Can Bézier curves be generalized to any number of dimensions or are they purely 2D?

Question

Can you generalize a Bezier curve to a function $\acute e : [\mathbb R^n] \to C_n$ where $C_n$ represents a curve in $\mathbb R^n$ and $[x]$ is a list of $x$?

Answer 1

You can carry it over to $\mathbb R^n$ (or $\mathbb C^n$ for that matter). Just use control points in $\mathbb R^n$ and carry over the algorithm. There is a nice animation that shows how Bézier curves can be constructed; it is not limited to $\mathbb R²$ in any way.

Each pair of consecutive points $(P_j,P_{j+1})$ defines a line in $\mathbb R^n$ (gray in the animation) which you travel along using the same parameter for all lines. This gives a new set of points $P'_j$ with one point less, and you use the same rule and parameter to travel the lines (green in the animation) between these points (also green). Then you get a set of points $P''_j$ and lines (blue) etc. until you finally reach at 1 Point (black) which travels along the Bézier (red) as the line-parameter travels from 0 to 1.

Image source (public domain)