A curve, such as 'S', is definitively one-dimensional, despite being represented in a two-dimensional Cartesian plane. This conclusion is based on the understanding that a curve locally resembles a line, allowing movement only forward or backward, which corresponds to one degree of freedom. While curves can be embedded in higher-dimensional spaces, their intrinsic dimensionality remains one-dimensional. The discussion highlights the distinction between the curve itself and its embedding in a larger space.
PREREQUISITESMathematicians, physicists, students studying geometry, and anyone interested in the properties of curves and their dimensionality.
kini.Amith said:Is a curve (Say 'S') 1d or 2d? I ask this question because for so long i was under the impression that it was 2d, since we need a 2d cartesian plane to draw and describe a curve. But then i read in a popular book that it was 1D, which is hard to believe. so which is it?
Lets say the curve is positively sloped. If you move forward (to the right), you would also be moving 'up'. Doesnt that means its 2D?Landau said:Intuitively: imagine living on a curve (forgetting about the "embedding", thinking of the curve as everything there is), then you can only go forward or backwards, i.e. moving along the curve. Phycisists would say something like "there is only one degree of freedom".