ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ

ᏔᎵ ᏗᎦᎾᏗᏫᏍᏗ ᎦᏃᎯᎵᏙ ᎭᏫᎾᏗᏢ ᏦᎢ-dimensional ᎤᏜᏅᏛ

ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᎦᏃᎯᎵᏙ ᎨᏒᎢ ᏄᎬᏫᏳᏒ ᏗᎳᏏᏙᏗ ᏔᎵ-dimensional ᏗᎦᏘᎴᏍᏗ. ᎤᎾᎵᎪᏒ, ᎾᏍᎩ ᎠᎾᏍᎬᏘ ᎾᏍᏋ visualized ᏥᏄᏍᏗ ᎤᏩᎾᏕᏍᎩ ᎢᎪᎯᏓ ᎨᏒ ᏗᎬᎯᏓ Ꭺ. ᎾᎿᎢ ᎠᎴ ᎯᎸᏍᎩ ᏩᏎᏍᏗ ᎾᎿ ᎯᎠ ᎦᏃᎯᎵᏙ, ᎢᏗᎦᏘᎭ ᎭᏫᎾᏗᏢ ᎯᎠ ᎤᏓᏅᏖᏗ Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᎠᎴ ᎦᏙ ᎤᏍᏗ ᏰᎵᏇ ᎾᏍᏋ ᎧᏁᏉᏛ ᎭᏫᎾᏗᏢ ᏄᏓᎴᎿᎥ ᏫᎦᎶᎯᏍᏗ ᎧᏁᎢᏍᏔᏅᎯ ᏓᎦᏘᎴᎦ ᎭᏫᎾᏗᏢ ᏐᎢ ᎡᏍᎦᏂ ᏗᏎᏍᏗ ᎤᎬᏩᎵ.

ᎭᏫᎾᏗᏢ ᎢᎦᏛ ᎡᏍᎦᏂ ᏗᏎᏍᏗ ᎤᎬᏩᎵ, ᏯᏛᎿ ᏥᏄᏍᏗ ᎦᏃᎯᎵᏙ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎠᎴ 2D ᎡᎵᏍ ᎠᏨᏍᏛ, ᎯᎠ ᏂᎦᏛ ᎤᏜᏅᏛ ᎭᏫᎾᏗᏢ ᎦᏙ ᎤᏍᏗ ᎯᎠ ᏗᎦᎸᏫᏍᏓᏁᏗ ᎨᏒᎢ ᎠᏫᏛᏓ atsinuqowisgv ᎨᏒᎢ ᏏᏴᏫ ᎦᏃᎯᎵᏙ. ᎭᏫᎾᏗᏢ ᏯᏛᎿ ᏄᏍᏗᏓᏅ ᎯᎠ ᎾᎿ ᎢᏴ ᎤᏓᏡᎬ ᎨᏒᎢ ᎬᏔᏅᎯ: ᎯᎠ ᎦᏃᎯᎵᏙ. ᎤᎪᏗᏗ ᏄᎬᏫᏳᏒ ᏗᎳᏏᏙᏗ ᎢᏯᏛᏁᏗ ᎠᎲ ᎭᏫᎾᏗᏢ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ, ᏓᏍᏓᏅᏅ ᏚᎷᏨ, ᎠᎴ graphing ᎠᎴ ᏄᏛᏁᎸ ᎭᏫᎾᏗᏢ ᎯᎠ ᏔᎵ dimensional ᎤᏜᏅᏛ, ᎠᎴ ᎭᏫᎾᏗᏢ ᏐᎢ ᎤᏂᏁᏨ, ᎭᏫᎾᏗᏢ ᎯᎠ ᎦᏃᎯᎵᏙ.

Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ

ᎦᏃᎯᎵᏙ ᎨᏒᎢ ᎦᏚᎢ ᏯᏛᎿ Ꮎ, ᎠᏓᏁᎸ ᏂᎦᎵᏍᏗᏍᎬᎫ ᏔᎵ ᎪᏍᏓᏱᎾᎿ ᎯᎠ ᎦᏚᎢ, ᎯᎠ ᎦᏚᎢ ᎾᏍᎩ ᎾᏍᏇ ᎢᎦᎢ ᎨᏐ ᎯᎠ ᎦᏥᏃᏍᏛ ᎠᏍᏓᏅᏅ Ꮎ ᏓᏂᎶᏍᎬᎢ ᏗᎬᏩᎶᏒ ᎯᎠ ᏔᎵ ᎠᏂ. ᏌᏊ ᏰᎵᏇ ᎬᏂᎨᏒ ᎢᏯᏓᏛᏁᏗ Cartesian ᎢᏗᎦᏘᎭ iyahdvnelidasdi ᎾᎿ ᎠᏓᏁᎸ ᎦᏃᎯᎵᏙ ᎭᏫᎾᏗᏢ ᎠᏓᏅᏍᏗ ᏚᏙᎥ ᎦᏯᎸ ᏂᎦᎥᎢ ᎪᏍᏓᏱ ᎾᎿ ᎾᏍᎩ ᏧᏓᎴᎿᎢ ᎬᏙᏗ ᏔᎵ ᏗᏎᏍᏗ, ᎯᎠ ᎪᏍᏓᏱ ᎤᏤᎵ ᎠᏟᎶᏍᏗ ᎦᏙᎯ.

