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Concentric
In geometry, two or more objects are said to be ''concentric'' when they share the same center. Any pair of (possibly unalike) objects with well-defined centers can be concentric, including circles, spheres, regular polygons, regular polyhedra, parallelograms, cones, conic sections, and quadrics. Geometric objects are '' coaxial'' if they share the same axis (line of symmetry). Geometric objects with a well-defined axis include circles (any line through the center), spheres, cylinders, conic sections, and surfaces of revolution. Concentric objects are often part of the broad category of '' whorled patterns'', which also includes '' spirals'' (a curve which emanates from a point, moving farther away as it revolves around the point). Geometric properties In the Euclidean plane, two circles that are concentric necessarily have different radii from each other.. However, circles in three-dimensional space may be concentric, and have the same radius as each other, but nevert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coaxial
In geometry, coaxial means that several three-dimensional linear or planar forms share a common axis. The two-dimensional analog is ''concentric''. Common examples: A coaxial cable has a wire conductor in the centre (D), a circumferential outer conductor (B), and an insulating medium called the dielectric (C) separating these two conductors. The outer conductor is usually sheathed in a protective PVC outer jacket (A). All these have a common axis. The dimension and material of the conductors and insulation determine the cable's characteristic impedance and attenuation at various frequencies. Coaxial rotors are a three-dimensional planar structure: a pair of helicopter rotors (wings) mounted one above the other on concentric shafts, with the same axis of rotation (but turning in opposite directions). In loudspeaker design, coaxial speakers are a loudspeaker system in which the individual drivers are mounted close to one another on the same axis, and thus radiate sound along t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midsphere
In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every Edge (geometry), edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedron, uniform polyhedra, including the regular polyhedron, regular, Quasiregular polyhedron, quasiregular and Semiregular polyhedron, semiregular polyhedra and their Dual polyhedron, duals (Catalan solid, Catalan solids) all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere. When a polyhedron has a midsphere, one can form two perpendicular circle packing theorem, circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its dual polyhedron, polar polyhedron, which has the same midsphere. The length of each polyhedron edge is the sum of the distances from its two endpoints to their correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annulus (mathematics)
In mathematics, an annulus (: annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word ''anulus'' or ''annulus'' meaning 'little ring'. The adjectival form is ''annular'' (as in annular eclipse). The open annulus is topologically equivalent to both the open cylinder and the punctured plane. Area The area of an annulus is the difference in the areas of the larger circle of radius and the smaller one of radius : :A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right) = \pi (R+r)(R-r) . The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord tangent to the inner circle, in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so and are sides of a right-angled triangle with hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whorl
A whorl ( or ) is an individual circle, oval, volution or equivalent in a whorled pattern, which consists of a spiral or multiple concentric objects (including circles, ovals and arcs). In nature File:Photograph and axial plane floral diagram of Friesodielsia desmoides.jpg, Botanical whorls: sepals, petals, leaves, or branches radiating from a single point (photo of flower of Friesodielsia desmoides, family Annonaceae, juxtaposed with diagram of axial cross-section) File:Anisus septegyrus1pl.jpg, Mollusc whorls: Each complete 360° turn in the spiral growth of the shell of the mollusc Anisus septemgyratus, family Planorbidae. File:Baby hairy head DSCN2483.jpg, A hair whorl is a patch of hair growing in a circular direction around a visible center point. File:Fingerprint Whorl.jpg, In a fingerprint, a whorl is each ridge arranged circularly around a central point on the finger. File:Bovine Bone Sample and 430 times Magnification.jpg, In histopathologic architecture, a wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spiral
In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects. Two-dimensional A two-dimensional, or plane, spiral may be easily described using polar coordinates, where the radius r is a monotonic continuous function of angle \varphi: * r=r(\varphi)\; . The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant). In ''x-y-coordinates'' the curve has the parametric representation: * x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\; . Examples Some of the most important sorts of two-dimensional spirals include: * The Archimedean spiral: r=a \varphi * The hyperbolic spiral: r = a/ \varphi * Fermat's spiral: r= a\varphi^ * The lituus: r = a\varphi^ * The logarithmic spiral: r=ae^ * The Cornu spiral or ''clothoid'' * The Fibonacci spiral and golden spiral * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicentric Polygon
In geometry, a bicentric polygon is a tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triangles and all regular polygons are bicentric. On the other hand, a rectangle with unequal sides is not bicentric, because no circle can be tangent to all four sides. Triangles Every triangle is bicentric. In a triangle, the radii ''r'' and ''R'' of the incircle and circumcircle respectively are related by the equation :\frac+\frac=\frac where ''x'' is the distance between the centers of the circles.. This is one version of Euler's triangle formula. Bicentric quadrilaterals Not all quadrilaterals are bicentric (having both an incircle and a circumcircle). Given two circles (one within the other) with radii ''R'' and ''r'' where R>r, there exists a convex quadrilateral inscribed in one of them and tangent to the other if and only if th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apollonian Circles
In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned ancient Greek geometer. Definition The Apollonian circles are defined in two different ways by a line segment denoted . Each circle in the first family (the blue circles in the figure) is associated with a positive real number , and is defined as the locus of points such that the ratio of distances from to and to equals , \left\. For values of close to zero, the corresponding circle is close to , while for values of close to , the corresponding circle is close to ; for the intermediate value , the circle degenerates to a line, the perpendicular bisector of . The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the center (geometry), ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is spherical Earth, often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circles
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, 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, astronomy and calculus. Terminology * 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 centre allows for two arcs that together make up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumsphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle''. As in the case of two-dimensional circumscribed circles (circumcircles), the radius of a sphere circumscribed around a polyhedron is called the circumradius of , and the center point of this sphere is called the circumcenter of . Existence and optimality When it exists, a circumscribed sphere need not be the smallest sphere containing the polyhedron; for instance, the tetrahedron formed by a vertex of a cube and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator. However, the smallest sphere containing a given polyhedron is always the circumsphere of the convex hull of a subset of the vertices of the polyhedron.. In ''De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Shell
In geometry, a spherical shell (a ball shell) is a generalization of an annulus to three dimensions. It is the region of a ball between two concentric spheres of differing radii. Volume The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : \begin V &= \tfrac43\pi R^3 - \tfrac43\pi r^3 \\ mu &= \tfrac43\pi \bigl(R^3 - r^3\bigr) \\ mu &= \tfrac43\pi (R-r)\bigl(R^2 + Rr + r^2\bigr) \end where is the radius of the inner sphere and is the radius of the outer sphere. Approximation An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). The total surface area of the spherical shell is 4 \pi r^2. See also * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere A sphere (from Ancient Greek, Greek , ) is a sur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of distinct non- antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the sphere is called the ''minor arc'', and is the shortest surface-path between them. Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
