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.

Modelling spatial reasoning systems with shape algebras and formal logic

1997 ◽  
Vol 11 (4) ◽  
pp. 273-285 ◽  
Author(s):  
Scott C. Chase
Keyword(s):  
Spatial Reasoning ◽  
Predicate Logic ◽  
Spatial Relations ◽  
Large Set ◽  
First Order ◽  
Reasoning Systems ◽  
Geometric Objects ◽  
Small Set ◽  
Definition Of ◽  
High Level

AbstractThe combination of the paradigms of shape algebras and predicate logic representations, used in a new method for describing designs, is presented. First-order predicate logic provides a natural, intuitive way of representing shapes and spatial relations in the development of complete computer systems for reasoning about designs. Shape algebraic formalisms have advantages over more traditional representations of geometric objects. Here we illustrate the definition of a large set of high-level design relations from a small set of simple structures and spatial relations, with examples from the domains of geographic information systems and architecture.

scholarly journals ТЕОРИЈE РАЗВОЈA ГЕОМЕТРИЈСКОГ МИШЉЕЊА ПРЕМА ВАН ХИЛУ, ФИШБАЈНУ И УДЕМОН-КУЗНИАКУ

TEME ◽  
2017 ◽  
pp. 623 ◽  
Author(s):  
Оливера Ђокић ◽  
Маријана Зељић
Keyword(s):  
Theoretical Analysis ◽  
Mathematics Curriculum ◽  
Theoretical Framework ◽  
Geometric Object ◽  
Geometric Reasoning ◽  
Theoretical Frameworks ◽  
Teaching Geometry ◽  
Content Domain ◽  
Geometric Objects ◽  
Development Of Students

This research is a pedagogical study of theoretical frameworks of development of students’ geometrical thinking in various forms, particularly students’ geometric reasoning in teaching geometry: 1) model of van Hieles’ levels of understanding of geometry, 2) theory of figural concepts of Fischbein and 3) paradigms of Houdement-Kuzniak development of geometrical thinking. The aim of our research was to analyze the three theoretical framework and explain the reasons for their choice and expose them in terms of finding opportunities to permeate and connect them into one complete theory. The study used a descriptive-analytical and analytical-critical method of theoretical analysis. The results show that from each of the three theoretical frameworks we can clearly notice and distinguish geometric objects, as the students do not see them. They see them blended and structured in a series of procedures, and for that very reason we can say that they are poorly linked. We also opened questions for further research of geometric object as an important element for content domain geometry within mathematics curriculum.

On the Definition of Geometric Objects. I

1953 ◽  
Vol 56 ◽  
pp. 208-215 ◽  
Author(s):  
J. Haantjes ◽  
G. Laman
Keyword(s):  
Geometric Objects ◽  
Definition Of

On the Definition of Geometric Objects. II

1953 ◽  
Vol 56 ◽  
pp. 216-222 ◽  
Author(s):  
J. Haantjes ◽  
G. Laman
Keyword(s):  
Geometric Objects ◽  
Definition Of

and Spaces

2017 ◽  
Author(s):  
Matt Clay ◽  
Dan Margalit
Keyword(s):  
Group Theory ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Group Action ◽  
Metric Spaces ◽  
Geometric Group Theory ◽  
Geometric Object ◽  
Geometric Objects ◽  
Group Of Symmetries

This chapter discusses the notion of space, first by explaining what it means for a group to be a group of symmetries of a geometric object. This is the idea of group action, and some examples are given. The chapter proceeds by defining, for any group G, the Cayley graph of G and shows that the symmetric group of of this graph is precisely the group G. It then introduces metric spaces, which formalize the notion of a geometric object, and highlights numerous metric spaces that groups can act on. It also demonstrates that groups themselves are metric spaces; in other words, groups themselves can be thought of as geometric objects. The chapter concludes by using these ideas to frame the motivating questions of geometric group theory. Exercises relevant to each idea are included.

A Geometric Illustration of Limits

1918 ◽  
Vol 11 (1) ◽  
pp. 34-35
Author(s):  
Carl Eben Stromquist
Keyword(s):  
High School ◽  
High School Teachers ◽  
Random Selection ◽  
Plane Geometry ◽  
School Teachers ◽  
Freshman Mathematics ◽  
Definition Of ◽  
Selection Of

Teachers of freshman mathematics, the calculus, or review courses for high school teachers often experience some difficulty when the topic of limits comes up for discussion. The difficulty arises from the fact that the teacher must first “unteach” an erroneous definition of limits which is still to be found in many of our high school texts. The following definition, taken from a rather widely used plane geometry text, is typical of the error. “The limit of a variable is a constant from which the variable can be made to differ by less than any assigned quantity, but to which it can never be made equal.” In a random selection of ten texts five gave essentially this definition. One of these was published as recently as 1915. The mischief is caused by the last part of the statement, viz., “but to which it can never be made equal.”

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.

scholarly journals Approximation the geometric objects of multidimensional space using arcs of curves passing through the given points

2019 ◽  
Author(s):  
Евгений Конопацкий ◽  
Evgeniy Konopatskiy ◽  
Сергей Ротков ◽  
Sergey Rotkov
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Influence Factors ◽  
Geometric Object ◽  
Multidimensional Space ◽  
Global Coordinate System ◽  
Factors Affecting ◽  
Geometric Objects ◽  
Basic Ideas ◽  
The Given

The paper presents the basic ideas of geometric objects approximation in multidimensional space by means the arcs of algebraic curves passing through given points, which is as follows. A special network of points with a dimension one less than the dimension of the space in which the simulated geometric object is located is formed. Taking into account the special properties the arcs of algebraic curves passing through the given points, a linear relationship between the parameters of the geometric object and the influence factors corresponding to the axes of the global coordinate system is established. Next, the nodes of the network are calculated such values of the response function, which provide the minimum value of the quadratic residual function. The proposed method allows to perform the generalization the method of least squares in the direction of increasing space dimension and, consequently, the number of investigated factors affecting the response function, which is especially important for modeling and optimization of multifactorial processes and phenomena.

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