Theorem of the curve

The Theorem of the Curve states that if you have a smooth, continuous curve, then the total curvature (how much it bends) along the entire length of the curve can be exactly calculated by summing the angles of the "turns" it makes as you move along it. In simpler terms, it relates the overall bending of a shape to the accumulated measure of how much it changes direction at every point. This theorem helps mathematicians understand the relationship between a curve's shape and its total bending, linking local changes in direction to the global geometric property.