In differential geometry, a curve is formally defined as a smooth, continuous map γ: I → ℝⁿ from an interval I of a real parameter t, so that curvature-free straight lines, constant-curvature circles, and variable-curvature spirals are all instances of the same generic object. Curves admit two complementary definitions: the implicit (level-curve) definition, where a curve is the set of points satisfying f(x,y) = c for constant c, which specifies geometric shape but no notion of position, direction, or rate of traversal; and the explicit/parametric definition, γ(t) = (x(t), y(t), ...), which supplies a coordinate function of a parameter t and thereby enables derivative-based quantities such as velocity, speed, acceleration, and curvature.
In differential geometry, a curve is formally defined as a smooth, continuous map γ: I → ℝⁿ from an interval I of a real parameter t, so that curvature-free straight lines, constant-curvature circles…