Abstract
The physical world around us is inherently curvy, dynamic, and variable, yet modern industrial civilization is grounded in the planar, rigid paradigms of science and technology. This fundamental disconnect between two-dimensional (2D) techniques and three-dimensional (3D) realities significantly restricts our ability to fully perceive and to understand the complexity of real-world objects. Over the past several decades, driven by application demands across various industries, advancements in high-speed, high-accuracy, and high-resolution sensors, as well as ever-increasing AI algorithms and computational power, curvy surface reconstruction that can reconstruct continuous, smooth geometrical and physical fields from discrete data by algorithms and mathematics have experienced tremendous developments. However, previous reviews in this field have primarily focused on geometric shapes, optical measurement techniques, or reconstruction algorithms, leaving a comprehensive overview that integrates both geometric and physical dimensions still lacking. Here, for the first time, we bridge this gap by expanding the scope from special curvy imaging to general curvy reconstruction incorporating physical fields, with a particular emphasis on measurement techniques, especially the emerging opportunities from advanced techniques. Initially, a brief overview starts with introducing the theoretical underpinnings and primary issues of curvy surface reconstruction. Next, an in-depth discussion of the main non-contact and contact measurement methods is presented, detailing their operational principles, progress, merits and demerits, and future efforts. Following that, several reconstruction algorithms and their applications are discussed. Finally, our insights on the ongoing challenges and opportunities in this field are summarized.