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How did Leonhard Euler extend the study of curvature to curves in space?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

In the 18th century, Leonhard Euler significantly advanced differential geometry by extending the study of curvature from plane curves to curves in space. While Isaac Newton and Gottfried Leibniz had previously developed formulas for the curvature of plane curves in the 17th century, Euler provided the necessary mathematical framework to calculate the curvature of curves existing within three-dimensional space.1

Euler’s approach relied on the use of calculus to determine the rate of change, or derivative, of the curve's tangent as one moves along the path. This work was foundational, as it allowed for a precise, analytical description of how a curve bends in space, moving beyond the earlier, more limited geometric descriptions.1

Furthermore, in 1760, Euler expanded these concepts to the study of surfaces by introducing the concept of normal curvatures. By examining cross sections of a surface made by planes containing the normal line at a given point, he defined the principal curvatures, which remain a cornerstone of modern differential geometry.21