Geometry of Curves and Surfaces

How do we mathematically describe the curvature of a winding road, the shape of a dome, or the complex surfaces used in modern computer graphics? Understanding the geometry of curves and surfaces is essential for fields ranging from physics and engineering to computer-aided design and data science. This course provides a clear, step-by-step guide to the foundational mathematical principles that govern shapes in three-dimensional space. You will transition from basic coordinate geometry to analyzing complex, curved structures with confidence and precision. What you will learn: Understand the core concepts of arc length, curvature, and torsion for spatial curves; Analyze surfaces using first and second fundamental forms; Calculate normal, principal, mean, and Gaussian curvature; Apply the Gauss-Bonnet theorem to relate local geometry to global topology; Explore modern computational applications of differential geometry. This written course begins with foundational definitions of vector calculus and parameterized curves before advancing to surface theory and intrinsic geometry. It is designed for beginners, students, and professionals in mathematics, physics, or engineering with no prior background in differential geometry. Start your journey into the mathematical beauty of shapes and spatial analysis today.