Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system; the Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute metaphysical, sense.
Today, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, Euclidean space is a good approximation for it only over short distances. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects; this is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is a systematization of earlier knowledge of geometry, its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones, they are now nearly all lost. There are 13 books in the Elements: Books I–IV and VI discuss plane geometry.
Many results about plane figures are proved, for example "In any triangle two angles taken together in any manner are less than two right angles." and the Pythagorean theorem "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced, it is proved. Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base; the platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of simple axioms; until the advent of non-Euclidean geometry, these axioms were considered to be true in the physical world, so that all the theorems would be true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.
Near the beginning of the first book of the Elements, Euclid gives five postulates for plane geometry, stated in terms of constructions: Let the following be postulated:To draw a straight line from any point to any point. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance; that all right angles are equal to one another.: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique; the Elements include the following five "common notions": Things that are equal to the same thing are equal to one another. If equals are added to equals the wholes are equal. If equals are subtracted from equals the differences are equal.
Things that coincide with one another are equal to one another. The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than the others, they aspired to create a system of certain propositions, to them it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible, since one can construct consistent systems of geometry in which the parallel postulate is true, others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated. For example, Playfair's axiom states: In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the giv
Hyperrectangle
In geometry, an n-orthotope is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals. A three-dimensional orthotope is called a right rectangular prism, rectangular cuboid, or rectangular parallelepiped. A special case of an n-orthotope, where all edges are equal length, is the n-cube. By analogy, the term "hyperrectangle" or "box" refers to Cartesian products of orthogonal intervals of other kinds, such as ranges of keys in database theory or ranges of integers, rather than real numbers; the dual polytope of an n-orthotope has been variously called a rectangular n-orthoplex, rhombic n-fusil, or n-lozenge. It is constructed by 2n points located in the center of the orthotope rectangular faces. An n-fusil's Schläfli symbol can be represented by a sum of n orthogonal line segments: + +... +. A 1-fusil is a line segment. A 2-fusil is a rhombus, its plane cross selections in all pairs of axes are rhombi. Minimum bounding box Cuboid Coxeter, Harold Scott MacDonald.
Regular Polytopes. New York: Dover. Pp. 122–123. ISBN 0-486-61480-8. Weisstein, Eric W. "Rectangular parallelepiped". MathWorld. Weisstein, Eric W. "Orthotope". MathWorld
Circle
A circle is a simple closed shape. It is the set of all points in a plane; the distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior. A circle may be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations. A circle is a plane figure bounded by one line, such that all right lines drawn from a certain point within it to the bounding line, are equal; the bounding line is called the point, its centre. Annulus: a ring-shaped object, the region bounded by two concentric circles.
Arc: any connected part of a circle. Specifying two end points of an arc and a center allows for two arcs that together make up a full circle. Centre: the point equidistant from all points on the circle. Chord: a line segment whose endpoints lie on the circle, thus dividing a circle in two sements. Circumference: the length of one circuit along the circle, or the distance around the circle. Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; this is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, its length is twice the length of a radius. Disc: the region of the plane bounded by a circle. Lens: the region common to two overlapping discs. Passant: a coplanar straight line that has no point in common with the circle. Radius: a line segment joining the centre of a circle with any single point on the circle itself. Sector: a region bounded by two radii of equal length with a common center and either of the two possible arcs, determined by this center and the endpoints of the radii.
Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the center of the circle to which their arc belongs to. Secant: an extended chord, a coplanar straight line, intersecting a circle in two points. Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as center. In non-technical common usage it may mean the interior of the two dimensional region bounded by a diameter and one of its arcs, technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. Tangent: a coplanar straight line that has one single point in common with a circle. All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries; the word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, meaning "hoop" or "ring".
The origins of the words circus and circuit are related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand; the circle is the basis for the wheel, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Early science geometry and astrology and astronomy, was connected to the divine for most medieval scholars, many believed that there was something intrinsically "divine" or "perfect" that could be found in circles; some highlights in the history of the circle are: 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of π. 300 BCE – Book 3 of Euclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed explanation of the circle.
Plato explains the perfect circle, how it is different from any drawing, definition or explanation. 1880 CE – Lindemann proves that π is transcendental settling the millennia-old problem of squaring the circle. The ratio of a circle's circumference to its diameter is π, an irrational constant equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by: C = 2 π r = π d; as proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to π multiplied by the radius squared: A r e a = π r 2. Equivalently, denoting diameter by d, A r e
Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. In other words, there is only one plane that contains that triangle, every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; this article is about triangles in Euclidean geometry, in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has all sides the same length. An equilateral triangle is a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. An isosceles triangle has two angles of the same measure, namely the angles opposite to the two sides of the same length; some mathematicians define an isosceles triangle to have two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.
The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of short line segments in the form of tally marks. In a triangle, the pattern is no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, a scalene triangle has different patterns on all sides since no sides are equal. Patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, a scalene triangle has different patterns on all angles since no angles are equal.
Triangles can be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°; the side opposite to the right angle is the longest side of the triangle. The other two sides are called the catheti of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, 5 are a Pythagorean triple; the other one is an isosceles triangle. Triangles that do not have an angle measuring 90° are called oblique triangles. A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.
