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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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 ...
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 ...
... 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 ...
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 ...
... 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 ...
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 ...
... 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 ...
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 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. ...
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.