Ring

See also: Circle (homonymy)

The term of circle has several directions derived from its initial geometrical direction.

In its direction first, the circle is the “round”, the ideal figure to which one reduces the shape of many natural or artificial objects: sun, an eye, the circumference of a tree, a wheel.

For a long time, the language running employed this term as much to name the curve (Circonférence) that surface which it delimits. Nowadays, in Mathematical, the circle indicates the curve exclusively; surface being called disc .

Geometry

A circle is a plane Courbe consisted of the points located at equal distance from a point named center . The value of this distance is called radius circle. This one being infinitely variable, there thus exists an infinity of circles for an unspecified center, in each plan of space.

The circle is a ellipse whose hearths are confused in the center of the circle; the length of the main roads is equal to the length of the small axis. It is a Conique whose eccentricity $e$ is worth 0. It can be obtained by the intersection of a plan with a cone of revolution when the plan is perpendicular to the axis of revolution of the cone (one speaks sometimes about “cross-section” of the cone).

In an Euclidean space, it is about the round which is associated in French at the end of circle. In a nonEuclidean space or the case of the definition of a nonEuclidean distance, the form can be more complex.

In a plan provided with a orthonormé reference mark, the Circle unit or trigonometrical Cercle is the circle whose center is the origin of the reference mark, and whose ray is worth 1.

In Draftsmanship, a circle is generally represented with its horizontal axis and its vertical axis (in features of axis: feature fine compound of long and short indents), or simply with its center materialized by a right cross “+” in fine features. A form of revolution, full or digs (Cylindre, cone, Sphère) and seen according to the axis of revolution is represented by a circle.

Definitions

• a cord is a segment of right-hand side whose ends are on the circle.
• a arc is a portion of circle delimited by two points.
• a arrow is the segment connecting the mediums of an arc and a cord defined by two same points.
• a ray is a segment of right-hand side uniting the center at a point of the circle.
• a Diamètre is a cord passing by the center; it is a segment of right-hand side which delimits the disc in two equal shares. The diameter is composed of two rays colinéaires; its length is $2r$.

Geometrical properties of the circle

Here some geometrical properties of the circle.

Measurements

The Length of an arc underlain by an angle $\backslash \; alpha$, expressed in Radian S, is equal to $\backslash \; alpha\; r$. Thus, for an angle of $2\; \backslash \; pi$ (a full rotation), the perimeter (the circumference) of the circle is worth $2\; \backslash \; pi\; r$.

The length of a cord underlain by an angle $\backslash \; alpha$ is equal to $2r\; \backslash \; sin\; (\backslash \; alpha/2)$.

The surface of the disc delimited by a circle of radius $r$ is worth $\backslash \; pi\; r^2$; if one takes a cord length $l$ given and that one makes use of it to delimit a closed surface, surface having the largest surface is delimited by a circle.

According to the legend of the foundation of Carthage, the sovereign had allowed Phéniciens to found a city whose circumference would be delimited by a skin of Vache; Didon made a large thin strap of it and chooses a circular form to have more large surface.

Tangent

The tangent in a point of the circle is perpendicular to the ray in this point.

This property has applications in geometrical Optique: a luminous ray passing by the center of a spherical Miroir sets out again in opposite direction according to the same direction (there is a reflection perpendicular to the mirror). If one puts a bulb at the center of a spherical mirror, the light is returned other side, which makes it possible for example “to fold back” the light towards a parabolic mirror (principle of the against-mirror).

Mediator

One can show that the mediator of a cord passes by the center of the circle. This makes it possible to find the center of a circle: it is enough to trace two nonparallel cords and to seek the intersection of their mediating.

One can as show as the three mediating ones of a triangle are convergent and that the point of contest is the center of the circle passing by the three tops, called Cercle circumscribed with the Triangle.

Ring and right-angled triangle

Let us take three points of the circle $A$, $B$ and $C$, of which two - $A$ and $C$ - are diametrically opposite (i.e. are the intersections of the circle with a diameter). Then, $ABC$ is a right-angled triangle in $B$.

This rises owing to the fact that the median of the right angle is worth half of the hypotenuse (there is a ray and a diameter); this is a property of the triangle called in the Anglo-Saxon countries the theorem of Thalès.

Inscribed angle, angle in the center

See also: Theorem of the inscribed angle, Theorem of the inscribed angle and the angle in the center

Let us take two distinct points $A$ and $B$ circle. $O$ is the center of the circle and $C$ is another point of the circle. Then, one has

$\backslash \; widehat\; \{AOB\}\; =\; 2\; \backslash \; cdot\; \backslash \; widehat\; \{ACB\}$ For the angle in the center $\backslash \; widehat\; \{AOB\}$, it is necessary to consider the angular sector which intercepts the arc opposed to the arc containing $C$.

This property is used in the apparatuses of spectral analysis by dispersion of Wavelength, it is the concept of circle of focusing or circle of Rowland .

