Plane Geometry: An Account of the More Elementary Properties of the Conic Sections, Treated by the Methods of Coordinate Geometry, and of Modern Projective Geometry, with Applications to Practical Drawing

1922 ◽  
Vol 11 (159) ◽  
pp. 128
Author(s):  
L. B. Benny
Keyword(s):  
Projective Geometry ◽  
Plane Geometry ◽  
Conic Sections ◽  
Coordinate Geometry

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 III. On scalar and clinant algebraical coordinate geometry, introducing a new and more general theory of analytical geometry, including the received as a particular case, and explaining ‘imaginary points,’ ‘intersections,’ and ‘lines.’

1860 ◽  
Vol 10 ◽  
pp. 415-426 ◽  
Keyword(s):  
General Theory ◽  
Present Theory ◽  
Plane Geometry ◽  
Analytical Geometry ◽  
Abstract Equation ◽  
Unit Of Length ◽  
The Right ◽  
Coordinate Geometry

Scalar Plane Geometry .— With O as a centre describe a circle with a radius equal to the unit of length. Let OA, OB be any two of its unit radii, termed ‘coordinate axes.’ From any point P in the plane AOB draw PM parallel to BO, so as to cut OA, produced either way if necessary, in M. Then there will exist some ‘scalars’ (‘real’ or ‘possible quantities’) u, v such that OM = u . OA, and Mp = v . OB, all lines being considered in respect both to magnitude and direction. Hence OP, which is the ‘appense’ or ‘geometrical sum’ of OM and MP, or = OM + MP, will = u . OA + v . OB. By varying the values of the 'coordinate scalars’ u, v P may be made to assume any position whatever on the plane of AOB. The angle AOB may be taken at pleasure, but greater symmetry is secured by choosing OI and OJ as coordinate axes, where IOJ is a right angle described in the right-handed direction. If any number of lines OP, OQ, OR, &c., be thus represented, the lengths of the lines PQ, QR, &c., and the sines and cosines of the angles IOP, POQ, QOR, &c., can be immediately furnished in terms of the unit of length and the coordinate scalars. If OP = x . OI + y . OJ, and any relation be assigned between the values of x and y , such as y = fx or ϕ ( x, y ) = 0 , then the possible positions of P are limited to those in which for any scalar value of x there exists a corresponding scalar value of y . The ensemble of all such positions of P constitutes the ‘ locus ’ of the two equations, viz. the ‘concrete equation’ OP = x . Ol + y .OJ, and the ‘abstract equation’ y = f. x. The peculiarity of the present theory consists in the recognition of these two equations to a curve, of which the ordinary theory only furnishes the latter, and inefficiently replaces the former by some convention respecting the use of the letters, whereby the coordinates themselves are not made a part of the calculation.

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.

Circular Graphs: Vehicles for Conic and Polar Connections

1995 ◽  
Vol 88 (1) ◽  
pp. 26-28
Author(s):  
Yvelyne Germain-McCarthy
Keyword(s):  
Trigonometric Functions ◽  
Conic Sections ◽  
Polar Plane ◽  
Circular Functions ◽  
Coordinate Geometry

A unified treatment of conic sections and polar equations of conics can be found in most calculus books where the reciprocals of limafçons are shown to be conic sections. The treatment, however, is from an algebraic standpoint and does not refer to the inherent connection between polar graphs and the graphs of trigonometric functions and conics. Beginning with information gained from the graphs of circular functions of the form y = A + B sin x, students can be guided to graph conic sections on the polar plane without using a table of values. This approach helps students to appreciate the roles that both algebra and coordinate geometry play in weaving various sections of mathematics into a meaningful whole.

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.

On proving certain optimisation theorems in plane geometry

2013 ◽  
Vol 97 (538) ◽  
pp. 75-80 ◽  
Author(s):  
I. Grattan-Guinness
Keyword(s):  
Plane Geometry ◽  
Integral Calculus ◽  
Spatial Geometry ◽  
Coordinate Geometry

A pleasurable aspect of mathematics and its teaching is to review the diversity of ways in which theorems are proved. Especially in elementary branches, there are various kinds of proof: using (or avoiding) spatial geometry, analytic or coordinate geometry, common algebra, vectors, abstract algebras, matrices, determinants, the differential and integral calculus, and maybe mixtures thereof. Further, sometimes a proof of one kind is elegant while another is clumsy, or one proof of a theorem suggests why it follows while another proof is not perspicuous. There is also the question of whether a proof is direct or indirect (for example, proofby contradiction).

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