ᎭᏫᏂᏗᏢ ᏂᎦᎵᏍᏗᏍᎬᎫ Euclidean ᎤᏜᏅᏛ, ᎦᏃᎯᎵᏙ ᎨᏒᎢ ᏧᏓᎴᎿᎢ ᏕᎫᎪᏔᏅ ᎾᎥᎢ ᏂᎦᎵᏍᏗᏍᎬᎫ ᎯᎠ ᎠᏍᏓᏩᏗᏒ ᎤᎾᏓᏟᏌᏅ:

• ᏦᎢ ᎬᏙᏗ-collinear ᏗᎪᏍᏓᏱ (ᎾᏍᎩ ᏂᎨᏒᎾ ᎠᏝᎥ ᎾᎿ ᎯᎠ ᎤᏠᏱ ᎠᏍᏓᏅᏅ)
• ᎠᏍᏓᏅᏅ ᎠᎴ ᎪᏍᏓᏱ ᎾᏍᎩ ᏂᎨᏒᎾ ᎾᎿ ᎯᎠ ᎠᏍᏓᏅᏅ
• ᏔᎵ ᏄᏓᎴᎿᎥ ᎤᎵᏍᏕᎸᏗ ᎦᏙ ᎤᏍᏗ ᏗᎦᎾᏗᏫᏍᏗ
• ᏔᎵ ᏄᏓᎴᎿᎥ ᎤᎵᏍᏕᎸᏗ ᎦᏙ ᎤᏍᏗ ᎠᎴ ᏚᏦᏔᏩᏘ

ᎦᏃᎯᎵᏙ embedded ᎭᏫᎾᏗᏢ R³

ᎪᎯ ᎾᎿ ᎨᏒ ᎨᏒᎢ ᎧᎵ ᏗᎪᎥᎯ ᎡᎯᏍᏛ ᎬᏙᏗ ᎦᏃᎯᎵᏙ embedded ᎭᏫᎾᏗᏢ ᏦᎢ ᎢᎦᎢ: ᎧᎵ ᏗᎪᎥᎯ, ᎭᏫᎾᏗᏢ R³.

Properties

ᎭᏫᎾᏗᏢ ᏦᎢ-dimensional ᎤᏜᏅᏛ, ᎢᏧᎳ ᎠᎾᏍᎬᏘ ᎠᎵᏖᎸᏗ ᎯᎠ ᎠᏍᏓᏩᏗᏒ ᏱᏓᏛᎾ ᎤᏙᎯᏳ Ꮎ ᎿᏛᎦ ᎾᏍᎩ ᏂᎨᏒᎾ ᏚᏂᏴ ᎭᏫᎾᏗᏢ ᎦᎸᎳᏗᎨᏍᏙᏗ ᎢᎦᎢ:

• ᏔᎵ ᎦᏃᎯᎵᏙ ᎠᎴ ᎢᏳᏍᏗᏊ ᏚᏦᏔᏩᏘ ᎠᎴ ᎤᏅᏌ ᏗᎦᎾᏗᏫᏍᏗ ᎭᏫᎾᏗᏢ ᎠᏍᏓᏅᏅ.
• ᎠᏍᏓᏅᏅ ᎨᏒᎢ ᎢᏳᏍᏗᏊ ᏚᏦᏔᏩᏘ ᎦᏃᎯᎵᏙ ᎠᎴ ᎤᏅᏌ ᏗᎦᎾᏗᏫᏍᏗ ᎾᎾᎢ ᏏᏴᏫ ᎪᏍᏓᏱ.
• ᏔᎵ ᎤᎵᏍᏕᎸᏗ ᏄᎶᏒᏍᏛᎾ ᎯᎠ ᎤᏠᏱ ᎦᏃᎯᎵᏙ ᎠᏎ ᎾᏍᏋ ᏚᏦᏔᏩᏘ ᎠᏂᏏᏴᏫᎭ ᏐᎢ.
• ᏔᎵ ᎦᏃᎯᎵᏙ ᏄᎶᏒᏍᏛᎾ ᎯᎠ ᎤᏠᏱ ᎠᏍᏓᏅᏅ ᎠᏎ ᎾᏍᏋ ᏚᏦᏔᏩᏘ ᎠᏂᏏᏴᏫᎭ ᏐᎢ.

