On the in-and-Circumscribed Triangles of the Plane Rational Quartic Curve (Classic Reprint) - Couverture souple
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Excerpt from On the in-and-Circumscribed Triangles of the Plane Rational Quartic Curve
The last and most difficult case is when the six curves are all of them one and the same carve.
It is to be noted that this formula gives the same number of triangles as has been found by the method used later. For example, in the case of the rational quartic, where a=4, A=6, a=18, the number of triangles is 8, which corresponds to that found on page 18. For the cuspidal quartic, where a=4, A=5, a=16, the number is two, which also corresponds to the number found on page 22.
In this paper it is proposed to look into the existence and actual number of such triangles for the following types of rational quartics:
I. Quartic with three double points.
II. Quartic with one double point and a tacnode.
III. Quartic with a triple point.
IV. Quartic with two double points and a cusp.
This discussion was led up to by preliminary work on the three-cusped rational quintic. Upon subjection to a quadratic transformation this curve goes into a rational quartic, which, it will be shown, has triangles of the kind here mentioned. Accordingly, it will first be proved that the quintic can have certain conditions imposed upon its coefficients so that it may acquire an additional cusp or a tacnode without degenerating. It will also be shown that it cannot have a triple point.
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On the in-and-Circumscribed Triangles of the Plane Rational Quartic Curve
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Paperback. Etat : New. Print on Demand. This book explores the fascinating world of in-and-circumscribed triangles of the plane rational quartic curve, providing an in-depth analysis of their existence, properties, and geometric constructions. The author delves into the historical context of these curves, tracing their origins and development within the broader field of mathematics. By examining different types of rational quartics, including those with double points, cusps, and tacnodes, the book reveals the intricate relationships between the curves' algebraic equations and their geometric properties. Through a series of carefully crafted examples, the author demonstrates the techniques used to construct triangles that are both inscribed within and circumscribed about these quartic curves. The book's insights contribute to our understanding of the geometry of rational curves and provide a valuable resource for mathematicians, researchers, and anyone interested in exploring the beauty and complexity of mathematical forms. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. N° de réf. du vendeur 9781330209219_0
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