He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere. Apollonius of Perga (c. 262–190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone. He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").
historian of mathematicsmathematicshistory
Cézanne advanced the theory that all images can be built up from the sphere, the cone, and the cylinder. This is still used in art theory today, although the exact list of shapes varies from author to author. Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design. Applications of geometry to architecture include the use of projective geometry to create forced perspective, the use of conic sections in constructing domes and similar objects, the use of tessellations, and the use of symmetry.
Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hipparchus of Nicaea (2nd century BC), and the beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.
In analogy with the conic sections, the set of points whose cartesian coordinates satisfy the general equation of the second degree, namely, :where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero is called a quadric surface. There are six types of non-degenerate quadric surfaces: The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane and all the lines of ℝ 3 through that conic that are normal to ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.
For example, a conic section is degenerate if and only if it has singular points (e.g., point, line, intersecting lines ). A degenerate conic is a conic section (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. * A degenerate triangle has collinear vertices and zero area, and thus coincides with a segment covered twice (if the three vertices are not all equal; otherwise, the triangle degenerates to a single point). If the three vertices are pairwise distinct, it has two 0° angles and one 180° angle. If two vertices are equal, it has one 0° angle and two undefined angles. where and are constant (with for all i).
Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, and parabolas. Among the most common 3-dimensional shapes are polyhedra, which are shapes with flat faces; ellipsoids, which are egg-shaped or sphere-shaped objects; cylinders; and cones. If an object falls into one of these categories exactly or even approximately, we can use it to describe the shape of the object. Thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk.
Conic section. Focus. Circle. List of circle topics. Thales' theorem. Circumcircle. Concyclic. Incircle and excircles of a triangle. Orthocentric system. Monge's theorem. Power center. Nine-point circle. Circle points segments proof. Mrs. Miniver's problem. Isoperimetric theorem. Annulus. Ptolemaios' theorem. Steiner chain. Eccentricity. Ellipse. Semi-major axis. Hyperbola. Parabola. Matrix representation of conic sections. Dandelin spheres. Curve of constant width. Reuleaux triangle. Frieze group. Golden angle. Holditch's theorem. Interactive geometry software. Parallel postulate. Polygon. Star polygon. Pick's theorem. Shape dissection. Bolyai–Gerwien theorem. Poncelet–Steiner theorem.
The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T. All circles are similar. A circle's circumference and radius are proportional. The area enclosed and the square of its radius are proportional. The constants of proportionality are 2 and, respectively. The circle that is centred at the origin with radius 1 is called the unit circle. Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
analytical geometryCartesian geometrycoordinate geometry
John Casey (1885) Analytic Geometry of the Point, Line, Circle, and Conic Sections, link from Internet Archive. John Casey (1885) Analytic Geometry of the Point, Line, Circle, and Conic Sections, link from Internet Archive. John Casey (1885) Analytic Geometry of the Point, Line, Circle, and Conic Sections, link from Internet Archive.
In the treatise by this name, written c. 225 BCE, Archimedes obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a sphere by exploiting the relationship between a sphere and its circumscribed right circular cylinder of the same height and diameter. The sphere has a volume two-thirds that of the circumscribed cylinder and a surface area two-thirds that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius r is 4⁄3r 3 = 2⁄3 (2r 3 ) . The surface area of this sphere is 4r 2 = 2⁄3 (6r 2 ) .
Symmetric n \times n matrices of real functions appear as the Hessians of twice continuously differentiable functions of n real variables. Every quadratic form q on can be uniquely written in the form with a symmetric n \times n matrix A. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of, "looks like" : with real numbers \lambda_i. This considerably simplifies the study of quadratic forms, as well as the study of the level sets which are generalizations of conic sections.
The centers for the remaining vertices are found by symmetry. With help of a French curve one draws a curve, which has smooth contact to the osculating circles. The following method to construct single points of an ellipse relies on the Steiner generation of a conic section: : Given two pencils of lines at two points U,\, V (all lines containing U and V, respectively) and a projective but not perspective mapping \pi of B(U) onto B(V), then the intersection points of corresponding lines form a non-degenerate projective conic section. For the generation of points of the ellipse one uses the pencils at the vertices V_1,\, V_2. Let be an upper co-vertex of the ellipse and.
Archimedes of SyracuseArchimedeanArchimedes Heat Ray
In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids. On Floating Bodies (two volumes). In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape. The Quadrature of the Parabola.
