What is the difference between De Castel's algorithm and Bernstein's polynomial?

The short answer is one analytical expression, the other a geometric algorithm. So you probably meant "What's the difference between drawing a Bezier curve using De Castelau's algorithm and just calculating the Bernstein polynomial?"

The short answer to this is: on a perfect computer, there is no difference. These are two seemingly different ways to achieve the same result. You use whatever is easier to use in the context in which you need them.

Long answer: you can work out the math and see that the Berstein polynomial can be expressed as a nested sequence of linear interpolations. This expression and its geometric interpretation are called the De Castellau algorithm. On most hardware, the evaluation speed is the t

same for a given one . Although both approaches have rounding errors (due to IEEE floating point numbers), the polynomial solution generates different rounding errors than the geometric expression solution. However, this is noticeably worse than the other.

So, to answer "Why do we need De Castelau's algorithm if we know the Bernstein polynomial?": No "we". There is only "you", according to individual assessment. Check out which is the most accurate and fastest, and then you use that for your specific use case.

Of course, sometimes it makes a lot of difference. For example, a CnC machine cannot estimate polynomials of order 3 or higher, but it can trivially perform a sequence of linear interpolations. So ... again, context. You make a challenge, nobody else.

Of course, in terms of explaining Bezier curves, there is a huge difference. Geometric interpretation is very easy to obtain, whereas analytical interpretation requires an understanding of high school calculus. So, again, context.