Examples : Real plane curves are examples of real algebraic sets and Pierce-Birkhoff conjecture ) are also semialgebraic mappings.
22.
In birational geometry, Noether introduced the fundamental technique of blowing up in order to prove resolution of singularities for plane curves.
23.
Eventually this produces a plane curve whose only singularities are ordinary multiple points ( all tangent lines have multiplicity 1 ).
24.
Informally Gauss defined the curvature of a surface in terms of the curvatures of certain plane curves connected with the surface.
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The envelope of this family of lines is a plane curve whose inverse image under the development is the edge of regression.
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A basic result, in contrast to the case of plane curves, is that not every immersed surface lifts to a knotted surface.
27.
Every immersed plane curve lifts to an embedded space curve via separating the intersection points, which is not true in higher dimensions.
28.
While immersed plane curves, up to regular homotopy, are determined by their turning number, knots have a very rich and complex structure.
29.
In inverse Mach reflection, the Mach stem is convex toward the oncoming flow, and the slip plane curves away from the reflecting surface.
30.
He devoted himself to the classification of higher plane curves, improving results obtained by Isaac Newton, James Stirling, Leonhard Euler, and Gabriel Cramer.
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