If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 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, start color #6495ed, A, end color #6495ed and start color #6495ed, B, end color #6495ed, then we can define a point, P, left parenthesis, t, right parenthesis on that line.
The equation for the point is:
P, left parenthesis, t, right parenthesis, equals, left parenthesis, 1, minus, t, right parenthesis, start color #6495ed, A, end color #6495ed, plus, t, start color #6495ed, B, end color #6495ed
A line between points A and B
As t goes from 0 to 1, P, left parenthesis, t, right parenthesis traces out the line from start color #6495ed, A, end color #6495ed and start color #6495ed, B, end color #6495ed. The equation is linear, so the line can be considered a degree 1 curve.

### Degree $2$2 curves

When we create a degree 2 curve (a parabola), we use three points, start color #6495ed, A, end color #6495ed, start color #6495ed, B, end color #6495ed, and start color #6495ed, C, end color #6495ed
A parabolic arc defined by points A, B and C
Now we get this equation for a point on the curve:
P, left parenthesis, t, right parenthesis, equals, left parenthesis, 1, minus, t, right parenthesis, squared, start color #6495ed, A, end color #6495ed, plus, 2, left parenthesis, 1, minus, t, right parenthesis, t, start color #6495ed, B, end color #6495ed, plus, t, squared, start color #6495ed, C, end color #6495ed

### Degree $3$3 curves

If we create a degree 3 curve using four points, start color #6495ed, A, end color #6495ed, start color #6495ed, B, end color #6495ed, start color #6495ed, C, end color #6495ed, and start color #6495ed, D, end color #6495ed, is the equation for a point on the curve in terms of start color #6495ed, A, end color #6495ed, start color #6495ed, B, end color #6495ed, start color #6495ed, C, end color #6495ed, and start color #6495ed, D, end color #6495ed?
P, left parenthesis, t, right parenthesis, equals

### Degree $4$4 curves

What about if we create a degree 4 curve using five points, start color #6495ed, A, end color #6495ed, start color #6495ed, B, end color #6495ed, start color #6495ed, C, end color #6495ed, start color #6495ed, D, end color #6495ed, and start color #6495ed, E, end color #6495ed?
P, left parenthesis, t, right parenthesis, equals

### Degree $n$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, plus, 1 points, start color #6495ed, A, start subscript, 0, end subscript, end color #6495ed, comma, start color #6495ed, A, start subscript, 1, end subscript, end color #6495ed, comma, point, point, point, comma, start color #6495ed, A, start subscript, n, minus, 1, end subscript, end color #6495ed, comma, start color #6495ed, A, start subscript, n, end subscript, end color #6495ed, 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 start color #6495ed, A, start subscript, 0, end subscript, end color #6495ed 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 start color #6495ed, A, start subscript, n, end subscript, end color #6495ed 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 parenthesis, 1, minus, t, right parenthesis raised to a power
3. t raised to a power
For example, for a degree 2 curve, the start color #6495ed, A, start subscript, 1, end subscript, end color #6495ed term is 2, left parenthesis, 1, minus, t, right parenthesis, t, so the constant term is 2, the exponent on left parenthesis, 1, minus, t, right parenthesis is 1, and the exponent on t is 1.
In the coefficient for the start color #6495ed, A, start subscript, i, end subscript, end color #6495ed term in an equation for an n degree curve:
What is the exponent on left parenthesis, 1, minus, t, right parenthesis?

What is the exponent on t?

### Extra Super Bonus Challenge

Can you find a formula for the constant term for start color #6495ed, A, start subscript, i, end subscript, end color #6495ed? Once you have done that, can you combine all these parts into an equation for P, left parenthesis, t, right parenthesis for an n degree curve?