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This is a collection of formulas for the mathematical sub-area of analytical geometry .
Preliminary remarks on the notation
In the following, consecutively numbered Cartesian coordinates (equivalent to ), (equivalent to ), (equivalent to ) are used. Vectors are noted in arrows. Position vectors are denoted with the same capital letter as the corresponding points. The scalar product is expressed by, the cross product (vector product) by .
![x_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![x_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
![x_ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/766d09a498699be10e276ad49145c921f8cbe335)
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![\ cdot](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba)
![\ times](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703)
Analytical geometry of the Euclidean plane
Designations
In the following the point has the coordinates ; the items in that order![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
![(p_1, p_2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa988e111a7acf37bf178e4a067e7db92a7a9fe)
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6)
Points
Points are described by Cartesian coordinates or position vectors .
Coordinate representation of a point
-
or
Position vector of the point :
![P (p_1 | p_2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/386d17da4a2e2c3bc6b85b75a9f3265416d74ed5)
![\ vec {P} = \ begin {pmatrix} p_1 \\ p_2 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abbc08e669ab991389e80ec3b3dd499534fb0003)
Connection vector of two points :
![FROM](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b)
![\ overrightarrow {AB} = \ vec {B} - \ vec {A} = \ begin {pmatrix} b_1-a_1 \\ b_2-a_2 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/875c8528a2121033f3e9077a49fddcfa35635600)
Center of the route (as position vector):
![FROM](https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd)
![\ vec {M} = \ tfrac {1} {2} \ left (\ vec {A} + \ vec {B} \ right) = \ tfrac {1} {2} \ begin {pmatrix} a_1 + b_1 \\ a_2 + b_2 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34bcef864f38b7ff6c9941d6d2f41b376330c2ef)
Division point : The point that divides the line in proportion :
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![\ vec {T} = \ frac {1} {1+ \ lambda} \ left (\ vec {A} + \ lambda \ vec {B} \ right) = \ frac {1} {1+ \ lambda} \ begin {pmatrix} a_1 + \ lambda b_1 \\ a_2 + \ lambda b_2 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca81f7ad96f300257604a5ce50a317f5e06280d3)
Center of gravity of a triangle :
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![\ vec {S} = \ tfrac {1} {3} \ left (\ vec {A} + \ vec {B} + \ vec {C} \ right) = \ tfrac {1} {3} \ begin {pmatrix } a_1 + b_1 + c_1 \\ a_2 + b_2 + c_2 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40d7072411d1f3633a5f782ce65b172abecdf8e5)
Straight lines
Parametric equation of the straight line (point-directional form) through the point with the direction vector :
![A (a_1 | a_2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0346e8fd54c6787d2401169d4114cba891ae774c)
![\ vec {u} = \ begin {pmatrix} u_1 \\ u_2 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27ea3a8e0f73a77838239f67da2fe17e0bcfee51)
![\ vec {X} = \ vec {A} + \ lambda \ vec {u} = \ begin {pmatrix} a_1 \\ a_2 \ end {pmatrix} + \ lambda \ begin {pmatrix} u_1 \\ u_2 \ end {pmatrix }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b162392070bfa89468cf6c1076792d22b1289a6)
The parameter can take all real numbers as a value and must not be the zero vector.
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![{\ vec {u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43)
Parametric equation of the straight line (two-point form) through the points :
![FROM](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b)
![\ vec {X} = \ vec {A} + \ lambda \ left (\ vec {B} - \ vec {A} \ right) = \ begin {pmatrix} a_1 \\ a_2 \ end {pmatrix} + \ lambda \ begin {pmatrix} b_1-a_1 \\ b_2-a_2 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/457d66f8d6ab714ffe09ee4790dcc5bc499c89e0)
The parameter can take all real numbers as value and. and must be different.
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
Normal equation of the straight line through the point with the normal vector in vector notation:
![\ vec {n} = \ begin {pmatrix} n_1 \\ n_2 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6acfd2532e0b02f2517e52da1a1ec94be221e2)
-
or.
