Poligon
Dari Wikipedia bahasa Melayu
Satu poligon (secara literal "banyak sudut", sila lihat Wiktionary untuk takrif yang lengkap) adalah satu ditutup menyatah jalan terdiri daripada sebilangan yang terhingga berurutan tembereng garisan. Garis lurus segmen-segmen yang membentuk poligon telah dipanggil nya tepi atau segi-segi dan titik-titik di mana tepi memenuhi adalah polygon's bucu-bucu. Andai satu poligon adalah mudah, kemudiannya tepinya (dan bucu-bucu) terdiri daripada sempadan satu rantau bersudut, dan tempoh poligon kadangkala juga menjelaskan pedalaman rantau bersudut (kawasan terbuka bahawa jalan ini memagari) atau kesatuan kedua-dua rantau dan sempadannya.
Jadual isi kandungan |
[Sunting] Nama-nama dan jenis-jenis
Poligon adalah dinamakan mengikut pada jumlah tepi, bergabung satu yunani-terbitan awalan angka dengan akhiran -gon. Contoh pentagon, dodekagon. Segi tiga, sisi empat, dan nonagon adalah pengecualian-pengecualian. Untuk nombor-nombor lebih besar, ahli matematik menulis angka sendiri, contoh 17-gon. Satu variabel boleh juga digunakan, biasanya n gon. Ini adalah jika jumlah berguna bagi tepi adalah digunakan dalam satu formula.
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| Name | Sides |
|---|---|
| henagon (or monogon) | 1 |
| digon | 2 |
| triangle (or trigon) | 3 |
| quadrilateral (or tetragon) | 4 |
| pentagon | 5 |
| heksagon (atau sexagon) | 6 |
| heptagon (avoid "septagon" = Latin [sept-] + Greek) | 7 |
| oktagon | 8 |
| nonagon (or enneagon) | 9 |
| dekagon | 10 |
| hendecagon (avoid "undecagon" = Latin [un-] + Greek) | 11 |
| dodecagon (avoid "duodecagon" = Latin [duo-] + Greek) | 12 |
| tridecagon or triskaidecagon (MathWorld) | 13 |
| tetradecagon or tetrakaidecagon interal angle approx 154.2857 degrees.(MathWorld) | 14 |
| pentadecagon (or quindecagon) or pentakaidecagon | 15 |
| hexadecagon or hexakaidecagon | 16 |
| heptadecagon or heptakaidecagon | 17 |
| octadecagon or octakaidecagon | 18 |
| enneadecagon or enneakaidecagon or nonadecagon | 19 |
| icosagon | 20 |
| triacontagon | 30 |
| tetracontagon | 40 |
| pentacontagon | 50 |
| hexacontagon (MathWorld) | 60 |
| heptacontagon | 70 |
| octacontagon | 80 |
| nonacontagon | 90 |
| hectagon (also hectogon) (avoid "centagon" = Latin [cent-] + Greek) | 100 |
| chiliagon | 1000 |
| myriagon | 10,000 |
| decemyriagon | 100,000 |
| hecatommyriagon (or hekatommyriagon) | 1,000,000 |
[Sunting] Penamaan poligon
To construct the name of a polygon with more than 20 and less than 100 sides, combine the prefixes as follows
| Tens | and | Ones | final suffix | ||
|---|---|---|---|---|---|
| -kai- | 1 | -hena- | -gon | ||
| 20 | icosa- | 2 | -di- | ||
| 30 | triaconta- | 3 | -tri- | ||
| 40 | tetraconta- | 4 | -tetra- | ||
| 50 | pentaconta- | 5 | -penta- | ||
| 60 | hexaconta- | 6 | -hexa- | ||
| 70 | heptaconta- | 7 | -hepta- | ||
| 80 | octaconta- | 8 | -octa- | ||
| 90 | enneaconta- | 9 | -ennea- | ||
That is, a 42-sided figure would be named as follows:
| Tens | and | Ones | final suffix | full polygon name |
|---|---|---|---|---|
| tetraconta- | -kai- | -di- | -gon | tetracontakaidigon |
and a 50-sided figure
| Tens | and | Ones | final suffix | full polygon name |
|---|---|---|---|---|
| pentaconta- | -gon | pentacontagon | ||
But beyond nonagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons).
[Sunting] Taxonomic classification
The taxonomic classification of polygons is illustrated by the following graph:
Polygon
/ \
Simple Complex
/ \ /
Convex Concave /
/ \ / /
Cyclic Equilateral
\ /
Regular
- A polygon is called simple if it is described by a single, non-intersecting boundary (hence has an inside and an outside); otherwise it is called complex.
- A simple polygon is called convex if it has no internal angles greater than 180°; otherwise it is called concave or non-convex.
- A simple polygon is called equilateral if all edges are of the same length. (A 5 or more sided polygon can be concave and equilateral)
- A convex polygon is called concyclic or a cyclic polygon if all the vertices lie on a single circle.
- A cyclic and equilateral polygon is called regular; for each number of sides, all regular polygons with the same number of sides are similar.
- A simple polygon may also be defined as regular if it is cyclic and equilateral.
[Sunting] Ciri-ciri
We will assume Euclidean geometry throughout.
An n-gon has 2n degrees of freedom, including 2 for position and 1 for rotational orientation, and 1 for over-all size, so 2n-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 (Ck), there are 2n-2 degrees of freedom for the shape. With additional mirror-image symmetry (Dk) there are n-1 degrees of freedom.
[Sunting] Sudut
Any polygon, regular or irregular, complex or simple, has as many angles as it has sides. The sum of the inner angles of a simple n-gon is (n−2)π radians (or (n−2)180°), and the inner angle of a regular n-gon is (n−2)π/n radians (or (n−2)180°/n, or (n−2)/(2n) 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).
[Sunting] Luas
The area A of a simple polygon can be computed if the cartesian coordinates (x1, y1), (x2, y2), ..., (xn, yn) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is
- A = ½ · (x1y2 − x2y1 + x2y3 − x3y2 + ... + xny1 − x1yn)
- = ½ · (x1(y2 − yn) + x2(y3 − y1) + x3(y4 − y2) + ... + xn(y1 − yn−1))
The formula was described by Meister in 1769 and by 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.
[Sunting] Concyclic
All regular polygons are concyclic, as are all triangles and rectangles (see circumcircle).
[Sunting] Titik dalam ujian polygon
In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x0,y0) lies inside a simple polygon given by a sequence of line segments. It is known as Point in polygon test.
[Sunting] Kotak istimewa
Some special cases are:
- Angle of 0° or 180° (degenerate case)
- Two non-adjacent sides are on the same line
- Equilateral polygon: a polygon whose sides are equal (Williams 1979, pp. 31-32)
- Equiangular polygon: a polygon whose vertex angles are equal (Williams 1979, p. 32)
A triangle is equilateral iff it is equiangular.
An equilateral quadrilateral is a rhombus, an equiangular quadrilateral is a rectangle or an "angular eight" with vertices on a rectangle.
[Sunting] Pautan luar
- Mathworld: Polygon
- Draw n Polygons Applet to draw a polygon with n vertices.
- [1] More information
[Sunting] Lihat juga
- Cyclic polygon
- Geometric shape
- Polygon triangulation
- Polyform
- Polyhedron
- Polytope
- Regular polygon
- Simple polygon
- Star polygon
- Synthetic geometry
- Tiling
- Tiling puzzle
- Golygon
