Dimension of a Spline Space

What is the dimension of a given spline space S(k,t)? What does the dimension of spline space refer to? It seems like it might refer to the dimensionality of the control points, but maybe I'm missing something.

I'm reading about geometric modeling with splines and this question came up.

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1 Answer

A "spline space" is a vector space. The elements of this vector space are the spline curves (or piecewise polynomials, if you prefer). So, like any finite-dimensional vector space, this space has a "dimension". That's what they're talking about.

The dimension of a vector space is the number of elements in any basis. Very roughly speaking, it's a measure of the "size" or "expressive power" of the space. In the case of a spline space, the dimension is mostly dependent on the number of entries in its knot sequence.

It's not related to the dimensionality (2D or 3D or whatever) of the control points.

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care to give the exact formula? –  nbubis Jan 6 '13 at 3:08
If the splines are of order $k$ (degree $k-1$), and you have $n$ knots, then the dimension of the spline space is $n-k$, I think. It's messy because the spline representations that are typically used in geometric modeling have a knot value at each end that's really not used, and could be omitted. For example, to represent a cubic Bezier curve, many people would use a knot sequence of $\{0,0,0,0,1,1,1,1\}$. But $\{a,0,0,0,1,1,1,b\}$ will give you exactly the same curve provided $a\le0$ and $b\ge1$. –  bubba Jan 6 '13 at 5:35
I believe this is incorrect. The size is $n+k$. Think about it like this: there are $n$ intervals each with a polynomial of degree $k$. This gives $n(k+1)$ free coefficients. However, we impose continuity in derivatives $0,...,k-1$ at the $n-1$ internal nodes. This is $(n-1)k$ constraints. $n(k+1) - (n-1)k = n+k$. –  Chester May 5 at 1:21
Consider a very simple case -- cubic Bezier curves (i.e. cubic splines that have a single segment and 4 control points). The usual knot sequence would be $\{0,0,0,0,1,1,1,1\}$, so $n=8$. Also $k=4$. So $n-k=4$, which is correct. On the other hand, $n+k =12$, which seems obviously wrong, to me. The flaw in your reasoning is that $n$ knots does not imply $n$ intervals. –  bubba May 9 at 8:53