Coordinate equation, explicit form of the straight line with the slope through the point of the -axis:
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![(0 | t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3aae4c886e1e9e9a01d671a11d10f4a5ecb071)
![x_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766)
![\, x_2 = m x_1 + t](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ce1499b7b1acbcc69babcacc7c6a525917f00a)
Restriction: The straight line must not be parallel to the axis.
![x_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766)
Coordinate equation, intercept form of the straight line through the points (on the -axis) and (on the -axis):
![(s | 0)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb3ac81c9e4b2ec2b056810c4e7623ed8549661)
![x_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308)
![(0 | t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3aae4c886e1e9e9a01d671a11d10f4a5ecb071)
![x_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766)
![\ frac {x_1} {s} + \ frac {x_2} {t} = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d04bf312624c6a7f544df1e7af891a5b813ebd6)
Restriction: The given points must not coincide with the origin, i. H. it must and apply.
![s \ ne 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/b696e0b457eecf592be8e2fda417810807613c99)
![t \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e528af863fd84d0edc808c3d9f53be7687a83d3)
distances
Distance between points :
![FROM](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b)
![\ overline {AB} = \ left | \ vec {B} - \ vec {A} \ right | = \ sqrt {(b_1-a_1) ^ 2 + (b_2-a_2) ^ 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/503e1f73afeabf572bba1f6aba81db962774c234)
Distance of the point from the straight line with the normal equation (see Hessian normal form ):
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![n_1 x_1 + n_2 x_2 + n_0 = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce72ab6f211328fbd7a2312a6e25bce3f6ddab73)
![d (P, g) = \ frac {\ left | n_1 p_1 + n_2 p_2 + n_0 \ right |} {\ sqrt {{n_1} ^ 2 + {n_2} ^ 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9842151b9b194e4d782c9dd3662dd69e85402a20)
Distance between two parallel straight lines and with the normal equations or :
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![G'](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a53c0df5d85b36e3fd327c74db998f679f4f55)
![n_1 x_1 + n_2 x_2 + n_0 = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce72ab6f211328fbd7a2312a6e25bce3f6ddab73)
![n_1 x_1 + n_2 x_2 + n_0 '= 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/b05ab967ce3725c0076f9e7ba9bc8f90c350fbd4)
![d (g, g ') = \ frac {\ left | n_0 - n_0 '\ right |} {\ sqrt {{n_1} ^ 2 + {n_2} ^ 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6712b7e1752f0dfb8e2091a33fb9b5a3f492aff4)
Projections
Orthogonal projection of a point onto a straight line in parametric form :
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![\ vec {X} = \ vec {A} + \ lambda \ vec {u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d42a315688468a9b76f98f6a15574ce9f0d44cc9)
![\ vec {P} _g = \ vec {A} + \ frac {(\ vec {B} - \ vec {A}) \ cdot \ vec {u}} {\ vec {u} \ cdot \ vec {u} } \, \ vec {u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f16d8ba26efd16c626be45fcef8b1f7a90a393a5)
Orthogonal projection of a point onto a straight line in normal form :
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![(\ vec {X} - \ vec {A}) \ cdot \ vec {n} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e76a36666dcbd61201052ff82d85d8b266ff63)
![\ vec {P} _g = \ vec {B} - \ frac {(\ vec {B} - \ vec {A}) \ cdot \ vec {n}} {\ vec {n} \ cdot \ vec {n} } \, \ vec {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da22393e6468130b9855ed03b66b884bc951b44d)
Parallel projection in the direction of a point onto a straight line in normal form :
![{\ vec {v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![(\ vec {X} - \ vec {A}) \ cdot \ vec {n} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e76a36666dcbd61201052ff82d85d8b266ff63)
![\ vec {P} _ {g, \ vec {v}} = \ vec {B} - \ frac {(\ vec {B} - \ vec {A}) \ cdot \ vec {n}} {\ vec { v} \ cdot \ vec {n}} \, \ vec {v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea26f7c8a55105332260e1566c334c32e02173c)
angle
Intersection angle (smaller angle) between two straight lines
with the direction vectorsand(compare scalar product ):
