This is a list of some of the most commonly used coordinate transformations.
Daftar transformasi koordinat berikut memuat transformasi sistem-sistem koordinat yang paling umum digunakan.
x&=e^{\rho }\cos \theta ,\\y&=e^{\rho }\sin \theta .\end{aligned}}} By using complex numbers ( x , y )x + i y ′ , the transformation can be written as x + i ye ρ + i θ } That is, it is given by the complex exponential function.
x=e^{\rho }\cos \theta ,\\y=e^{\rho }\sin \theta .\end{cases}}} Dengan menggunakan bilangan kompleks ( x , y )x + i y ′ , transformasi dapat ditulis sebagai x + i ye ρ + i θ \,} yaitu diberikan oleh fungsi eksponensial kompleks.
To find θ , one must refer to the original Cartesian coordinate, determine the quadrant in which θ lies (for example, (3,−3)
Untuk menghitung θ , harus dirujuk pada koordinat Kartesius semua, tentukan kuadran di mana θ terletak (misalnya (3,-3)
This means that the inverse function will only give values in the domain of the function, but restricted to a single period.
Artinya fungsi invers hanya menghasilkan nilai dalam domain fungsi itu, tetapi terbatas pada satu periode saja.
Note that one can also use rx 2 + y 2 θ ′2 arctan ⁡ y x + r r&={\sqrt {x^{2}+y^{2}}}\\\theta '&=2\arctan {\frac {y}{x+r}}\end{aligned}}}
Perhatikan bahwa dapat pula digunakan rx 2 + y 2 +y^{2}}}} θ ′2 arctan ⁡ y x + r2\arctan {\frac {y}{x+r}}} { xe ρ cos ⁡ θ , ye ρ sin ⁡ θ .
rr 1 2 + r 2 2 − 2 c 2 2 θarctan ⁡ [ 8 c 2 ( r 1 2 + r 2 2 − 2 c 2 ) r 1 2 − r 2 2 − 1 ] r&={\sqrt {\frac {r_{1}^{2}+r_{2}^{2}-2c^{2}}{2}}}\\\theta &=\arctan \left[{\sqrt {{\frac {8c^{2}(r_{1}^{2}+r_{2}^{2}-2c^{2})}{r_{1}^{2}-r_{2}^{2}}}-1}}\right]\end{aligned}}} Where 2c is the distance between the poles.
xr 1 2 − r 2 2 4 c ^{2}-r_{2}^{2}}{4c}}} y± 1 4 c 16 c 2 r 1 2 − ( r 1 2 − r 2 2 + 4 c 2 ) 2 {4c}}{\sqrt {16c^{2}r_{1}^{2}-(r_{1}^{2}-r_{2}^{2}+4c^{2})^{2}}}} rr 1 2 + r 2 2 − 2 c 2 2 ^{2}+r_{2}^{2}-2c^{2}}{2}}}} θarctan ⁡ [ 8 c 2 ( r 1 2 + r 2 2 − 2 c 2 ) r 1 2 − r 2 2 − 1 ] (r_{1}^{2}+r_{2}^{2}-2c^{2})}{r_{1}^{2}-r_{2}^{2}}}-1}}\right]} di mana 2c adalah jarak antara kutub-kutub.
As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken.
Sebagaimana φ mempunyai rentang 360° pertimbangan yang sama dengan koordinat polar (2 dimensi) diterapkan bilamana diambil suatu arctangen.
θ has a range of 180°, running from 0° to 180°, and does not pose any problem when calculated from an arccosine, but beware for an arctangent.
θ mempunyai rentang 180°, dari 0° ke 180°, dan tidak bermasalah jika dihitung dari suatu arckosinus, tetapi perhatikan untuk suatu arctangen.
If, in the alternative definition, θ is chosen to run from −90° to +90°, in opposite direction of the earlier definition, it can be found uniquely from an arcsine, but beware of an arccotangent.
Jika, dalam definisi alternatif, θ dipilih untuk rentang dari −90° ke +90°, dengan arah yang berlawanan dibandingkan definisi sebelumnya, maka dapat dihitung secara unik dari suatu arcsinus, tetapi hati-hati dengan arckotangen.
d ρ d θ d φρ 2 sin ⁡ θ d ρ d θ d φ {\partial (\rho ,\theta ,\varphi )}}\,d\rho \,d\theta \,d\varphi\rho ^{2}\sin
d ρ d θ d ϕρ 2 sin ⁡ θ d ρ d θ d ϕ {\partial (\rho ,\theta ,\phi )}}d\rho \;d\theta \;d\phi\rho ^{2}\sin \theta \;d\rho \;d\theta \;d\phi \;} xr cos ⁡ θ{r}\,\cos \theta }
So for the element: d ρ d θ d φdet ∂ ( ρ , θ , φ ) ∂ ( x , y , z )
d ρ d θ d ϕdet ∂ ( ρ , θ , ϕ ) ∂ ( x , y , z )
d x d y d z1 x 2 + y 2 x 2 + y 2 + z 2 d x d y d z {\partial (x,y,z)}}\,dx\,dy\,dz={\frac {1}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\,dx\,dy\,dz} ρr 2 + h 2 θarctan ⁡ r h φφ ∂ ( ρ , θ , φ ) ∂ ( r , h , φ )
d x d y d z1 x 2 + y 2 x 2 + y 2 + z 2 d x d y d z {\partial (x,y,z)}}dx\ dy\ dz={\frac {1}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}dx\ dy\ dz} ρr 2 + h 2{\sqrt {r^{2}+h^{2}}}}
( x x 2 + y 2 y x 2 + y 2 0 − y x 2 + y 2 x x 2 + y 2 0 0 0 1 ) {\partial (x,y,z)}}={\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}&0\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}&0\\0&0&1\end{pmatrix}}} rρ sin ⁡ φ hρ cos ⁡ φ θθ ∂ ( r , h , θ ) ∂ ( ρ , φ , θ )( sin ⁡ φ ρ cos ⁡ φ 0 cos ⁡ φ − ρ sin ⁡ φ 0 0 0 1 ) det ∂ ( r , h , θ ) ∂ ( ρ , φ , θ )
( x x 2 + y 2 y x 2 + y 2 0 − y x 2 + y 2 x x 2 + y 2 0 0 0 1 ) {\partial (x,y,z)}}={\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}&0\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}&0\\0&0&1\end{pmatrix}}} rρ sin ⁡ ϕ θθ hρ cos ⁡ ϕ ∂ ( r , θ , h ) ∂ ( ρ , θ , ϕ )