Splines

[Will]

What does it mean to parameterize at 0 and 1? Does this mean S1 starts
at 0 and S2 ends at 1?

It seems like I'm missing something obvious.

Thanks!

[Ziv Bar-Joseph]

Hi, it means that, at the point they meet the spline on the left has a x
value of 1 and on the right of 0.
Ziv Bar-Joseph

[Matt]

Still not entirely clear on what's being asked here. As far as I can
tell, we are being asked to fit the point (i, v) using splines S1 and
S2. I'm not sure what you mean by your previous response, because both
splines should have the same x-value (and y-value) at the spot they
meet. Also, the problem says that "both splines start at 0 and end at
1", but S1 is only to the left of i and S2 is only to the right of i.
Or are 0 and 1 y-coordinates here? Are we to assume that i is just a
point label, or an x-coordinate between 0 and 1? It seems that there
would be an infinite number of splines that could be fit to just a
single point i. Are we also fitting to the endpoints (0, ?) and
(1, ?). If that's the case, what are the y-values corresponding to
these endpoints? Thanks.

[Ziv Bar-Joseph]

We are fitting a set of points (n) with a few splines (k). The question
does not ask about the fitting but about the constraints of splines which
make them smooth. I mentioned a number of such requirements in class and
these apply to the intersection points of two splines (where one ends and
the other starts). To determine the number of DOF for this point I have
asked you to derive the explicit constraints these requirements lead to. I
said that it may be easier if you assume that each spline is indexed
between 0 and 1 on the X axis so that at the point they meet the one to
the left has a x value of 1 and the one to the right has a x value of 0.
Of course, there are a number of requirements for this intersection point
(related to, for example, equality of the y values) and these should be
used to derive the equations. You do not have to use this indexing scheme
if you prefer something else.
Ziv Bar-Joseph