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.
historian of mathematicsmathematicshistory
In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry, including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.
Shear mapping. 2D computer graphics. 2D geometric model. Altitude. Brahmagupta's formula. Bretschneider's formula. Compass and straightedge constructions. Squaring the circle. Complex geometry. 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.
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.
Solid geometry. Region (mathematics).
Archimedes of SyracuseArchimedeanArchimedes Heat Ray
Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola. Other mathematical achievements include deriving an accurate approximation of pi, defining and investigating the spiral bearing his name, and creating a system using exponentiation for expressing very large numbers.
infinitesimal calculusdifferential and integral calculusclassical calculus
ISBN: 0-8218-2830-4 Differential and Integral Calculus, American Mathematical Society. Robert A. Adams. (1999). ISBN: 978-0-201-39607-2 Calculus: A complete course. Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. (1986) Undergraduate Programs in the Mathematics and Computer Sciences: The 1985–1986 Survey, Mathematical Association of America No. 7. John Lane Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998. ISBN: 978-0-521-62401-5. Uses synthetic differential geometry and nilpotent infinitesimals. Florian Cajori, "The History of Notations of the Calculus." Annals of Mathematics, 2nd Ser., Vol. 25, No. 1 (Sep. 1923), pp. 1–46. Leonid P.
In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. 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.
plane geometryEuclideanEuclidean plane geometry
Apollonius of Perga (c. 262 BCE – c. 190 BCE) is mainly known for his investigation of conic sections. René Descartes (1596–1650) developed analytic geometry, an alternative method for formalizing geometry which focused on turning geometry into algebra. In this approach, a point on a plane is represented by its Cartesian (x, y) coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.
complex algebraic geometryalgebraiccomputational algebraic geometry
Menaechmus (circa 350 BC) considered the problem geometrically by intersecting the pair of plane conics ay = x 2 and xy = ab. The later work, in the 3rd century BC, of Archimedes and Apollonius studied more systematically problems on conic sections, and also involved the use of coordinates. The Muslim mathematicians were able to solve by purely algebraic means certain cubic equations, and then to interpret the results geometrically. This was done, for instance, by Ibn al-Haytham in the 10th century AD.
hyperbolic planehyperbolichyperbolic surface
The idea used a conic section or quadric to define a region, and used cross ratio to define a metric. The projective transformations that leave the conic section or quadric stable are the isometries. "Klein showed that if the Cayley absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..." For more history, see article on non-Euclidean geometry, and the references Coxeter and Milnor. The discovery of hyperbolic geometry had important philosophical consequences.
A Schwarz triangle is a spherical triangle that can be used to tile a sphere. It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry. A uniform tiling in the hyperbolic plane (which may be regular, quasiregular or semiregular) is an edge-to-edge filling of the hyperbolic plane, with regular polygons as faces; these are vertex-transitive (transitive on its vertices), and isogonal (there is an isometry mapping any vertex onto any other). A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells.
metricmetric spacesmetric geometry
Space (mathematics). Metric (mathematics). Metric signature. Metric tensor. Metric tree. Norm (mathematics). Normed vector space. Measure (mathematics). Hilbert space. Hilbert's fourth problem. Product metric. Aleksandrov–Rassias problem. Category of metric spaces. Classical Wiener space. Glossary of Riemannian and metric geometry. Isometry, contraction mapping and metric map. Lipschitz continuity. Triangle inequality. Ultrametric space. Victor Bryant, Metric Spaces: Iteration and Application, Cambridge University Press, 1985, ISBN: 0-521-31897-1. Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN: 0-8218-2129-6.
A square matrix A that is equal to its transpose, that is, A = A T, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −A T, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A ∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A. By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.
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.
EuclideanspaceEuclidean vector space
As for affine spaces, projective spaces are defined over any field, and are fundamental spaces of algebraic geometry. Non-Euclidean geometry refers usually to geometrical spaces where the parallel postulate is false. They include elliptic geometry, where the sum of the angles of a triangle is less than 180°, and hyperbolic geometry, where this sum is more than 180°.
They may, depending on the application, be called symmetric with respect to that transformation. For example, objects with translational symmetry are invariant under certain translations. The integral of the Gaussian curvature K of a 2-dimensional Riemannian manifold (M,g) is invariant under changes of the Riemannian metric g. This is the Gauss–Bonnet theorem.
In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, meaning that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space.
This formalism is used in geometric algebra and, more generally, in the Clifford algebra representation of Lie groups. In the case of a positive-definite Euclidean quadratic form, the double covering group of the isometry group is known as the Spin group,. It can be conveniently described in terms of a Clifford algebra. Unit quaternions give the group. In spherical geometry, a direct motion of the n-sphere (an example of the elliptic geometry) is the same as a rotation of (n + 1) -dimensional Euclidean space about the origin ( SO(n + 1) ).
Usually any such degeneracy indicates some underlying symmetry in the system. Degeneracy (graph theory). Degenerate form. Trivial (mathematics). Pathological (mathematics). Vacuous truth.
Is math invented or discovered? (millennia-old question, raised among others by Mario Livio). Pure mathematics. Applied mathematics. Theoretical physics. Mathematical physics. Non-Euclidean geometry. Fourier series. Conic section. Kepler's laws of planetary motion. Saving the phenomena. Positron#History. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematical universe hypothesis. Zeno's paradoxes. Axiomatic system. Mathematical model. Hilbert's sixth problem. Empiricism. Formalism (mathematics). Mathematics of general relativity. Bourbaki. Experimental mathematics. History of Maxwell's equations. Philosophy of mathematics#Platonism. History of astronomy.
Euclid of AlexandriaEuklidGreek Mathematician
Phaenomena, a treatise on spherical astronomy, survives in Greek; it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Conics was a work on conic sections that was later extended by Apollonius of Perga into his famous work on the subject. It is likely that the first four books of Apollonius's work come directly from Euclid. According to Pappus, "Apollonius, having completed Euclid's four books of conics and added four others, handed down eight volumes of conics." The Conics of Apollonius quickly supplanted the former work, and by the time of Pappus, Euclid's work was already lost.
In classical geometry, a surface is generally defined as a locus of a point or a line. For example, a sphere is the locus of a point which is at a given distance of a fixed point, called the center; a conical surface is the locus of a line passing through a fixed point and crossing a curve; a surface of revolution is the locus of a curve rotating around a line. A ruled surface is the locus of a moving line satisfying some constraints; in modern terminology, a ruled surface is a surface, which is a union of lines. In this article, several kinds of surfaces are considered and compared. An unambiguous terminology is thus necessary to distinguish them.
manifoldsboundarymanifold with boundary
Here is another example, applying this method to the construction of a sphere: A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of R 3 : : The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of R 2. Consider the northern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). The function χ defined by : maps the northern hemisphere to the open unit disc by projecting it on the (x, y) plane. A similar chart exists for the southern hemisphere.
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law.