Report/ratio of the inscribed circles

• Ray $R\text{'}$ of the 2 larger circles inscribed in the circle of radius $R$ and surfaces $S$

  $R\text{'}\; =\; \backslash \; frac\; \{R\}\; \{2\}$
• Rayon $R\text{'}$ and surfaces $S\text{'}$ of the 3 larger circles inscrits$R\text{'}\; =\; \backslash \; frac\; \{R\}\; \{1+\; \backslash \; sqrt\; \{\backslash \; frac\; \{4\}\; \{3\}\}\}\; \backslash \; qquad\; 3S\text{'}\; =\; \backslash \; frac\; \{S\}\; \{\backslash \; left\; (\backslash \; frac\; \{2+\; \backslash \; sqrt\; \{3\}\}\; \{3\}\; \backslash \; right)\; ^2\}$
• Rayon $R\text{'}$ and surfaces $S\text{'}$ of the 4 larger circles inscrits$R\text{'}\; =\; \backslash \; frac\; \{R\}\; \{1+\; \backslash \; sqrt\{2\}\}\; \backslash \; qquad\; 4S\text{'}=\; \backslash \; frac\; \{S\}\; \{\backslash \; frac\; \{3+\; \backslash \; sqrt\; \{8\}\}\; \{4\}\}$
• Ray $R\text{'}$ of the 5 larger circles inscrits$R\text{'}\; =\; \backslash \; frac\; \{R\}\; \{1+\; \backslash \; sqrt\; \{2+\; \backslash \; sqrt\; \{\backslash \; frac\; \{4\}\; \{5\}\}\}\}$
• Ray $R\text{'}$ of the 7 larger inscribed circles (1 circle in the center surrounded by 6) $R\text{'}\; =\; \backslash \; frac\; \{R\}\; \{3\}$

Power of a point compared to a circle

If $M$ is a point and $\backslash \; Gamma$ is a circle of center $O$ and ray $R$, then, for any line passing by $M$ and meeting the circle in $A$ and $B$, one has

$MA\; \backslash \; times\; MB\; =\; |OM^2\; -\; R^2|$. This value does not depend the line chosen, but only on the position of $M$ compared to the circle.

One can notice that

• if $M$ is outside the circle, $MA\; \backslash \; times\; MB\; =\; OM^2\; -\; R^2$;
• if $M$ is inside the circle, $OM^2\; -\; R^2\; =\; -\; MY\; \backslash \; times\; MB$; ce product corresponds to the product of the algebraic measurements and.

One then calls power of the point $M$ compared to the circle $\backslash \; Gamma$ produces it algebraic measurements and. This product is independent the selected line and is worth always $OM^2\; -\; R^2$.

When the point $M$ is outside the circle, it is possible to lead tangents to the circle. By calling $T$ the contact point of one of these tangents, according to the Theorem of Pythagore in the triangle $OMT$, the power of $M$ is $MT^2$. Equality

$MA\; \backslash \; times\; MB\; =\; MT^2$ is sufficient to affirm that the line ($MT$) is tangent with the circle.

The power of a point makes it possible to check that four points are cocyclic: indeed, if

• $A$, $B$, $C$, $D$ is four points such as ($AB$) and ($CD$) is cut in $M$ and
• (in algebraic measurements),

then the four points are cocyclic.

Equations

In a plan provided with a orthonormé reference mark, the equation of the circle of center $C\; (has,\; b)$ and of ray $r$ is:

$(X\; -\; a)^2\; +\; (there\; -\; b)^2\; =\; r^2$ this equation is in fact an application of the Théorème of Pythagore for the right-angled Triangle formed by the point of the circle and its projection on the two rays parallel with the axes; the equation of the Cercle unit is thus $x^2\; +\; y^2\; =\; 1$. By highlighting $y$, one obtains the Cartesian equations of the circle: $y\; =\; B\; \backslash \; pm\; \backslash \; sqrt\; \{r^2\; -\; (x-a)\; ^2\}$.

The parametric equations of the circle are

$\backslash \; begin\; \{boxes\}\; x=a+r\; \backslash \; cos\; \backslash \; theta\; \backslash \; \backslash \; y=b+r\; \backslash \; sin\; \backslash \; theta\; \backslash \; end\; \{boxes\}$ maybe for the circle unit $\backslash \; begin\; \{boxes\}\; x=\; \backslash \; cos\; \backslash \; theta\; \backslash \; \backslash \; y=\; \backslash \; sin\; \backslash \; theta\; \backslash \; end\; \{boxes\}$

One can also determine an equation for the circle of diameter $$:

$(X\; -\; x\_A)\; (X\; -\; x\_B)\; +\; (there\; -\; y\_A)\; (there\; -\; y\_B)\; =\; 0$, that is to say still $x^2\; +\; y^2\; -\; (x\_A\; +\; x\_B)\; X\; -\; (y\_A\; +\; y\_B)\; there\; +\; x\_A\; x\_B\; +\; y\_A\; y\_B\; =\; 0$.

One can finally express the ray, the cord and the arrow according to two of them:

$C\; =\; 2\; \backslash \; sqrt\; \{F\; (2R\; -\; F)\}$ $R\; =\; \backslash \; frac\; \{4F^2+C^2\}\; \{8F\}$ $F\; =\; R\; -\; \backslash \; sqrt\; \{R^2\; -\; \backslash \; tfrac\; \{C^2\}\; 4\}$

See too

• Polar circle
• Sphere
• Algorithm of layout of arc of circle of chromatic Bresenham
• Circle
• Problem of Napoleon
• Vicious circle

Simple: Circle Zh-min-nan: Îⁿ-hêng

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Opening for an academic festival | Vespa 400 | JoWooD Productions | Chèvrerie (Haute-Savoie) | A.K. Antony | 1570

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