ᎪᏍᏓᏱ ᎠᎴ ᏄᎶᏒᏍᏛᎾ ᎢᏚᏳᎪᏛ

ᎭᏫᎾᏗᏢ ᏦᎢ-dimensional ᎦᏃᎴᏍᎬ ᎤᏜᏅᏛ, ᎾᎿᎢ ᎨᏒᎢ ᏄᏓᎴ ᎤᎵᏍᎨᏛ ᎦᎶᎯᏍᏗ defining ᎦᏃᎯᎵᏙ:

• ᎪᏍᏓᏱ ᎠᎴ ᎠᏍᏓᏅᏅ, ᎦᏙ ᎤᏍᏗ ᎨᏒᎢ ᏄᎶᏒᏍᏛᎾ (ᏄᎶᏒᏍᏛᎾ) ᎯᎠ ᎦᏃᎯᎵᏙ

ᎢᏧᎳ ᏰᎵᏇ ᎦᏛᎬᎢ ᏄᏍᏛ ᎧᏃᎮᏗ ᎯᎠ ᏄᎵᏍᏔᏅ ᎦᏃᎯᎵᏙ; ᎤᎵᏍᎪᎸᏔᏅ $\vec p$ ᎾᏍᏋ ᎯᎠ ᎪᏍᏓᏱ ᎢᏧᎳ ᎠᏚᎳᏗ ᎦᎾᎬᎢ ᎭᏫᎾᏗᏢ ᎯᎠ ᎦᏃᎯᎵᏙ, ᎠᎴ ᎤᎵᏍᎪᎸᏔᏅ $\vec n$ ᎾᏍᏋ nonzero ᎢᏚᏳᎪᏛ ᏚᏦᏔᏩᏘ ᎯᎠ ᎠᏍᏓᏅᏅ ᎢᏧᎳ ᎠᏚᎳᏗ ᎾᏍᏋ ᏄᎶᏒᏍᏛᎾ ᎯᎠ ᎦᏃᎯᎵᏙ. ᎯᎠ ᎠᏚᎸᏓ ᎦᏃᎯᎵᏙ ᎨᏒᎢ ᎯᎠ ᎠᏫᏒᏗ ᏂᎦᏛ ᏗᎪᏍᏓᏱ $\vec r$ ᏯᏛᎿ Ꮎ

$\vec n\cdot(\vec r-\vec p)=0.$

ᎢᏳᏃ ᎢᏧᎳ ᎪᏪᎶᏗ $\vec n = (a, b, c)$, $\vec r = (x, y, z)$, ᎠᎴ $\vec n\cdot\vec p=-d$, ᎾᎯᏳᎢ ᎯᎠ ᎦᏃᎯᎵᏙ ᎨᏒᎢ ᏕᎫᎪᏔᏅ ᎾᎥᎢ ᎯᎠ ᏄᏍᏗᏓᏅ

$ax + by + cz + d = 0$,

ᎭᏢᏃ a, b, c ᎠᎴ d ᏰᎵᏇ ᎾᏍᏋ ᏂᎦᎵᏍᏗᏍᎬᎫ ᎤᏙᎯᏳ ᎾᏍᎩ ᏎᏍᏗᏯᏛᎿ Ꮎ ᎾᏍᎩ ᏂᎨᏒᎾ ᏂᎦᏛ a, b, c ᎠᎴ ᏭᎶᏒᏍᏛ ᎤᏴᏜ.

