Concentric objects

Last updated

An archery target, featuring evenly spaced concentric circles that surround a "bullseye". WA 80 cm archery target.svg
An archery target, featuring evenly spaced concentric circles that surround a "bullseye".
Kepler's cosmological model formed by concentric spheres and regular polyhedra Kepler-solar-system-2.png
Kepler's cosmological model formed by concentric spheres and regular polyhedra

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. [1]

Contents

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, [2] 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. [3] However, circles in three-dimensional space may be concentric, and have the same radius as each other, but nevertheless be different circles. For example, two different meridians of a terrestrial globe are concentric with each other and with the globe of the earth (approximated as a sphere). More generally, every two great circles on a sphere are concentric with each other and with the sphere. [4]

By Euler's theorem in geometry on the distance between the circumcenter and incenter of a triangle, two concentric circles (with that distance being zero) are the circumcircle and incircle of a triangle if and only if the radius of one is twice the radius of the other, in which case the triangle is equilateral. [5] :p. 198

The circumcircle and the incircle of a regular n-gon, and the regular n-gon itself, are concentric. For the circumradius-to-inradius ratio for various n, see Bicentric polygon#Regular polygons. The same can be said of a regular polyhedron's insphere, midsphere and circumsphere.

The region of the plane between two concentric circles is an annulus, and analogously the region of space between two concentric spheres is a spherical shell. [6]

For a given point c in the plane, the set of all circles having c as their center forms a pencil of circles. Each two circles in the pencil are concentric, and have different radii. Every point in the plane, except for the shared center, belongs to exactly one of the circles in the pencil. Every two disjoint circles, and every hyperbolic pencil of circles, may be transformed into a set of concentric circles by a Möbius transformation. [7] [8]

Applications and examples

The ripples formed by dropping a small object into still water naturally form an expanding system of concentric circles. [9] Evenly spaced circles on the targets used in target archery [10] or similar sports provide another familiar example of concentric circles.

Coaxial cable is a type of electrical cable in which the combined neutral and earth core completely surrounds the live core(s) in system of concentric cylindrical shells. [11]

Johannes Kepler's Mysterium Cosmographicum envisioned a cosmological system formed by concentric regular polyhedra and spheres. [12]

Concentric circles have been used on firearms surfaces as means of holding lubrication or reducing friction on components, similar to jewelling. [13]

Concentric circles are also found in diopter sights, a type of mechanic sights commonly found on target rifles. They usually feature a large disk with a small-diameter hole near the shooter's eye, and a front globe sight (a circle contained inside another circle, called tunnel). When these sights are correctly aligned, the point of impact will be in the middle of the front sight circle.

See also

Related Research Articles

<span class="mw-page-title-main">Circle</span> Simple curve of Euclidean geometry

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.

In geometry, a polygon is a plane figure made up of line segments connected to form a closed polygonal chain.

<span class="mw-page-title-main">Triangle</span> Shape with three sides

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle. The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex; the shortest segment between base and apex is the height. The area of a triangle equals one-half the product of height and base length.

<span class="mw-page-title-main">Hexagon</span> Shape with six sides

In geometry, a hexagon is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

<span class="mw-page-title-main">Nine-point circle</span> Circle constructed from a triangle

In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:

<span class="mw-page-title-main">Equilateral triangle</span> Shape with three equal sides

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.

<span class="mw-page-title-main">Decagon</span> Shape with ten sides

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

<span class="mw-page-title-main">Concyclic points</span> Points on a common circle

In geometry, a set of points are said to be concyclic if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its circumscribing circle or circumcircle. All concyclic points are equidistant from the center of the circle.

<span class="mw-page-title-main">Circumscribed sphere</span> Sphere touching all of a polyhedrons vertices

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 P is called the circumradius of P, and the center point of this sphere is called the circumcenter of P.

<span class="mw-page-title-main">Pencil (geometry)</span> Family of geometric objects with a common property

In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane.

<span class="mw-page-title-main">Inscribed figure</span> Geometric figure which is "snugly enclosed" by another figure

In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. A polygon inscribed in a circle, ellipse, or polygon has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure. An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed figure gives another inscribed figure that is congruent to the original one.

<span class="mw-page-title-main">Centre (geometry)</span> Middle of the object in geometry

In geometry, a centre or center of an object is a point in some sense in the middle of the object. According to the specific definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of isometry groups, then a centre is a fixed point of all the isometries that move the object onto itself.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.

<span class="mw-page-title-main">Bicentric polygon</span>

In geometry, a bicentric polygon is a tangential polygon 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.

