I would like to know, out of curiosity, if there is any useful or interesting application of approximating vector-valued functions defined on $\Bbb R$ with curves whose component functions are polynomials.
Thank you in advance for any answer!
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What is the context? – Wuestenfux Apr 16 '22 at 11:27
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@Wuestenfux I found an analogous of Taylor’s Theorem for vector-valued functions and I wanted to know if it were of any use – Matteo Menghini Apr 16 '22 at 13:35
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The spline curves that are used in CAD (computer-aided design) systems are vector-valued piecewise polynomials. In the particular case where the spline has only a single polynomial piece, it’s called a Bézier curve. These are just vector-valued polynomials expressed in the Bernstein basis.
In CAD systems, spline curves are used extensively to construct approximations of more complex curves. For example, the curve of intersection of two surfaces is typically approximated by a spline curve.
Drawing packages typically use cubic Bézier curves, which are polynomial mappings from $[0,1]$ to $\mathbb R^2$. So, these kinds of curves are present in a great many places. See here.
In doing this work, Taylor series expansions are not very useful, because they give you an approximation that’s very good at a single point, and pretty bad elsewhere. In engineering/manufacturing, you generally want approximations that are uniformly good over an entire interval, which leads to Chebyshev-style approximation, rather than Taylor series.
bubba
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Reading your answers is a pleasure. You are one of the very few people on Math SE who actually has experience in industry. – Rodrigo de Azevedo Apr 21 '22 at 08:17
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Thank you very much for this amazing answer! I didn't expect that there would such interesting things involved with vector-valued functions approximation. – Matteo Menghini Apr 21 '22 at 09:03
