Let me add a comment here -- in practice, it is easiest to define our polynomials p(u) for 0 <= u <= 1. What that means for this slide is that we can keep this notion of knots occurring at different times. However, it might be easier to reparameterize our f(t) where t_i <= t <= t_{i+1} in terms of parameter u, such that 0 <= u <= 1. This is pretty simple -- u is just (t-t_i) / (t_{i+1} - t_i). This reparameterization makes our polynomials easier to work with, derive, and think about. We will make the assumption of this parameterization in the class.
Let me add a comment here -- in practice, it is easiest to define our polynomials p(u) for 0 <= u <= 1. What that means for this slide is that we can keep this notion of knots occurring at different times. However, it might be easier to reparameterize our f(t) where t_i <= t <= t_{i+1} in terms of parameter u, such that 0 <= u <= 1. This is pretty simple -- u is just (t-t_i) / (t_{i+1} - t_i). This reparameterization makes our polynomials easier to work with, derive, and think about. We will make the assumption of this parameterization in the class.