Plane Geometry: An Account of the More Elementary Properties of the Conic Sections, treated by the Methods of Co-ordinate Geometry, and of Modern Projective Geometry, with Applications to Practical Drawing

Nature ◽  
1923 ◽  
Vol 112 (2816) ◽  
pp. 587-587
Author(s):  
S. B.
Keyword(s):  
Projective Geometry ◽  
Plane Geometry ◽  
Conic Sections

An Easy Proof for Some Classical Theorems in Plane Geometry

1992 ◽  
Vol 35 (4) ◽  
pp. 560-568 ◽  
Author(s):  
C. Thas
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
The Other ◽  
Plane Geometry ◽  
The Real ◽  
Easy Proof ◽  
Special Cases ◽  
Euclidean Planes ◽  
Straight Lines ◽  
Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

Lines as “Foci” for Conic Sections

2019 ◽  
Vol 112 (4) ◽  
pp. 312-316
Author(s):  
Wayne Nirode
Keyword(s):  
Spherical Geometry ◽  
Geometric Object ◽  
Plane Geometry ◽  
Conic Sections ◽  
Geometric Objects ◽  
Definition Of ◽  
Taxicab Geometry

One of my goals, as a geometry teacher, is for my students to develop a deep and flexible understanding of the written definition of a geometric object and the corresponding prototypical diagram. Providing students with opportunities to explore analogous problems is an ideal way to help foster this understanding. Two ways to do this is either to change the surface from a plane to a sphere or change the metric from Pythagorean distance to taxicab distance (where distance is defined as the sum of the horizontal and vertical components between two points). Using a different surface or metric can have dramatic effects on the appearance of geometric objects. For example, in spherical geometry, triangles that are impossible in plane geometry (such as triangles with three right or three obtuse angles) are now possible. In taxicab geometry, a circle now looks like a Euclidean square that has been rotated 45 degrees.

scholarly journals XXXVI. Properties of the conic sections; deduced by a compendious method. Being a work of the late William Jones, Esq; F. R. S. which he formerly communicated to Mr. John Robertson, Libr. R. S. who now addresses it to the Reverend Nevil Maskelyne, F. R. S. Astronomer Royal.

1773 ◽  
Vol 63 ◽  
pp. 340-360
Keyword(s):  
Plane Geometry ◽  
Conic Sections ◽  
Nevil Maskelyne

Sir, You well know that the curves formed by the sections of a cone, and therefore called Conic Sections, have, from the earliest ages of geometry, engaged the attention of mathematicians, on account of their extensive utility in the solution of many problems, which were incapable of being constructed by an possible combination of right lines and circles, the magnitudes used in plane geometry.

Historically Speaking,—: Johan de Witt's kinematical constructions of the conics

1963 ◽  
Vol 56 (8) ◽  
pp. 632-635
Author(s):  
Joy B. Easton
Keyword(s):  
Plane Geometry ◽  
Conic Sections ◽  
Quadratic Equations ◽  
Second Year

In second-year algebra courses the conic sections are defined as type-quadratic equations and the graphs of the curves plotted. It is difficult at this level to relate these equations to the usual geometric definitions of the curves. There arc, however, a number of little-known locus definitions which lead immediately to the canonical equations through the application of elementary theorems of proportion which should be familiar to students of plane geometry.

scholarly journals Essay on Conic Sections

KÜLÖNBSÉG ◽  
1970 ◽  
Vol 13 (1) ◽  
Author(s):  
Blaise Pascal
Keyword(s):  
History Of Science ◽  
Projective Geometry ◽  
Euclidean Geometry ◽  
Early Age ◽  
Conic Sections ◽  
Finite Form ◽  
Parallel Lines ◽  
History Of

Blaise Pascal’s two papers on mathematics, Essay on Conic Sections and The generation of conic sections, are considered basic texts in the history of projective geometry. The two essays are not only important from the perspective of the history of science but are also significant from the perspective of Pascal’s subsequent thinking. When Pascal interpreted conic sections projectively, he encountered the problem of the mathematical infinite in several places. In projective geometry one needs to presuppose that parallel lines cross each other in the infinite, which is not evident in Euclidean geometry. Also, while generating conic sections projectively, often the picture of a finite form will be infinite, like a parabola or a hyperbola, while they are images of the cone’s base, the circle. Pascal had to handle mathematical paradoxes connected to the infinite at an early age, and he tried to integrate these problems into his work rather than reject them. This attitude to the infinite would characterize his subsequent mathematical and philosophical works.

Treatise on Conic Sections

2009 ◽  
Author(s):  
Apollonius of Perga
Keyword(s):  
Conic Sections

The Conic Sections

1891 ◽  
Vol 31 (803supp) ◽  
pp. 12836-12837
Author(s):  
C. W. MacCord
Keyword(s):  
Conic Sections
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