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A generalization of the first Plücker formula Sparling, George William

Abstract

The first Plücker formula from algebraic geometry gives the class of an algebraic curve in terms of the order and the singularities of the curve. Here a study is made of real, differentiable curves with a view to finding the corresponding result for such curves. The class of a point P with respect to a real, differentiable curve C is defined to be the number of tangents of C which pass through P. First it is shown how the class of P depends on its position relative to C, then it is shown how the class of P depends on the nature, numbers, and relative positions of the singularities of C. In the last Chapter the results are applied to classify real, differentiable curves of class three. It is found that a curve of class three must contain one of the following three combinations of singularities: (1) One cusp and one inflection point. (2) One cusp and one double tangent. (3) Three cusps.

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