ApolloniusApollonius of PergeApollonian
Conjugate diameters (Greek suzugeis diametroi, where suzugeis is “yoked together”), however, are symmetric in two dimensions. The figures to which they apply require also an areal center (Greek kentron), today called a centroid, serving as a center of symmetry in two directions. These figures are the circle, ellipse, and two-branched hyperbola. There is only one centroid, which must not be confused with the foci. A diameter is a chord passing through the centroid, which always bisects it.
locuslocilocus of points
All conic sections are loci:. Parabola: the set of points equidistant from a fixed point (the focus) and a line (the directrix). Circle: the set of points for which the distance from a single point is constant (the radius). The set of points for each of which the ratio of the distances to two given foci is a positive constant (that is, not 1) is referred to as a circle of Apollonius. Hyperbola: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant. Ellipse: the set of points for each of which the sum of the distances to two given foci is a constant.
In the Cartesian coordinate system, an elliptic cone is the locus of an equation of the form It is an affine image of the right-circular unit cone with equation From the fact, that the affine image of a conic section is a conic section of the same type (ellipse, parabola,...) one gets: *Any plane section of an elliptic cone is a conic section. Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see circular section). In projective geometry, a cylinder is simply a cone whose apex is at infinity.
metricmetric spacesmetric geometry
No. 1 (2002), 1–37. * * Far and near — several examples of distance functions at cut-the-knot. the distance from a point to itself is zero. the distance between two distinct points is positive. the distance from A to B is the same as the distance from B to A, and. the distance from A to B (directly) is less than or equal to the distance from A to B via any third point C. 1. || || identity of indiscernibles. 2. || || symmetry. 3. || || subadditivity or triangle inequality. }. 2. || || symmetry. 3. || || subadditivity or triangle inequality. }. style="width:250px"|. by triangle inequality. by symmetry. by identity of indiscernibles. we have non-negativity. }. by symmetry. by identity of
projective planeFlat projective planeprojective manifolds
This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in R 3. The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map. It follows that the fundamental group of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2. One can take the loop AB from the figure above to be the generator. Because the sphere covers the real projective plane twice, the plane may be represented as a closed hemisphere around whose rim opposite points are similarly identified.
semi-major axissemimajor axissemi-major axes
The semi-minor axis (more properly, minor semi-axis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle. The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e and the semi-latus rectum \ell, as follows: The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either vertex of the hyperbola.
*Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon. The converse is also true: the Braikenridge–Maclaurin theorem states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic, which may be degenerate as in Pappus's hexagon theorem.
The plane may be given a spherical geometry by using the stereographic projection. This can be thought of as placing a sphere on the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point). This is one of the projections that may be used in making a flat map of part of the Earth's surface. The resulting geometry has constant positive curvature. Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the hyperbolic plane.
In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean planes passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry. Given a line and any point A on it, we may consider A as decomposing this line into two parts. Each such part is called a ray (or half-line) and the point A is called its initial point.
M.C. EscherEscherMaurits Cornelis Escher
These are the classification of regular tilings using the edge relationships of tiles: two-color and two-motif tilings (counterchange symmetry or antisymmetry); color symmetry (in crystallography); metamorphosis or topological change; covering surfaces with symmetric patterns; Escher's algorithm (for generating patterns using decorated squares); creating tile shapes; local versus global definitions of regularity; symmetry of a tiling induced by the symmetry of a tile; orderliness not induced by symmetry groups; the filling of the central void in Escher's lithograph Print Gallery by H. Lenstra and B. de Smit.
Gauss curvaturecurvatureLiebmann's theorem
On the other hand, since a sphere of radius R has constant positive curvature R −2 and a flat plane has constant curvature 0, these two surfaces are not isometric, even locally. Thus any planar representation of even a part of a sphere must distort the distances. Therefore, no cartographic projection is perfect. The Gauss–Bonnet theorem links the total curvature of a surface to its Euler characteristic and provides an important link between local geometric properties and global topological properties. C 2 ) closed surfaces in R 3 with constant positive Gaussian curvature are spheres. If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid.
The axes of symmetry or principal axes are the transverse axis (containing the segment of length 2a with endpoints at the vertices) and the conjugate axis (containing the segment of length 2b perpendicular to the transverse axis and with midpoint at the hyperbola's center). As opposed to an ellipse, a hyperbola has only two vertices:. The two points on the conjugate axes are not on the hyperbola. It follows from the equation that the hyperbola is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin.