Understanding Curve Curvature
Curvature shows how quickly a curve turns. A straight path has zero curvature. A tight bend has high curvature. For a parametric curve, the value depends on velocity and acceleration at the selected parameter. This calculator focuses on that local turning behavior. It does not need the full graph. It needs derivative data near the point.
Why Curvature Matters
Curvature is useful in geometry, design, robotics, physics, roads, animation, and path planning. A vehicle path with large curvature needs tighter steering. A tool path with changing curvature may need slower motion. A particle moving along a curve has normal acceleration. That part of acceleration points toward the curve center. Curvature helps connect the path shape with motion data.
Parametric Curve Inputs
A curve can be written as r(t). In two dimensions, r(t) has x(t) and y(t). In three dimensions, it also has z(t). The first derivative r'(t) is the velocity vector. The second derivative r''(t) is the acceleration vector. This page accepts those components directly. It can also estimate them from three nearby points using central differences.
Reading The Result
The main result is k(t). It is always nonnegative in the standard formula. For two dimensional curves, the page also shows signed curvature. The sign tells orientation. A positive sign usually indicates left turning under the chosen parameter direction. A negative sign indicates right turning. The radius of curvature is the inverse of curvature. Large radius means a gentle turn. Small radius means a sharp turn.
Practical Notes
Curvature depends on parameter direction only through orientation, not through speed scaling. If the velocity vector is zero, curvature is undefined. The curve is not regular at that point. Very small speed can also make results unstable. Use consistent units for all components. If position is measured in meters, curvature is measured as inverse meters. The radius then appears in meters.
Best Use
Start with derivative mode when exact derivatives are known. Use point mode for measured paths. Choose a small step value. Check the speed and cross product. Export results when building reports, tables, or study notes. Use the example table to compare cases. Save each run before changing inputs. Later checks remain clear and repeatable for teams.