If c is the length of the longest side a2 + b2 > c2, where a and b are the lengths of the other sides. A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side a2 + b2 < c2, where a and b are the lengths of the other sides. A triangle with an interior angle of 180° is degenerate. A right degenerate triangle has collinear vertices. A triangle that has two angles with the same measure has two sides with the same length, therefore it is an isosceles triangle, it follows that in a triangle where all angles have the same measure, all three sides have the same length, such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures. In rigorous treatments, a triangle is therefore called a 2-simplex. Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC; the sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees.
This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle, a linear pair to an interior angle; the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it. The sum of the measures of the three exterior angles of any triangle is 360 degrees. Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle; the corresponding sides of similar triangles have lengths that are in the same proportion, this property is sufficient to establish similarity. Some basic theorems about similar triangles are: If and only if one pair of internal angles of two triangles have the sam
Polygon
In elementary geometry, a polygon is a plane figure, described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon; the segments of a polygonal circuit are called its edges or sides, the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides. A simple polygon is one. Mathematicians are concerned only with the bounding polygonal chains of simple polygons and they define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes; the word polygon derives from the Greek adjective πολύς "much", "many" and γωνία "corner" or "angle".
It has been suggested. Polygons are classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Convex: any line drawn through the polygon meets its boundary twice; as a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. Concave. Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge; the polygon must be simple, may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself.
The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon can not be both star-shaped. Equiangular: all corner angles are equal. Cyclic: all corners lie on a single circle, called the circumcircle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit; the polygon is cyclic and equiangular. Equilateral: all edges are of the same length; the polygon need not be convex. Tangential: all sides are tangent to an inscribed circle. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit; the polygon is equilateral and tangential. Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both equilateral, or both equilateral and equiangular. A non-convex regular polygon is called a regular star polygon. Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
Monotone with respect to a given line L: every line orthogonal to L intersects the polygon not more than twice. Euclidean geometry is assumed throughout. Any polygon has as many corners; each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees; this is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is 180 − 360 n degrees; the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular p q -gon, each interior angle is π p radians or 180 p degrees. Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°.
This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or starriness of the polygon. See orbit. In this section, the vertices of the polygon under consideration are taken to be, ( x 1
Simple polygon
In geometry a simple polygon is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pair-wise to form a closed path. If the sides intersect the polygon is not simple; the qualifier "simple" is omitted, with the above definition being understood to define a polygon in general. The definition given above ensures the following properties: A polygon encloses a region which always has a measurable area; the line segments that make-up a polygon meet only at their endpoints, called vertices or less formally "corners". Two edges meet at each vertex; the number of edges always equals the number of vertices. Two edges meeting at a corner are required to form an angle, not straight. Mathematicians use "polygon" to refer only to the shape made up by the line segments, not the enclosed region, however some may use "polygon" to refer to a plane figure, bounded by a closed path, composed of a finite sequence of straight line segments. According to the definition in use, this boundary may not form part of the polygon itself.
Simple polygons are called Jordan polygons, because the Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions, the region inside it and the region outside it. A polygon in the plane is only if it is topologically equivalent to a circle, its interior is topologically equivalent to a disk. If a collection of non-crossing line segments forms the boundary of a region of the plane, topologically equivalent to a disk this boundary is called a weakly simple polygon. In the image on the left, ABCDEFGHJKLM is a weakly simple polygon according to this definition, with the color blue marking the region for which it is the boundary; this type of weakly simple polygon can arise in computer graphics and CAD as a computer representation of polygonal regions with holes: for each hole a "cut" is created to connect it to an external boundary. Referring to the image above, ABCM is an external boundary of a planar region with a hole FGHJ; the cut ED connects the hole with the exterior and is traversed twice in the resulting weakly simple polygonal representation.
In an alternative and more general definition of weakly simple polygons, they are the limits of sequences of simple polygons of the same combinatorial type, with the convergence under the Fréchet distance. This formalizes the notion. However, this type of weakly simple polygon does not need to form the boundary of a region, as its "interior" can be empty. For example, referring to the image above, the polygonal chain ABCBA is a weakly simple polygon according to this definition: it may be viewed as the limit of "squeezing" of the polygon ABCFGHA. In computational geometry, several important computational tasks involve inputs in the form of a simple polygon. Point in polygon testing involves determining, for a simple polygon P and a query point q, whether q lies interior to P. Simple formulae are known for computing polygon area. Polygon partition is a set of primitive units, which do not overlap and whose union equals the polygon. A polygon partition problem is a problem of finding a partition, minimal in some sense, for example: a partition with a smallest number of units or with units of smallest total side-length.
A special case of polygon partition is Polygon triangulation: dividing a simple polygon into triangles. Although convex polygons are easy to triangulate, triangulating a general simple polygon is more difficult because we have to avoid adding edges that cross outside the polygon. Bernard Chazelle showed in 1991 that any simple polygon with n vertices can be triangulated in Θ time, optimal; the same algorithm may be used for determining whether a closed polygonal chain forms a simple polygon. Boolean operations on polygons: Various Boolean operations on the sets of points defined by polygonal regions; the convex hull of a simple polygon may be computed more efficiently than the convex hull of other types of inputs, such as the convex hull of a point set. Voronoi diagram of a simple polygon Medial axis/topological skeleton/straight skeleton of a simple polygon Offset curve of a simple polygon Minkowski sum for simple polygons Star domain Weisstein, Eric W. "Simple polygon". MathWorld