Ncurve

Ncurve is a term used in mathematical visualization and computational geometry to denote a family of planar curves indexed by a positive integer N. A specific N-curve is defined by an equation or parametric form that depends on N, so changing N alters the curve’s symmetry and complexity. Common realizations include polar forms r = rN(θ) with N controlling rotational symmetry, and implicit polynomial forms FN(x,y) = 0 where FN is a function of (x,y) and the parameter N.

Construction: One typical construction uses r = a + b cos(Nθ), yielding curves with N-fold symmetry and a varying number of petals or lobes as N changes. Another approach uses parametric equations x = XN(t), y = YN(t) with t in an interval, where the path’s curvature increases with N.

Properties: N-curves often exhibit symmetry corresponding to the index N, and their topology can vary from simple closed loops to curves with multiple lobes or self-intersections. Their smoothness depends on the chosen definitions; some N-curves are analytic, others are piecewise smooth.

Applications: They appear in mathematical visualization, graphics, and education to demonstrate symmetry, topology, and curve behavior under parameter variation. They can also serve as test cases in numerical methods for curve rendering and intersection problems.

See also: Rose curve, Lissajous curve, plane curve, polar coordinate systems, parametric curve.