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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?

## Want to join the conversation?

• thanks for those really interesting courses.
Maybe I missed something but how can we get the value from a spline curve given a t value from the timeline (not the t parametric value going from 0 to 1).
In the first video, it's easy with th y=mx+b equation but I can't get it with the Casteljau's equation.
Maybe I missed something ?
• If I understand the question correctly, you want to know, for example, if you have a Bezier spline that goes from frame 1 to frame 10, how do you find the value at, say, frame 5. It's tempted to say that you put in the value of t = 0.5, but that doesn't quite work since the x-coordinate along the curve does not change linearly with t.

I don't know if you can do it directly via deCasteljau's equation. I believe it's done by finding the equation as a cubic, and solving for the cubic, which is quite difficult and involves some fairly intense algebra.
• (1-t)^4 A + 4(1-t)^3 t B + 4(1-t)^2 t^2 C + 4(1-t) t^3 D + t^4 E
I typed this into the 4th-degree question and it says it is wrong. Is it because of spacing maybe? From the rest of the notes, it seems it would be the correct answer.
• You were so close!! if you look at my answer to Favour Umoru you will see the pascal's triangle at the end. In your case you have got the coeficient of the middle term wrong it should be a 6 and not a 4. This ought to be what you are looking for
(1-t)^4 A + 4(1-t)^3 t B + 6(1-t)^2 t^2 C + 4(1-t) t^3 D + t^4 E
• Lessons have a Pixar part and a math part.
It would be nice to have a programming part as well; for example, drawing n-degree curves.
• this is cabrasing my brain
• I'm unclear on the terms A(sub)0 or A(sub)1 or A(sub)i etc. Are these sequential terms in the equation or do they refer to specific locations in the equation? I have everything correct until what is the exponent for (1-t)? part.
• A0, A1 etc. refer to points to the control points in order. For a degree 2 curve, we used points A, B, C. But for a degree n curve, we use A0, A1, ... An. We use this instead of A, B, C to make it easier to refer to a point.

So, they do refer to sequential terms in the equation (but the order doesn't really matter - you could write P = (1 - t)A + tB or P = tB + (1 - t)A). With the new naming scheming for the points, the equation for the degree 2 curve is P = (1 - t)A0 + tA1.
• was i supposed to do this because i am only in 4 grade