Piecewise Domain and Range Calculator: Understanding the Power of Bézier Curves

In various fields such as computer graphics, animation, and user interface design, animators and designers often refer to the 'physics' or 'feel' of an operation. What they are actually referring to is the specific Bézier curve used to control the velocity over time of a particular move. For instance, a Bézier curve can be employed to specify the velocity over time of an object moving from point A to point B, rather than simply moving at a fixed number of pixels per step.

Bézier curves have numerous applications in various domains. In image manipulation programs, paths are referred to as combinations of linked Bézier curves. In vector graphics, Bézier curves are used to model smooth curves that can be scaled indefinitely. The Bézier triangle is a special case of the latter. These curves can be combined to form a Bézier spline or generalized to higher dimensions to form Bézier surfaces.

The uses of Bézier curves extend beyond animation and graphics, as they are also used in the design of computer fonts. French engineer Pierre Bézier (1910-1999) first employed these curves in the 1960s for designing bodywork for Renault cars. The curve is typically intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated.

A set of discrete control points defines a smooth, continuous curve through the use of a formula. The formula is based on basis functions, which are used to calculate the position of the curve at any given time. In the case of cubic Bézier curves, there are four control points that define the curve's shape and position.

In conclusion, Bézier curves play a crucial role in various fields where smooth and continuous curves are required. The piecewise domain and range calculator is an essential tool for animators and designers to achieve the desired 'physics' or 'feel' of an operation. By understanding how Bézier curves work, designers can create more realistic and intuitive animations that capture the user's attention.

Piecewise domain and range calculator