<span class="mw-page-title-main">Nine-point hyperbola</span> Hyperbola constructed from a given triangle and point

In Euclidean geometry with triangle ABC, the nine-point hyperbola is an instance of the nine-point conic described by American mathematician Maxime Bôcher in 1892. The celebrated nine-point circle is a separate instance of Bôcher's conic:

<span class="mw-page-title-main">Right kite</span> Symmetrical quadrilateral

In Euclidean geometry, a right kite is a kite that can be inscribed in a circle. That is, it is a kite with a circumcircle. Thus the right kite is a convex quadrilateral and has two opposite right angles. If there are exactly two right angles, each must be between sides of different lengths. All right kites are bicentric quadrilaterals, since all kites have an incircle. One of the diagonals divides the right kite into two right triangles and is also a diameter of the circumcircle.

<span class="mw-page-title-main">Circumgon</span> Geometric figure which circumscribes a circle

In mathematics and particularly in elementary geometry, a circumgon is a geometric figure which circumscribes some circle, in the sense that it is the union of the outer edges of non-overlapping triangles each of which has a vertex at the center of the circle and opposite side on a line that is tangent to the circle. The limiting case in which part or all of the circumgon is a circular arc is permitted. A circumgonal region is the union of those triangular regions.

<span class="mw-page-title-main">Modern triangle geometry</span>

In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and their properties were the subject of investigation since at least the time of Euclid. In fact, Euclid's Elements contains description of the four special points – centroid, incenter, circumcenter and orthocenter - associated with a triangle. Even though Pascal and Ceva in the seventeenth century, Euler in the eighteenth century and Feuerbach in the nineteenth century and many other mathematicians had made important discoveries regarding the properties of the triangle, it was the publication in 1873 of a paper by Emile Lemoine (1840–1912) with the title "On a remarkable point of the triangle" that was considered to have, according to Nathan Altschiller-Court, "laid the foundations...of the modern geometry of the triangle as a whole." The American Mathematical Monthly, in which much of Lemoine's work is published, declared that "To none of these [geometers] more than Émile-Michel-Hyacinthe Lemoine is due the honor of starting this movement of modern triangle geometry". The publication of this paper caused a remarkable upsurge of interest in investigating the properties of the triangle during the last quarter of the nineteenth century and the early years of the twentieth century. A hundred-page article on triangle geometry in Klein's Encyclopedia of Mathematical Sciences published in 1914 bears witness to this upsurge of interest in triangle geometry.

References

  1. Circles: Alexander, Daniel C.; Koeberlein, Geralyn M. (2009), Elementary Geometry for College Students, Cengage Learning, p. 279, ISBN   9781111788599

    Spheres: Apostol (2013)

    Regular polygons: Hardy, Godfrey Harold (1908), A Course of Pure Mathematics, The University Press, p. 107

    Regular polyhedra: Gillard, Robert D. (1987), Comprehensive Coordination Chemistry: Theory & background, Pergamon Press, pp.  137, 139, ISBN   9780080262321 .

  2. Spurk, Joseph; Aksel, Nuri (2008), Fluid Mechanics, Springer, p. 174, ISBN   9783540735366 .
  3. Cole, George M.; Harbin, Andrew L. (2009), Surveyor Reference Manual, www.ppi2pass.com, §2, p. 6, ISBN   9781591261742 .
  4. Morse, Jedidiah (1812), The American universal geography;: or, A view of the present state of all the kingdoms, states, and colonies in the known world, Volume 1 (6th ed.), Thomas & Andrews, p. 19.
  5. Dragutin Svrtan and Darko Veljan (2012), "Non-Euclidean versions of some classical triangle inequalities", forumgeom.fau.edu, Forum Geometricorum, pp. 197–209
  6. Apostol, Tom (2013), New Horizons in Geometry, Dolciani Mathematical Expositions, vol. 47, Mathematical Association of America, p. 140, ISBN   9780883853542 .
  7. Hahn, Liang-shin (1994), Complex Numbers and Geometry, MAA Spectrum, Cambridge University Press, p. 142, ISBN   9780883855102 .
  8. Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (2011), Geometry, Cambridge University Press, pp. 320–321, ISBN   9781139503709 .
  9. Fleming, Sir John Ambrose (1902), Waves and Ripples in Water, Air, and Æther: Being a Course of Christmas Lectures Delivered at the Royal Institution of Great Britain, Society for Promoting Christian Knowledge, p. 20.
  10. Haywood, Kathleen; Lewis, Catherine (2006), Archery: Steps to Success, Human Kinetics, p. xxiii, ISBN   9780736055420 .
  11. Weik, Martin (1997), Fiber Optics Standard Dictionary, Springer, p. 124, ISBN   9780412122415 .
  12. Meyer, Walter A. (2006), Geometry and Its Applications (2nd ed.), Academic Press, p. 436, ISBN   9780080478036 .
  13. Elliot, Dave (2018), "Behind Enemy Lines: Sterling Hayden's Registered Magnum", American Handgunner