ᏅᎪᏢᎯᏐᏗᏱ, ᎦᏃᎯᎵᏙ ᎠᎾᏍᎬᏘ ᎾᏍᏋ ᏄᏍᏛ ᎧᏃᎮᎸ parametrically ᏥᏄᏍᏗ ᎯᎠ ᎠᏫᏒᏗ ᏂᎦᏛ ᏗᎪᏍᏓᏱ ᎯᎠ ᎤᏙᏢᏒ

$\vec{u} + s\vec{v} + t\vec{w},$

ᎭᏢᏃ s ᎠᎴ t ᎠᏍᏓᏅᏅ ᎦᏬᎯᎸᏙᏗ ᏂᎦᏛ ᎤᏙᎯᏳ ᎾᏍᎩ ᏗᏎᏍᏗ, ᎠᎴ $\vec{u}$, $\vec{v}$ ᎠᎴ $\vec{w}$ ᎠᎴ ᎠᏓᏁᎸ ᎢᏚᏳᎪᏛdefining ᎯᎠ ᎦᏃᎯᎵᏙ. $\vec{u}$ ᏗᎪᏍᏓᏱ ᏂᏛᎴᏅᏓ ᎯᎠ ᎠᏓᎴᏂᏍᎬ ᎬᏙᏗ ᎪᏍᏓᏱ ᎾᎿ ᎯᎠ ᎦᏃᎯᎵᏙ, ᎠᎴ $\vec{v}$ ᎠᎴ $\vec{w}$ ᏰᎵᏇ ᎾᏍᏋ visualized ᏥᏄᏍᏗ ᎠᏂᎩᏍᏗᏍᎬ ᎾᎾᎢ $\vec{u}$ ᎠᎴ ᎪᏍᏓᏱ ᎭᏫᎾᏗᏢ ᏄᏓᎴᎿᎥ ᏫᏂᏚᏳᎪᏛ ᎨᎳᏛᏍᏗ ᎯᎠ ᎦᏃᎯᎵᏙ. $\vec{v}$ ᎠᎴ $\vec{w}$ ᏰᎵᏇ, ᎠᎴ ᎿᏛᎦ ᎾᏍᎩ ᏂᎨᏒᎾ ᎤᎭ ᎾᏍᏋ ᏄᎶᏒᏍᏛᎾ.

ᎦᏃᎯᎵᏙ ᏗᎬᏩᎶᏒ ᏦᎢ ᏗᎪᏍᏓᏱ

ᎯᎠ ᎦᏃᎯᎵᏙ ᎦᎶᏍᎬ ᏗᎬᏩᎶᏒ ᏦᎢ ᏗᎪᏍᏓᏱ $\vec p_1 = (x_1,y_1,z_1)$, $\vec p_2 = (x_2,y_2,z_2)$ ᎠᎴ $\vec p_3 = (x_3,y_3,z_3)$ ᏰᎵᏇ ᎾᏍᏋ ᏕᎫᎪᏔᏅ ᎾᎥᎢ ᎯᎠ ᎠᏍᏓᏩᏗᏒ determinant ᎠᏓᏃᎮᏗ:

$\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1& z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} =\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x - x_2 & y - y_2 & z - z_2 \\ x - x_3 & y - y_3 & z - z_3 \end{vmatrix} = 0.$

ᎪᎯ ᎦᏃᎯᎵᏙ ᏰᎵᏇ ᎾᏍᎩ ᎾᏍᏇ ᎾᏍᏋ ᏄᏍᏛ ᎧᏃᎮᎸ ᎾᎥᎢ ᎯᎠ "ᎪᏍᏓᏱ ᎠᎴ ᏄᎶᏒᏍᏛᎾ ᎢᏚᏳᎪᏛ" lyudetiyvda ᎦᎸᎳᏗᏢ. ᏗᎾᏙᎳᎩ ᏄᎶᏒᏍᏛᎾ ᎢᏚᏳᎪᏛ ᎨᏒᎢ ᎠᏓᏁᎸ ᎾᎥᎢ ᎯᎠ ᏓᏓᎿᏩᏍᏛ ᎤᏛᏒᎯ $\vec n = ( \vec p_2 - \vec p_1 ) \times ( \vec p_3 - \vec p_1 ),$ ᎠᎴ ᎯᎠ ᎪᏍᏓᏱ $\vec p$ ᏰᎵᏇ ᎾᏍᏋ ᎠᎩᏒᎯ ᎾᏍᏋ $\vec p_1$.

ᎯᎠ ᎾᎿ ᎢᏴᎢ ᏂᏛᎴᏅᏓ ᎪᏍᏓᏱ ᎦᏃᎯᎵᏙ

ᎾᏍᎩᎾᎢ ᎦᏃᎯᎵᏙ $ax + by + cz + d = 0$ ᎠᎴ ᎪᏍᏓᏱ $\vec p_1 = (x_1,y_1,z_1)$ ᎾᏍᎩ ᏂᎨᏒᎾ ᎠᏎᎾᏍᎩ ᎢᏳᎵᏍᏙᏗ ᎠᏝᎥ ᎾᎿ ᎯᎠ ᎦᏃᎯᎵᏙ, ᎯᎠ ᎾᎿ ᎢᏴᎢ ᏂᏛᎴᏅᏓ $\vec p_1$ ᎯᎠ ᎦᏃᎯᎵᏙ ᎨᏒᎢ

