Curvature and Its Interpretation

Curvature describes how rapidly a curve changes direction at a particular point. A curve with a small curvature bends gently, while a curve with a large curvature turns sharply. For a space curve, the position of a moving object can be described by a vector-valued function r(t), where t often represents time. The direction of motion is determined by the tangent vector, and the unit tangent vector is obtained by normalizing the derivative of the position vector.

The unit tangent vector gives the instantaneous direction of motion along the curve. Curvature measures how quickly this direction changes as the object moves along the path. Since arc length measures distance traveled along the curve, curvature is defined as the magnitude of the rate of change of the unit tangent vector with respect to arc length.

When the curve is described using time, curvature can be found by comparing the time rate of change of the unit tangent vector with the speed of the moving object. In other words, curvature measures how much the direction changes per unit of distance traveled, not simply per unit of time. The derivative of the unit tangent vector is always perpendicular to the tangent direction and points toward the center of curvature, indicating the direction in which the curve is bending.

A helix is a common example of a space curve. It appears in objects such as coiled springs, spiral staircases, and screw threads. The shape of a helix is determined by its radius, which measures the distance from the central axis, and its pitch, which measures the vertical rise per radian.

For a helix, curvature depends on both the radius and the pitch. Increasing the pitch stretches the helix vertically, making it bend less sharply and decreasing its curvature. When the pitch is zero, the helix becomes a circle, whose curvature is the reciprocal of its radius. When the radius is zero, the path becomes a straight vertical line, and its curvature is zero.