![{\ vec {u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43)
![{\ vec {v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
![\ cos \ epsilon = \ frac {\ left | \ vec {u} \ cdot \ vec {v} \ right |} {\ left | \ vec {u} \ right | \ left | \ vec {v} \ right |} = \ frac {\ left | u_1 v_1 + u_2 v_2 \ right |} {\ sqrt {{u_1} ^ 2 + {u_2} ^ 2} \ sqrt {{v_1} ^ 2 + {v_2} ^ 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9bdf85d5eb25f9a74d9433659d1a566966f60f)
Surfaces
Area of the triangle (see cross product ):
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971)
![\ begin {align} F_ {ABC} & = \ tfrac {1} {2} \ left | \ overrightarrow {AB} \ times \ overrightarrow {AC} \ right | = \ tfrac {1} {2} \ left | \ left (\ vec {B} - \ vec {A} \ right) \ times \ left (\ vec {C} - \ vec {A} \ right) \ right | \\ & = \ tfrac {1} {2} \ left | (a_1 b_2 - a_2 b_1) + (b_1 c_2 - b_2 c_1) + (c_1 a_2 - c_2 a_1) \ right | \ end {align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c314bae2d27e6ee88182ebd572029a9e4ff18091)
Area of the polygon not overridden with the corners :
![P_ {1} (p _ {{11}} | p _ {{12}}), \ dotsc, P_ {n} (p _ {{n1}} | p _ {{n2}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c57fa7c77c012689853f7c908740e277014f5c5)
![{\ begin {aligned} A = {\ Big |} {\ tfrac {1} {2}} \ cdot & \ left (p _ {{11}} p _ {{22}} + p _ {{21}} p_ { {32}} + \ dotsb + p _ {{n-1,1}} p _ {{n2}} + p _ {{n1}} p _ {{12}} \ right. \\ & - \ left.p _ {{ 21}} p _ {{12}} - p _ {{31}} p _ {{22}} - \ dotsb -p _ {{n1}} p _ {{n-1,2}} - p _ {{11}} p_ {{n2}} \ right) {\ Big |} \ end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e69d6ff39b1389e9b9659d29c390b00dfaf3549)
Equation of the circle in Cartesian coordinates :
![{x_1} ^ 2 + {x_2} ^ 2 = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/372dc60dbad205fc375f3bc15118c16c249eff12)
- general: center in , radius
![(CD)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee6b94416101a5232787c5d1f20362c35adfd49)
![(xc) ^ 2 + (yd) ^ 2 = r ^ 2 \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dde2551b6c1997791391cb493e2062a176db394)
in parametric form (general):
-
With
Equation of the circle through three points
![\ begin {vmatrix} x ^ 2 + y ^ 2 & x & y & 1 \\ x_1 ^ 2 + y_1 ^ 2 & x_1 & y_1 & 1 \\ x_2 ^ 2 + y_2 ^ 2 & x_2 & y_2 & 1 \\ x_3 ^ 2 + y_3 ^ 2 & x_3 & y_3 & 1 \ end {vmatrix} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/97186091efae737400aa14449751f0c9ed02ee88)
Equation of the circle tangent at the point
- Unit circle
![\, b_1 x_1 + b_2 x_2 = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/969f7af0657fb0cbda0ded60f1b06af25a49b26f)
- General:
![(xc) (b_1-c) + (yd) (b_2-d) = r ^ 2 \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/515df57a8ce5a88c682d32ad05cdca79ea6db508)
Intersection of the straight line with the circle![y = mx + c](https://wikimedia.org/api/rest_v1/media/math/render/svg/3daca9fbcf73b8a2b575a10e2d5de25f9f1c6f49)
:
![x_ {1,2} = - \ frac {cm} {1 + m ^ 2} \ pm \ frac1 {1 + m ^ 2} \ sqrt {r ^ 2 (1 + m ^ 2) -c ^ 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a4914817527dc60774ab4b1dd42032d4d2653f5)
![y_ {1,2} = \ frac {c} {1 + m ^ 2} \ pm \ frac {m} {1 + m ^ 2} \ sqrt {r ^ 2 (1 + m ^ 2) -c ^ 2 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fed6e3a9cd88d3b93d24d4e0c06a0bb439a67385)
Center of the circle through three points
that are not on a straight line:
![P_1 (x_1, y_1), P_2 (x_2, y_2), P_3 (x_3, y_3)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3874abacd1b6e31ed49689e5f1329e556ca29cd9)
![\ vec {X} = \ left (\ begin {array} {c} x_ {1} \\ x_ {2} \ end {array} \ right) = \ frac {1} {2} \ left (\ begin { array} {ccc} x_ {1} -x_ {3} & y_ {1} -y_ {3} \\ x_ {2} -x_ {3} & y_ {2} -y_ {3} \ end {array} \ right) ^ {- 1} \ left (\ begin {array} {c} \ vec {P} _1 \ cdot \ vec {P} _1- \ vec {P} _3 \ cdot \ vec {P} _3 \\ \ vec {P} _2 \ cdot \ vec {P} _2- \ vec {P} _3 \ cdot \ vec {P} _3 \ end {array} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/da3a030d2b3e6d775fddbc44afd79de05ab6d168)
Conic section
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ellipse
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hyperbole
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parabola
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properties
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Definition: set of all points for which ...