$D = \frac{\left | a x_1 + b y_1 + c z_1+d \right |}{\sqrt{a^2+b^2+c^2}}.$

ᎯᎠ ᎠᏍᏓᏅᏅ ᏗᎦᎾᏗᏫᏍᏗ ᎠᏰᎵ ᏔᎵ ᎦᏃᎯᎵᏙ

ᎠᏓᏁᎸ ᏗᎦᎾᏗᏫᏍᏗ ᎦᏃᎯᎵᏙ ᏄᏍᏛ ᎧᏃᎮᎸ ᎾᎥᎢ $\vec n_1\cdot \vec r = h_1$ ᎠᎴ $\vec n_2\cdot \vec r = h_2$, ᎯᎠ ᎠᏍᏓᏅᏅ ᏗᎦᎾᏗᏫᏍᏗ ᎨᏒᎢ ᏄᎶᏒᏍᏛᎾ ᎢᏧᎳ $\vec n_1$ ᎠᎴ $\vec n_2$ ᎠᎴ ᎯᎠ ᎢᏴ ᏚᏦᏔᏩᏘ $\vec n_1 \times \vec n_2$ .

ᎢᏳᏃ ᎢᏧᎳ ᎤᏗᏗᏢ ᎢᏰᎵᏍᏗ Ꮎ $\vec n_1$ ᎠᎴ $\vec n_2$ ᎠᎴ orthonormal ᎾᎯᏳᎢ ᎯᎠ closest ᎪᏍᏓᏱ ᎾᎿ ᎯᎠ ᎠᏍᏓᏅᏅ ᏗᎦᎾᏗᏫᏍᏗ ᎯᎠ ᎠᏓᎴᏂᏍᎬ ᎨᏒᎢ $\vec r_0 = h_1\vec n_1 + h_2\vec n_2$ .

ᎯᎠ dihedral ᏓᏍᏓᏅᏅ ᏚᎷᏨ

ᎠᏓᏁᎸ ᏔᎵ ᏗᎦᎾᏗᏫᏍᏗ ᎦᏃᎯᎵᏙ ᏄᏍᏛ ᎧᏃᎮᎸ ᎾᎥᎢ $a_1 x + b_1 y + c_1 z + d_1 = 0$ ᎠᎴ $a_2 x + b_2 y + c_2 z + d_2 = 0$, ᎯᎠ dihedral ᏓᏍᏓᏅᏅ ᏚᎷᏨ ᎠᏰᎵ ᎠᏂ ᎨᏒᎢ ᎧᏁᎢᏍᏔᏅᎯ ᎾᏍᏋ ᎯᎠ ᏓᏍᏓᏅᏅ ᏚᎷᏨ $\alpha$ ᎠᏰᎵ ᎤᎾᏤᎵ ᏄᎶᏒᏍᏛᎾ ᏫᏂᏚᏳᎪᏛ:

$\cos\alpha = \hat n_1\cdot \hat n_2 = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.$

ᎯᎠ ᎦᏃᎯᎵᏙ ᎭᏫᎾᏗᏢ ᏐᎢ ᎡᏍᎦᏂ ᏗᏎᏍᏗ ᎤᎬᏩᎵ

ᎭᏫᎾᏗᏢ ᎦᏟᏐᏗᎩ Ꮝ ᎠᏓᏙᎳᎩ geometric ᎠᏛᎯᏍᏙᏗ, ᎬᏙᏗ ᎤᏙᏢᏒᎾ ᎠᎴ isometries ᎬᏙᏗ ᎪᎯᏳᎯ ᎯᎠ ᎤᏠᏱ ᎭᏫᎾᏗᏢ ᎤᏛᏒᎯ, ᎯᎠ ᎦᏃᎯᎵᏙ ᎠᎾᏍᎬᏘ ᎾᏍᏋ viewed ᎾᎾᎢ ᏧᏓᎴᏅᏓ ᏐᎢ ᎤᏩᎾᏕᏍᎩ ᎪᏣᎴᏛ. ᎠᏂᏏᏴᏫᎭ ᎤᏩᎾᏕᏍᎩ ᎪᏣᎴᏛ ᏗᏓᏛᎪᏗ ᏗᎪᏍᎩ ᎤᎾᏓᏟᏌᎲ.

