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.

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.

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.

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

Spacetime Geometry

2018 ◽  
Author(s):  
David M. Wittman
Keyword(s):  
Coordinate System ◽  
Special Relativity ◽  
Displacement Vector ◽  
Proper Time ◽  
Plane Geometry ◽  
Spatial Displacement ◽  
Spacetime Geometry ◽  
Space And Time ◽  
Displacement Vectors

This chapter shows that the counterintuitive aspects of special relativity are due to the geometry of spacetime. We begin by showing, in the familiar context of plane geometry, how a metric equation separates frame‐dependent quantities from invariant ones. The components of a displacement vector depend on the coordinate system you choose, but its magnitude (the distance between two points, which is more physically meaningful) is invariant. Similarly, space and time components of a spacetime displacement are frame‐dependent, but the magnitude (proper time) is invariant and more physically meaningful. In plane geometry displacements in both x and y contribute positively to the distance, but in spacetime geometry the spatial displacement contributes negatively to the proper time. This is the source of counterintuitive aspects of special relativity. We develop spacetime intuition by practicing with a graphic stretching‐triangle representation of spacetime displacement vectors.

Three-point perspective and plane geometry

2007 ◽  
Vol 1 (4) ◽  
pp. 213-223 ◽  
Author(s):  
Marc Frantz ◽  
Annalisa Crannell
Keyword(s):  
Plane Geometry

Static and dynamic response of sandwich beams with lattice and pantographic cores

2021 ◽  
pp. 109963622110338
Author(s):  
Yury Solyaev ◽  
Arseniy Babaytsev ◽  
Anastasia Ustenko ◽  
Andrey Ripetskiy ◽  
Alexander Volkov
Keyword(s):  
Mechanical Performance ◽  
Lattice Structure ◽  
Unit Length ◽  
Experimental Tests ◽  
Plane Geometry ◽  
Sandwich Beams ◽  
Static Tests ◽  
Three Point Bending ◽  
The Core ◽  
The Impact

Mechanical performance of 3d-printed polyamide sandwich beams with different type of the lattice cores is investigated. Four variants of the beams are considered, which differ in the type of connections between the elements in the lattice structure of the core. We consider the pantographic-type lattices formed by the two families of inclined beams placed with small offset and connected by stiff joints (variant 1), by hinges (variant 2) and made without joints (variant 3). The fourth type of the core has the standard plane geometry formed by the intersected beams lying in the same plane (variant 4). Experimental tests were performed for the localized indentation loading according to the three-point bending scheme with small span-to-thickness ratio. From the experiments we found that the plane geometry of variant 4 has the highest rigidity and the highest load bearing capacity in the static tests. However, other three variants of the pantographic-type cores (1–3) demonstrate the better performance under the impact loading. The impact strength of such structures are in 3.5–5 times higher than those one of variant 4 with almost the same mass per unit length. This result is validated by using numerical simulations and explained by the decrease of the stress concentration and the stress state triaxiality and also by the delocalization effects that arise in the pantographic-type cores.

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