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the sum of the distances to the focal points is constantly equal to 2a.
![F_1, F_2](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddbaefad7d000285a069031a623cdc454e4b79e4) |
the difference in the distances between the two focal points is constant 2a.
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the distance to a focal point and the guide line l is constant.
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Linear eccentricity
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Coordinates
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Cartesian coordinates
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Axis-parallel position
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Parametric shape
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With
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![x = \ frac a {\ cos (t)}; \; y = \ pm b \ tan (t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/aec47b7bd1a2673184a088e42aa4abb248919eb6)
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Straight lines
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Tangent in
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Normal through
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Intersection with the straight line
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![x_ {1,2} = a ^ 2m \ alpha \ pm \ beta \ cdot \ sqrt {D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4799f90245ca3006d4d2eb41c0519911dca83e8f)
![y_ {1,2} = b ^ 2 \ alpha \ pm m \ beta \ cdot \ sqrt {D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adf4e61f5332dacb2892aacc108eb6b1ca79b71b)
![\ alpha: = \ frac {C} {b ^ 2 + a ^ 2m ^ 2}; \ beta: = \ frac {ab} {b ^ 2 + a ^ 2m ^ 2};](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b9878873fd360d9651cf8971b9e33cb100c5f34)
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![x_ {1,2} = a ^ 2m \ alpha \ pm \ beta \ cdot \ sqrt {D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4799f90245ca3006d4d2eb41c0519911dca83e8f)
![y_ {1,2} = b ^ 2 \ alpha \ pm m \ beta \ cdot \ sqrt {D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adf4e61f5332dacb2892aacc108eb6b1ca79b71b)
![\ alpha: = \ frac {C} {b ^ 2-a ^ 2m ^ 2}; \ beta: = \ frac {ab} {b ^ 2-a ^ 2m ^ 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad37267d9c29f551ede305384d104124e94367f)
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![x_ {1,2} = \ frac {p-Cm} {m ^ 2} \ pm \ frac1 {m ^ 2} \ cdot \ sqrt {D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf5f852f5891b905577ccab96f5eac1e3f6327c)
![y_ {1,2} = \ frac pm \ pm \ frac1m \ cdot \ sqrt {D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f017a1043724c9fd140e0d11dad2f9b2cafe0c)
|
Area
|
Flat curves with excellent curvature
Since the geometric shape of a plane curve remains invariant under translation and rotation, an excellent (symmetrical) representation of its analytical description can be chosen. In particular, every flat curve, which can be continuously differentiated twice, is clearly described by specifying its curvature (at each point). In the following formulas there are arbitrary but fixed constants and always denotes the arc length (with natural parameterization).
![a, b \ in \ R ^ +](https://wikimedia.org/api/rest_v1/media/math/render/svg/0476090cd325a435f3a6e22b9f8982bf3e7dca85)
![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632)
Here and denote the Fresnel integrals .
![C_b (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d401953f67ae1682371530771b2d10f25fdce6c)
![S_b (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ef34da48ae08b987dad62ca92f21b2f64e5c7a)
Analytical geometry of three-dimensional Euclidean space
Designations
In the following, the points have the coordinates in this order .
