பல்கோணம்: திருத்தங்களுக்கு இடையிலான வேறுபாடு

உள்ளடக்கம் நீக்கப்பட்டது உள்ளடக்கம் சேர்க்கப்பட்டது
சி SieBotஆல் செய்யப்பட்ட கடைசித் தொகுப்புக்கு முன்நிலையாக்கப்பட்டது
வரிசை 61:
 
We will assume [[Euclidean geometry]] throughout.
 
An ''n''-gon has 2''n'' [[degrees of freedom]], including 2 for position and 1 for rotational orientation, and 1 for over-all size, so 2''n''-4 for [[shape]].
 
In the case of a line of symmetry the latter reduces to ''n''-2.
 
Let ''k''≥2. For an ''nk''-gon with ''k''-fold rotational symmetry (''C<sub>k</sub>''), there are 2''n''-2 degrees of freedom for the shape. With additional mirror-image symmetry (''D<sub>k</sub>'') there are ''n''-1 degrees of freedom.
 
=== கோணம் ===
ஒரு பல்கோணம், அது ஒழுங்கானதாயினும், ஒழுங்கற்றதாயினும், சிக்கலானதாயினும், எளிமையானதாயினும், அதன் பக்கங்களின் எண்ணிக்கையளவு கோணங்களைக் கொண்டிருக்கும். ''n'' பக்கங்களைக் கொண்ட ஒரு பல்கோணத்தின் உட்கோணங்களின் கூட்டுத்தொகை (''n''−2)[[Pi|π]] [[ஆரையன்|ஆரையன்கள்]] (அல்லது (''n''−2)180°), அத்துடன் ஒரு ஒழுங்கான பல்கோணத்தின் ஒரு உட்கோணத்தின் அளவு (''n''−2)π/''n'' ஆரையன்கள் (அல்லது (''n''−2)180°/''n'', அல்லது (''n''−2)/(2''n'') [[Turn (geometry)|turns. This can be seen in two different ways:
* Moving around a simple ''n''-gon (like a car on a road), the amount one "turns" at a vertex is 180° minus the inner angle. "Driving around" the polygon, one makes one full turn, so the sum of these turns must be 360°, from which the formula follows easily. The reasoning also applies if some inner angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the [[winding number]] of the orientation of the sides, where at every vertex the contribution is between -1/2 and 1/2 winding.)
* Any simple ''n''-gon can be considered to be made up of (''n''−2) triangles, each of which has an angle sum of π radians or 180°.
 
Moving around an n-gon in general, the total amount one "turns" at the vertices can be any integer times 360°, e.g. 720° for a [[pentagram]] and 0° for an angular "eight". See also [[orbit (dynamics)]].
 
=== பரப்பளவு ===
 
[[படிமம்:Apothem of hexagon.svg|thumb|right|Apothem of a [[hexagon]]]]
Several formulae give the area of a regular polygon:
:<math>A=\frac{nt^2}{4\tan(180^\circ/n)}</math>
: half the perimeter multiplied by the length of the apothem (the line drawn from the centre of the polygon perpendicular to a side)
 
The [[area (geometry)|area]] ''A'' of a simple polygon can be computed if the [[cartesian coordinate system|cartesian coordinates]] (''x''<sub>1</sub>, ''y''<sub>1</sub>), (''x''<sub>2</sub>, ''y''<sub>2</sub>), ..., (''x''<sub>''n''</sub>, ''y''<sub>''n''</sub>) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is
:''A'' = ½ · (''x''<sub>1</sub>''y''<sub>2</sub> − ''x''<sub>2</sub>''y''<sub>1</sub> + ''x''<sub>2</sub>''y''<sub>3</sub> − ''x''<sub>3</sub>''y''<sub>2</sub> + ... + ''x''<sub>''n''</sub>''y''<sub>1</sub> − ''x''<sub>1</sub>''y''<sub>''n''</sub>)
:&nbsp;&nbsp;= ½ · (''x''<sub>1</sub>(''y''<sub>2</sub> − ''y''<sub>''n''</sub>) + ''x''<sub>2</sub>(''y''<sub>3</sub> − ''y''<sub>1</sub>) + ''x''<sub>3</sub>(''y''<sub>4</sub> − ''y''<sub>2</sub>) + ... + ''x''<sub>''n''</sub>(''y''<sub>1</sub> − ''y''<sub>''n''−1</sub>))
The formula was described by Meister in 1769 and by [[Carl Friedrich Gauss|Gauss]] in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of [[Green's theorem]].
 
If the polygon can be drawn on an equally-spaced grid such that all its vertices are grid points, [[Pick's theorem]] gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.
 
If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the [[Bolyai-Gerwien theorem]].
 
=== அமைப்பு ===
 
All regular polygons are concyclic, as are all triangles and rectangles (see [[circumcircle]]).
 
A regular ''n''-sided polygon can be constructed with [[Ruler-and-compass construction|ruler and compass]] if and only if the [[odd number|odd]] [[prime number|prime]] factors of ''n'' are distinct [[Fermat prime]]s. See [[constructible polygon]].
 
== Point in polygon test ==
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