ᎾᎾᎢ ᏌᏊ ᎦᎶᏒᏍᏔᏅ, ᏂᎦᏛ geometrical ᎠᎴ ᎠᏟᎶᏍᏗ ᎦᏙᎯ ᎠᏓᏅᏖᏗ ᎠᎾᏍᎬᏘ ᎾᏍᏋ ᎤᏬᎭᏒᎩ ᎠᏓᏅᏍᏗ ᎯᎠ topological ᎦᏃᎯᎵᏙ, ᎦᏙ ᎤᏍᏗ ᎠᎾᏍᎬᏘ ᎾᏍᏋ ᏁᎵᏒ ᏥᏄᏍᏗ idealised homotopically ᏱᏓᏟᎶᏍᏔᏅ ᎢᎪᎯᏓ ᎨᏒ ᎠᎹ ᎦᏄᎪᎬ ᏗᎬᎯᏓ, ᎦᏙ ᎤᏍᏗ ᏗᎦᏂᏴᎯ ᎢᎬᏁᎸ ᎢᏰᎵᏍᏗ ᎾᎥᏂᎨᏍᏙᏗ, ᎠᎴ ᎤᎭ Ꮭ ᎾᎿ ᎢᏴᎢ. ᎯᎠ topological ᎦᏃᎯᎵᏙ ᎤᎭ ᎠᏓᏅᏖᏗ ᎦᏌᏆᎸ ᎤᏍᏗ ᎦᏅᏅ, ᎠᎴ Ꮭ ᎠᏓᏅᏖᏗ ᎦᏥᏃᏍᏛ ᎠᏍᏓᏅᏅ. ᎯᎠ topological ᎦᏃᎯᎵᏙ, ᎠᎴ Ꮝ ᎢᏗᎦᏘᎭ ᎯᎠ ᎠᏍᏚᎢᏛ ᏙᏯᏗᏢ, ᎨᏒᎢ ᎯᎠ ᏄᏦᏍᏛᎾ topological neighbourhood ᎬᏔᏅᎯ ᎠᏛᎯᏍᏙᏗ ᎦᏚᎢ(ᎠᎴ 2-ᏧᏓᎴᏅᏓ ᎦᏟᏌᏅ) classified ᎭᏫᎾᏗᏢ ᎡᎳᏗ-dimensional topology. ᎤᏙᏢᏒ ᎯᎠ topological ᎦᏃᎯᎵᏙ ᎠᎴ ᏂᎦᏛ ᏫᎬᎵᏱᎵᏒ bijection. ᎯᎠ topological ᎦᏃᎯᎵᏙ ᎨᏒᎢ ᎯᎠ ᏂᎬᏩᏍᏛ ᏄᏍᏗᏓᏅ ᎾᏍᎩᎾᎢ ᎯᎠ ᎤᏩᏂᎦᎸ graph ᎪᎷᏩᏛᏗ Ꮎ ᏗᏓᏅᏓᏁᏗᏱ ᎬᏙᏗ planar graphs, ᎠᎴ ᏂᏚᎵᏍᏔᏅ ᏯᏛᎿ ᏥᏄᏍᏗ ᎯᎠ ᏅᎩ ᏗᎧᏃᏗ ᎢᏳᏍᏗ ᎧᏃᎮᏗ.

ᎯᎠ ᎦᏃᎯᎵᏙ ᎠᎾᏍᎬᏘ ᎾᏍᎩ ᎾᏍᏇ ᎾᏍᏋ viewed ᏥᏄᏍᏗ affine ᎤᏜᏅᏛ, ᎾᏍᎩ ᎤᏤᎵ ᎤᏙᏢᏒ ᎠᎴ ᎤᎾᏓᏟᏌᏅ ᎠᏁᏢᏔᏅᎯ ᎠᎴ ᎬᏙᏗ-ᏫᏚᏳᎪᏛ ᎦᏌᏆᎸ ᎡᎶᎯ ᏗᏟᎶᏍᏔᏅ. ᏂᏛᎴᏅᏓ ᎪᎯ viewpoint ᎾᎿᎢ ᎠᎴ Ꮭ ᎾᎿ ᎢᏴᎢ, ᎠᎴ colinearity ᎠᎴ ᎠᎾᎵᏐᏈᎸᏍᎬ ᎾᎿ ᎢᏴᎢ ᎾᎿ ᏂᎦᎵᏍᏗᏍᎬᎫ ᎠᏍᏓᏅᏅ ᎠᎴ ᏄᎾᏰᎯᏍᏛᎾ.

ᏎᏍᏗ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᏫᏓᎧᏃᏗᏱ ᎦᏃᎯᎵᏙ ᏥᏄᏍᏗ 2-dimensional ᎤᏙᎯᏳ ᎾᏍᎩ ᏧᏓᎴᏅᏓ ᎦᏟᏌᏅ, topological ᎦᏃᎯᎵᏙ ᎦᏙ ᎤᏍᏗ ᎨᏒᎢ ᎠᏓᏁᎳᏅ ᎬᏙᏗ ᏎᏍᏗ ᎠᏛᎯᏍᏙᏗ. ᎠᏏᏇ ᎭᏫᎾᏗᏢ ᎪᎯ ᎦᎸᏛ ᎧᏁᏌᎢ, ᎾᎿᎢ ᎨᏒᎢ Ꮭ ᎢᏰᎵᏍᏗ ᎾᎿ ᎢᏴᎢ, ᎠᎴ ᎾᎿᎢ ᎨᏒᎢ ᎾᏊ ᎠᏓᏅᏖᏗ ᎦᏃᎯᎵᏙ ᎡᎶᎯ ᏗᏟᎶᏍᏔᏅ, ᎾᏍᎩᎾᎢ ᏱᏓᏟᎶᏍᏔᏅ differentiable ᎠᎴ ᎤᏩᎾᏕᏍᎩ ᎤᏍᏗ ᎦᏅᏅ (ᎥᎵᏍᎦᏍᏙᏗᏍᎬ ᎾᎿ ᎯᎠ ᏗᎦᎪᏗ ᏎᏍᏗ ᎠᏛᎯᏍᏙᏗ ᎢᎬᎾᏔᏅᎯ). ᎯᎠ ᎤᏙᏢᏒ ᎭᏫᎾᏗᏢ ᎪᎯ ᎦᎸᏛ ᎧᏁᏌᎢ ᎠᎴ bijections ᎬᏙᏗ ᎯᎠ ᎠᏑᏰᏛ ᎢᎦᎢ ᎢᎦᏘ differentiability.

