Curvas de bézier cúbicas e suas propriedades

Abstract

This work investigates the mathematical characteristics of Bézier curves, we start with the discussion of parametrically differentiable curves and curve reparametrization. Next, Bernstein polynomials, the main characteristics of Bézier curves, their historical origins, and modern applications are addressed. Mathematical definitions are presented, including Casteljau's algorithm, along with properties such as the endpoint interpolation property, control points and curve shape, convex hull property, and invariance under affine transformations. We demonstrate the convex hull property for the specific case of cubic Bézier curves and provide some examples showing the invariance of Bézier curves under affine transformations like translation, linear scaling, reflection, and rotation. The objective of this work is to explore the mathematical elements and properties of Bézier curves.