cubic hermite spline

[h4tt3n]

Hello,

 

While learning about splines I've run into some confusion about the wikipedia article on cubic hermite splines:

 

http://en.wikipedia.org/wiki/Cubic_Hermite_spline

 

A few lines from the top of the article it reads:

 

"The subinterval (xk,xk + 1) is normalized to (0,1) via t = (x − xk) / (xk + 1 − xk)."

 

This is causing me some trouble. What does the "x" represent in the (x - xk) term? Also, if xk and xk+1 are vectors, then what does the division mean, since dividing two vectors is undefined?

 

Is the t value a scalar or vector? In the article t seems to be used as a scalar term between (0, 1) which defines where the interpolated point on the spline is relative to the endpoints. A few pages down (Interpolating a data set) it says:

 

"A data set (tk, pk) for k = 1, ..., n (...)"

 

Here t seems to represent something different than the t in the equations in the first part of the article. What does the tk part of the data set represent? tangent vectors or scalar t values? I would appreciate if someone could clarify this a bit.

 

Cheers,

Mike

[DrRocket]

Hello,

 

While learning about splines I've run into some confusion about the wikipedia article on cubic hermite splines:

 

http://en.wikipedia...._Hermite_spline

 

A few lines from the top of the article it reads:

 

"The subinterval (xk,xk + 1) is normalized to (0,1) via t = (x − xk) / (xk + 1 − xk)."

 

This is causing me some trouble. What does the "x" represent in the (x - xk) term? Also, if xk and xk+1 are vectors, then what does the division mean, since dividing two vectors is undefined?

 

Is the t value a scalar or vector? In the article t seems to be used as a scalar term between (0, 1) which defines where the interpolated point on the spline is relative to the endpoints. A few pages down (Interpolating a data set) it says:

 

"A data set (tk, pk) for k = 1, ..., n (...)"

 

Here t seems to represent something different than the t in the equations in the first part of the article. What does the tk part of the data set represent? tangent vectors or scalar t values? I would appreciate if someone could clarify this a bit.

 

Cheers,

Mike

 

[math] (x_k,x_{k+1})[/math] is an interval on [math]\mathbb R[/math] The normaliization is a map [math](x_k,x_{k+1}) \rightarrow (0,1) [/math] given by [math]x \rightarrow t[/math] where [math] t= \frac{x-x_k}{x_{k+1}-x_k}[/math]

 

"t's are points on the real line, "p's" are given values at the point "m's" are tangents, according to the notation adopted in the article.

 

I don't quite see why this is not clear from the article.

[h4tt3n]

[math] (x_k,x_{k+1})[/math] is an interval on [math]\mathbb R[/math] The normaliization is a map [math](x_k,x_{k+1}) \rightarrow (0,1) [/math] given by [math]x \rightarrow t[/math] where [math] t= \frac{x-x_k}{x_{k+1}-x_k}[/math]

 

"t's are points on the real line, "p's" are given values at the point "m's" are tangents, according to the notation adopted in the article.

 

I don't quite see why this is not clear from the article.

 

Thanks for replying, DrRocket

 

First, a few words from the heart of a frustrated person trying to grasp a new field of math:

 

While probably beeing perfectly clear to someone already understanding the concepts of the article, I hope you understand that it may still be unclear to someone still in the process of trying to understand them - which probably includes a good part of the people reading this article. I've seen this too often - the author of an article assumes the reader to know as much or nearly as much about the article topic as he does himself and therefore doesn't see the importance of explaining the used notation or the meaning of used variables, and wether they represent scalars, vectors, matrices and so on. I think this is one mayor reason so many people find math a hard nut to crack, as it can make simple concepts look more difficult than they are. Please have this in mind.

 

And back on topic again:

 

What I still don't understand is this: In the first equation, t is used as a unit interval defining how close the spline point defined by function P(t) is to either of the end points P0 and P1. If t = 0 then P(t) = p0, if t = 1 then p(t) = p1. So, if you want to do a smooth interpolation of p0 and p1 by calculating the - say - 9 equally spaced points on the Hermite spline laying between p0 and p1, you solve p(t) for t = 1/10, ... 9/10.

Further down, the article mentions a data set (t_k, p_k), and this is what's confusing me. Here, t appears to represent something else or beeing used in a different way. The t_k values are used to calculate the tangents in different ways - finite difference, cardinal spline, and Catmull-Rom spline. But what exactly are they and how are they calculated?

 

Cheers,

Mike

[h4tt3n]

In the article both m and t seem to represent a tangent which defines the shape of the spline, but what is the difference between them?

 

In the data set, (t_k, P_k) for k = 1,...,n, does t_k represent a set of arbitrarily chosen tangent vectors?

 

Cheers,

Mike

Edited June 5, 2011 by h4tt3n