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Distinct distances between points and lines
Distinct distances between points and lines

Chapter 5: Poincare Models of Hyperbolic Geometry
Chapter 5: Poincare Models of Hyperbolic Geometry

... composition of two isometries. Note that M is first sent to O and then to ∞ by inversion. Thus, the image of Γ is a (Euclidean) line. Since the center of the circle is on the real axis, the circle intersects the axis at right angles. Since inversion preserves angles, the image of Γ is a vertical (Eu ...
Lines that intersect Circles
Lines that intersect Circles

... lie on a circle. Diameter: -a chord that contains the center -connects two points on the circle and passes through the center Secant: line that intersects a circle at two points ...
geometryylp1011 - MATH-at
geometryylp1011 - MATH-at

... distance from the center of the base to the common vertex where all lateral faces meet. G.G.15 Apply the properties of a right circular cone  “Slant height” refers to the distance along a G.G.16 Apply the properties of a sphere lateral face from the base to the common vertex where all lateral faces ...
NYS Mathematics Glossary* – Geometry
NYS Mathematics Glossary* – Geometry

A geometric view of complex trigonometric functions
A geometric view of complex trigonometric functions

Fundamentals 2
Fundamentals 2

MGS43 Geometry 3 Fall Curriculum Map
MGS43 Geometry 3 Fall Curriculum Map

... Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections)   Note: Use proper function notation.  Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections  Ide ...
MATH 120-04 - CSUSB Math Department
MATH 120-04 - CSUSB Math Department

SYNTHETIC PROJECTIVE GEOMETRY
SYNTHETIC PROJECTIVE GEOMETRY

... However, it is clear that (a) and (a∗ ) are rephrasings of (1∗ ) and (1) respectively, and likewise (b) and (b∗ ) are rephrasings of (2∗ ) and (2) respectively. Thus (1) − (1 ∗ ) and (2) − (2∗ ) for (P ∗ , P ∗∗ ) are logically equivalent to (1) − (1 ∗ ) and (2) − (2∗ ) for (P, P ∗ ). As indicated a ...
Geometry - BAschools.org
Geometry - BAschools.org

Co-incidence Problems and Methods
Co-incidence Problems and Methods

Geometric Relationship Sample Tasks with Solutions
Geometric Relationship Sample Tasks with Solutions

1 Lecture 7 THE POINCARÉ DISK MODEL OF HYPERBOLIC
1 Lecture 7 THE POINCARÉ DISK MODEL OF HYPERBOLIC

The SMSG Axioms for Euclidean Geometry
The SMSG Axioms for Euclidean Geometry

The SMSG Axioms for Euclidean Geometry
The SMSG Axioms for Euclidean Geometry

Reason Sheet Chapter 3
Reason Sheet Chapter 3

128 [Mar., A SET OF AXIOMS FOR LINE GEOMETRY* 1
128 [Mar., A SET OF AXIOMS FOR LINE GEOMETRY* 1

geometric congruence
geometric congruence

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A Simple Non-Desarguesian Plane Geometry

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Nikolai Lobachevsky (1792-1856)

Euclidean Geometry - UH - Department of Mathematics
Euclidean Geometry - UH - Department of Mathematics

... Axiom 5 introduces the third undefined term (plane), along with its relationship to points. The term “non-collinear” means “not lying on the same line.” Since there are at least 3 non-collinear points to a plane, a plane is much different from a line (see A1). Exercise: Suppose you take 3 non-collin ...
Sharing Joints, in Moderation A Grounshaking Clash between
Sharing Joints, in Moderation A Grounshaking Clash between

7 Foundations Practice Exam
7 Foundations Practice Exam

... Unit 1 Practice Exam: Foundations of Geometry Be sure to draw sketches and show work where needed in order to receive full credit. Use the diagram to the right for questions 1 – 3. Please separate items in a list with commas. 1. (2 pts.) List three collinear points. ...
Unit 2.1b
Unit 2.1b

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Conic section



In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, and as a quadric of dimension 1. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a non-circular conic consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity.Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well.The conic sections have been named and studied at least since 200 BC, when Apollonius of Perga undertook a systematic study of their properties.
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