piecewisekurva

A piecewise curve is a curve that is defined by multiple distinct functions over different intervals of its domain. Instead of a single mathematical expression describing the entire curve, a piecewise curve is constructed by joining together segments, each governed by its own unique function. These functions can be of any type, such as linear, quadratic, exponential, or trigonometric, and they are applied to specific ranges of the input variable (typically x).

The critical aspect of a piecewise curve is the points where the different function segments meet, known as "junction points" or "breakpoints." For a piecewise curve to be considered continuous at a junction point, the value of the function from the left segment must equal the value of the function from the right segment at that specific point. If these values do not match, the curve will have a "jump discontinuity" at that point. Furthermore, continuity can also refer to smoothness; a piecewise curve can be continuous but not differentiable at a junction point if the slopes of the adjacent segments do not match.

Piecewise curves are widely used in various fields, including mathematics, physics, engineering, and computer graphics. They are particularly useful for modeling real-world phenomena that exhibit different behaviors in different situations or over time. For example, a speed profile of a vehicle might be represented by a piecewise linear function, with different slopes indicating acceleration, constant speed, and deceleration. In computer graphics, they are fundamental for creating smooth and complex shapes and animations.