The characteristic curve of a katana blade is primarily a result of the differential hardening process and the unique design of the sword. The curve of a katana is intentional and purposeful, serving multiple functions:

1. Differential Hardening: During the swordmaking process, the blade of a katana undergoes differential hardening, also known as selective hardening. The edge of the blade is heated and then rapidly cooled, resulting in a harder edge that can hold a sharp edge and withstand cutting tasks. The back or spine of the blade is allowed to cool more slowly, remaining softer and providing flexibility. The contrasting hardness between the edge and the spine creates stress, causing the blade to naturally curve.
2. Structural Integrity: The curve in the katana helps distribute the force of a cutting or striking motion along the length of the blade. It enhances the strength and structural integrity of the sword, preventing the blade from bending or breaking during use. The curve allows the blade to absorb and disperse the impact, making the sword more durable and less prone to damage.
3. Cutting Efficiency: The curve of the katana contributes to its cutting efficiency. It enables a more effective and controlled slicing motion, allowing the sword to smoothly cut through targets with reduced resistance. The curve helps guide the blade along the path of the cut, optimizing its cutting performance.

It's important to note that the curvature of a katana is not a result of intentional bending or shaping after the blade has been hardened. Rather, it is a result of the blade's design and the properties derived from the differential hardening process.

The specific curvature of a katana, known as the sori, varies from blade to blade. The degree of curvature can depend on factors such as the preferences of the swordsmith, the intended use of the sword, and regional or historical variations in design.

The carefully crafted curve is one of the defining characteristics of a katana, contributing to its balance, cutting ability, and overall functionality as a highly efficient cutting weapon.