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It seems that when using cubic splines the most common thing to do is to specify that the first (or second) derivatives at the left and rightmost points equal prescribed numbers.

I have never seen anyone imposing a condition on the first and second derivative in one of these points. This would still yield the same number of equations as unknowns and I guess unless one is unlucky this would still yield a non-zero determinant.

Or am I missing something obvious?

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Taking out as many degrees of freedom as possible, suppose we want to interpolate from $(0,0)$ to $(1,j)$ with the first and second derivatives at the origin being $k$ and $l$ respectively. Then if the the cubic spline is $$f(x)=ax^3+bx^2+cx$$ so that $f'(x)=3ax^2+2bx+c$ and $f''(x)=6ax+2b$, the following equations are satisfied: $$f(1)=j:a+b+c=j$$ $$f'(0)=k:c=k$$ $$f''(0)=l:2b=l$$ Thus we have the simple expressions $a=j-\frac l2-k$, $b=\frac l2$, $c=k$ that uniquely determine $f$ for every choice of $j,k,l$. It is entirely possible to fit a unique cubic spline between two points given first and second derivative information at one point, provided that the other point is unconstrained.

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You are not missing anything. Theoretically, you can impose first/2nd derivative at any point. But traditionally, we only impose first or 2nd derivative at the end points simply because

  • we would like the interpolation to produce the same result even when the point order is reversed, and
  • we need to impose first or 2nd derivative at end points to make the spline "connect" to other curves with tangent or curvature continuity (e.g.: creating a transitional curve between two curves).
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