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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(t) on that line.
The equation for the point is:
P(t)=(1t)A+tB
A line between points A and B
As t goes from 0 to 1, P(t) 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
A parabolic arc defined by points A, B and C
Now we get this equation for a point on the curve:
P(t)=(1t)2A+2(1t)tB+t2C

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(t)=

Degree 4 curves

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

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, A0,A1,,An1,An, 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 A0 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 An 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. (1t) raised to a power
  3. t raised to a power
For example, for a degree 2 curve, the A1 term is 2(1t)t, so the constant term is 2, the exponent on (1t) is 1, and the exponent on t is 1.
In the coefficient for the Ai term in an equation for an n degree curve:
What is the exponent on (1t)?

What is the exponent on t?

Extra Super Bonus Challenge

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

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