Expert-level Explanation: Differentiable parametric curves are defined using parametric equations r(t)=(x(t),y(t),z(t),…) where the functions x(t),y(t),z(t),… are differentiable with respect to the parameter t. These curves are crucial in fields such as computer graphics, physics, and robotics, where smooth and continuous paths are required. The differentiability ensures smooth transitions and continuous changes in direction, which is important for applications like motion planning and trajectory optimization. In AI and machine learning, differentiable parametric curves can be used to model complex decision boundaries or to create smooth approximations of functions.

Historical Overview: The concept of parametric curves has been used since the development of analytic geometry in the 17th century, with further formalization and the notion of differentiability becoming more prominent in the 19th century through the work of mathematicians such as Augustin-Louis Cauchy and Karl Weierstrass. The specific focus on differentiable parametric curves has gained more prominence in the 20th and 21st centuries with the rise of computer graphics and automated control systems.

Key Contributors: Key contributors to the mathematical foundations include Pierre de Fermat and René Descartes, who laid the groundwork for analytic geometry. Augustin-Louis Cauchy and Karl Weierstrass significantly contributed to the formal theory of differentiability. In the context of computer graphics and modern applications, Bézier and De Casteljau’s work on Bézier curves is highly influential, with these curves being a practical subset of differentiable parametric curves widely used in graphic design and modeling.