point on the curve in terms of $A$, $B$, $C$, and $D$?
$P\left(t\right)=$

### Degree $4$‍  curves

What about if we create a degree $4$ curve using five points, $A$, $B$, $C$, $D$, and $E$?
$P\left(t\right)=$

### 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, ${A}_{0},{A}_{1},\text{…},{A}_{n-1},{A}_{n}$, to define an $n$ degree curve.
Look at the first term in each of the above equations and see if you can spot a pattern.
What would be the coefficient for ${A}_{0}$ in an $n$ degree curve?

Look at the last term in each of the above equations and see if you can spot a pattern.
What would be the coefficient for ${A}_{n}$ in an $n$ degree curve?

Now, the hardest part: look at the remaining terms in each of the above equations. Notice that each term includes:
1. a constant
2. $\left(1-t\right)$ raised to a power
3. $t$ raised to a power
For example, for a degree $2$ curve, the ${A}_{1}$ term is $2\left(1-t\right)t$, so the constant term is $2$, the exponent on $\left(1-t\right)$ is $1$, and the exponent on $t$ is $1$.
In the coefficient for the ${A}_{i}$ term in an equation for an $n$ degree curve:
What is the exponent on $\left(1-t\right)$?

What is the exponent on $t$?

### Extra Super Bonus Challenge

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