ᎭᏫᎾᏗᏢ ᎯᎠ ᎠᎿᏗᏢ ᎢᏚᏳᎪᏛ ᎪᏣᎴᏛ, ᎢᏧᎳ ᎠᎾᏍᎬᏘ ᎢᎬᎾᏙᏗ ᏓᏓᏁᏤᎸ ᏠᎨᏏ ᎠᏛᎯᏍᏙᏗ ᎯᎠ geometric ᎦᏃᎯᎵᏙ, ᎠᎾᏓᏁᎲ ᎤᎵᏌᎳᏙᏗ ᎯᎠ ᏂᎦᏛ ᎦᏃᎯᎵᏙ ᎠᎴ ᎯᎠ ᎠᏂᏯᏩᏍᎩ ᏄᎬᏫᏳᏒ ᎡᏍᎦᏂ ᏂᎦᏛ ᏗᎫᎪᏙᏗ ᎢᎬᏁᏗ. ᎯᎠ ᏂᎦᏛ ᏠᎨᏏ ᎤᎭ ᎾᏍᎩ ᎤᏩᏒ ᏔᎵ ᎤᏙᏢᏒ, ᎯᎠ ᎢᏗᎦᏘᎭ ᎠᎴ ᏔᎵ.

ᎭᏫᎾᏗᏢ ᎯᎠ ᎤᏠᏱ ᎦᎶᎯᏍᏗ ᏥᏄᏍᏗ ᎭᏫᎾᏗᏢ ᎯᎠ ᎤᏙᎯᏳ ᎾᏍᎩ ᎦᎸᏛ ᎧᏁᏌᎢ, ᎯᎠ ᎦᏃᎯᎵᏙ ᎠᎾᏍᎬᏘ ᎾᏍᎩ ᎾᏍᏇ ᎾᏍᏋ viewed ᏥᏄᏍᏗ ᎯᎠ simplest, ᏌᏊ-dimensional (ᎦᏬᎯᎸᏙᏗ ᎯᎠ ᏂᎦᏛ ᏗᏎᏍᏗ) ᏂᎦᏛ ᏧᏓᎴᏅᏓ ᎦᏟᏌᏅ, ᏱᏓᏟᎶᏍᏔᏅ ᎤᏯᏅᎲ ᎯᎠ ᏂᎦᏛ ᎠᏍᏓᏅᏅ. ᏱᏂᎬᏛᎾ, ᎪᎯ viewpoint ᏧᏓᎴᎾᎢ ᎪᏍᏓᏯ ᎬᏙᏗ ᎯᎠ ᎦᎸᏛ ᎧᏁᏌᎢ ᎯᎠ ᎦᏃᎯᎵᏙ ᏥᏄᏍᏗ 2-dimensional ᎤᏙᎯᏳ ᎾᏍᎩ ᏧᏓᎴᏅᏓ ᎦᏟᏌᏅ. ᎯᎠ ᎤᏙᏢᏒ ᎠᎴ ᏂᎦᏛ conformal bijections ᎯᎠ ᏂᎦᏛ ᎦᏃᎯᎵᏙ, ᎠᎴ ᎯᎠ ᎾᏍᎩ ᎤᏩᏒ possibilities ᎠᎴ ᎡᎶᎯ ᏗᏟᎶᏍᏔᏅ Ꮎ ᏗᏓᏛᎪᏗ ᎯᎠ ᎪᏪᎳᏅ ᎠᎪᎵᏰᏗ ᎠᏓᏃᎮᏗ ᎾᎥᎢ ᏂᎦᏛ ᏎᏍᏗ ᎠᎴ ᎠᏁᏢᏔᏅᎯ.

