Are Hermite Splines continuous beyond the first derivative? Bezier curves have continuous second derivatives. Does that make them superior to Hermite interpolation in applications that need a smooth curve?
DavidVakshlyak
To interpolate multiple points would we just interpolate P0, P1 then P1, P2 and so on?
zehric
I was thinking about this and the only way I can think of to make a function discontinuous at the second derivative is a piecewise function. If this is wrong, someone please correct me. Assuming this is right, we'd never have this problem here since we're guaranteed to get a smooth non-piecewise function here.
chenwnicole
Would like some clarifications on this: so basically, we just learned Hermit Basis Functions for 2 points. And what this slide is saying that we can do that for inputs with more than 2 points - we just simply compute hermit basis functions for each 2 adjacent points?
wangcynthia
Also confused about this. Does this mean we compute a bunch of different interpolations (one for each 2 adjacent points) as people mentioned above? Or do we treat the input as some n points and n derivatives, and do the same mathematical derivation we did for two points to find one n - 1 degree polynomial?
Jordanwyli
I think what was mentioned in lecture was that we consider the points in a pairwise fashion. The resulting curve would then be piecewise function. It would be mathematically complex to construct a curve with all n points, not to mention plotting it would take a long time.
afang-story
I think Hermite splines are only guaranteed to be continuous at the first derivative because they are calculated so that the first derivatives match in between the points. It is possible to make the second derivatives match as well, but at the same time you may need a higher degree polynomial in order to do so.
Are Hermite Splines continuous beyond the first derivative? Bezier curves have continuous second derivatives. Does that make them superior to Hermite interpolation in applications that need a smooth curve?
To interpolate multiple points would we just interpolate P0, P1 then P1, P2 and so on?
I was thinking about this and the only way I can think of to make a function discontinuous at the second derivative is a piecewise function. If this is wrong, someone please correct me. Assuming this is right, we'd never have this problem here since we're guaranteed to get a smooth non-piecewise function here.
Would like some clarifications on this: so basically, we just learned Hermit Basis Functions for 2 points. And what this slide is saying that we can do that for inputs with more than 2 points - we just simply compute hermit basis functions for each 2 adjacent points?
Also confused about this. Does this mean we compute a bunch of different interpolations (one for each 2 adjacent points) as people mentioned above? Or do we treat the input as some n points and n derivatives, and do the same mathematical derivation we did for two points to find one n - 1 degree polynomial?
I think what was mentioned in lecture was that we consider the points in a pairwise fashion. The resulting curve would then be piecewise function. It would be mathematically complex to construct a curve with all n points, not to mention plotting it would take a long time.
I think Hermite splines are only guaranteed to be continuous at the first derivative because they are calculated so that the first derivatives match in between the points. It is possible to make the second derivatives match as well, but at the same time you may need a higher degree polynomial in order to do so.