1 - 7 of 7 Chapters
[The study of arbitrary curves—not just conics—becomes possible during the 17th century, first via the consideration of polynomial equations. The development of differential calculus allows next the study of very general curves. We describe the first historical attempts for handling questions...
[Plane curves are studied both via their parametric equations or their Cartesian equation. We study the tangent to a curve and the related problem of the envelope of a family of curves; we exhibit some interesting applications in physics. After a careful study of the curvature of a plane curve,...
[We present a whole bunch of historically important plane curves and list their major characteristics and properties.]
[Curves in the three dimensional real space are studied from the points of view of their equations, their tangent, their curvature and their torsion. We establish the Frenet formulas and we investigate the more involved question of the intrinsic equations of a skew curve.]
[First, we study the equations and the tangent plane to a surface in the three dimensional real space. The central notion of the chapter is that of normal curvature, together with the related notions of umbilical point and principal directions. We establish the important results concerning these...
[The first purpose of this chapter is to provide a deep intuition of formal notions like the metric tensor, the Christoffel symbols, the Riemann tensor, vector fields, the covariant derivative, and so on: an intuition based on the consideration of surfaces in the three dimensional real space. We...
[Global theory of surfaces is interested in those properties which refer to wide pieces of the surface, not just to the neighborhood of each point. We study surfaces of revolution, ruled surfaces, developable surfaces. We study when two surfaces are just an “isometric deformation” of each other...
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