ᎭᏫᎾᏗᏢ ᎦᏟᏐᏗᎩ, ᎯᎠ Euclidean ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ (ᎦᏙ ᎤᏍᏗ ᎤᎭ ᏭᎶᏒᏍᏛ ᎤᏴᏜ ᎠᏗᏌᏓᏗᏍᏗ ᏂᎬᎾᏓ) ᎨᏒᎢ ᎾᏍᎩ ᏂᎨᏒᎾ ᎯᎠ ᎾᏍᎩ ᎤᏩᏒ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ Ꮎ ᎯᎠ ᎦᏃᎯᎵᏙ ᎠᎾᏍᎬᏘ ᎤᎭ. ᎯᎠ ᎦᏃᎯᎵᏙ ᎠᎾᏍᎬᏘ ᎾᏍᏋ ᎠᏓᏁᎸ ᎦᏐᏆᎸ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎾᎥᎢ ᎬᏗᏍᎬᎢ ᎯᎠ stereographic ᏂᎯ. ᎪᎯ ᏰᎵᏇ ᎾᏍᏋ ᏁᎵᏒ ᏥᏄᏍᏗ ᏩᎲᏍᎬ ᎢᎦᎢ ᎢᎦᏘ ᎾᎿ ᎯᎠ ᎦᏃᎯᎵᏙ (ᎣᏍᏛ ᎾᏍᎩᏯᎢ ᎠᎳᏍᎦᎶᏗ ᎾᎿ ᎯᎠ ᎠᏯᏖᏃᎯ), ᎠᏂᎲᏍᎬ ᎯᎠ ᎦᏚᎢ ᎪᏍᏓᏱ, ᎠᎴ ᎦᏛᎢ ᎯᎠ ᎢᎦᎢ ᎢᎦᏘ onto ᎯᎠ ᎦᏃᎯᎵᏙ ᏂᏛᎴᏅᏓ ᎪᎯ ᎪᏍᏓᏱ). ᎪᎯ ᎨᏒᎢ ᏌᏊ ᎯᎠ ᏂᎯ Ꮎ ᎠᎾᏍᎬᏘ ᎾᏍᏋ ᎬᏔᏅᎯ ᎭᏫᎾᏗᏢ ᎠᏃᏢᏍᎬ ᎤᏩᎾᏕᏍᎩ ᎡᎶᎯ ᏓᏟᎶᏍᏔᏅ ᎢᎦᏛ ᎯᎠ ᎡᎶᎯ ᎤᏤᎵ ᎦᏚᎢ. ᎯᎠ ᏄᎵᏍᏔᏅ ᏗᏎᏍᏗ ᏓᏍᏓᏅᏅ ᎤᎭ ᏂᎪᎯᎸ ᎤᏙᎯᏳ ᎠᏗᏌᏓᏗᏍᏗ.

ᏅᎪᏢᎯᏐᏗᏱ, ᎯᎠ ᎦᏃᎯᎵᏙ ᏰᎵᏇ ᎾᏍᎩ ᎾᏍᏇ ᎾᏍᏋ ᎠᏓᏁᎸ ᎠᏟᎶᏍᏗ ᎦᏙᎯ ᎦᏙ ᎤᏍᏗ ᎠᏓᏁᏗ ᎾᏍᎩ ᏂᎪᎯᎸ ᏗᎦᏘᎴᎩ ᏄᏍᏛ ᎧᏃᎮᏗ ᎠᏗᏌᏓᏗᏍᏗ ᎠᎾᏓᏁᎲ ᎯᎠ hyperbolic ᎦᏃᎯᎵᏙ. ᎯᎠ ᎣᏂᏯᎨ ᎤᏝᏅᏓᏕᎲ ᎠᏩᏛᏓ ᎠᏔᏲᏍᏙᏗ ᎭᏫᎾᏗᏢ ᎯᎠ ᎪᎷᏩᏛᏗ ᎤᏤᎵᏛ relativity ᎭᏫᎾᏗᏢ ᎯᎠ simplified ᎦᎸᏛ ᎧᏁᏌᎢ ᎭᏢᏃ ᎾᎿᎢ ᎨᏒᎢ ᏌᏊ ᏂᎦᏅᎯᏒ ᎤᏜᏅᏛ ᎠᎴ ᏌᏊ iyuwakodi.

ᎠᎪᏩᏛᏗ ᎾᏍᎩ ᎾᏍᏇ

• ᎠᏰᎵ ᎢᏴ-ᎦᏃᎯᎵᏙ
• Hyperplane
• ᎠᏍᏓᏅᏅ-ᎦᏃᎯᎵᏙ ᏗᎦᎾᏗᏫᏍᏗ

ᏙᏯᏗᏢ ᏗᏕᎬᏔᏛ

• Mathworld: Plane