![X, P, A, B, C](https://wikimedia.org/api/rest_v1/media/math/render/svg/a25e1910bca2605a760e211d78493b6e8eb57146)
![(x_1, x_2, x_3), (p_1, p_2, p_3), (a_1, a_2, a_3), (b_1, b_2, b_2), (c_1, c_2, c_3)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc997d8658b5c3e49ab23e282e0bb07f01fe45cc)
Points
Points are described by Cartesian coordinates or position vectors .
Coordinate representation
![P (p_1 | p_2 | p_3)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b06565d2cc0bd5ef4ffebe8f086c562df8687a7e)
Position vector
![\ vec {P} = \ begin {pmatrix} p_1 \\ p_2 \\ p_3 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd6887e11438318b3dbdf8ad6837d03ceea797)
Connection vector of two points :
![FROM](https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd)
![\ overrightarrow {AB} = \ vec {B} - \ vec {A} = \ begin {pmatrix} b_1-a_1 \\ b_2-a_2 \\ b_3-a_3 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b95699805fac4bcc1fbf96673130312129176bee)
Midpoint of the route :
![FROM](https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd)
![\ vec {M} = \ tfrac {1} {2} \ left (\ vec {A} + \ vec {B} \ right) = \ tfrac {1} {2} \ begin {pmatrix} a_1 + b_1 \\ a_2 + b_2 \\ a_3 + b_3 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df3d92aadd3675d161b4879ef2da68f9723d097e)
Division point that divides the distance in proportion :
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![\ vec {T} = \ frac {1} {1+ \ lambda} \ left (\ vec {A} + \ lambda \ vec {B} \ right) = \ frac {1} {1+ \ lambda} \ begin {pmatrix} a_1 + \ lambda b_1 \\ a_2 + \ lambda b_2 \\ a_3 + \ lambda b_3 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f032c7b3a2af6c1cd1f5010307cd1e1ba8c486ca)
Center of gravity of a triangle with the corners :
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6)
![\ vec {S} = \ tfrac {1} {3} \ left (\ vec {A} + \ vec {B} + \ vec {C} \ right) = \ tfrac {1} {3} \ begin {pmatrix } a_1 + b_1 + c_1 \\ a_2 + b_2 + c_2 \\ a_3 + b_3 + c_3 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49bd37367bf113a9f2c0171f1adaef6ea5095dea)
Straight lines
Parametric equation of a straight line (point-directional form) through the point with the direction vector :
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![\ vec {u} = \ begin {pmatrix} u_1 \\ u_2 \\ u_3 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd81795c6314da423b64b4da3ff57c8324bbb2a)
![\ vec {X} = \ vec {A} + \ lambda \ vec {u} = \ begin {pmatrix} a_1 \\ a_2 \\ a_3 \ end {pmatrix} + \ lambda \ begin {pmatrix} u_1 \\ u_2 \ \ u_3 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a43e6fb8d7d4474b7e3498d0b6c947674604e6d7)
The parameter can take all real numbers as a value and must not be the zero vector.
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![{\ vec {u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43)
Levels
Parametric equation of the plane (point-direction form) through the point with the direction vectors and :
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ vec {u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43)
![{\ vec {v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
![\ vec {X} = \ vec {A} + \ lambda \ vec {u} + \ mu \ vec {v} = \ begin {pmatrix} a_1 \\ a_2 \\ a_3 \ end {pmatrix} + \ lambda \ begin {pmatrix} u_1 \\ u_2 \\ u_3 \ end {pmatrix} + \ mu \ begin {pmatrix} v_1 \\ v_2 \\ v_3 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31ccfee6ab351a60c663d2055f9ed1e85828dfd2)
The parameters and can take any real number as value and the vectors must be linearly independent (i.e. and is not a scalar multiple of )
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![\ vec {u}, \ vec {v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa77ba827ad8f32691cfd9dc20162a61d96bdd1a)
![\ vec {u}, \ vec {v} \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/985130b176056ffadbfbd19fac4c66183f276e3e)
![{\ vec {u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43)
![{\ vec {v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
Parametric equation of a plane (three-point form) through the points :
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6)
![\ vec {X} = \ vec {A} + \ lambda \ left (\ vec {B} - \ vec {A} \ right) + \ mu \ left (\ vec {C} - \ vec {A} \ right ) = \ begin {pmatrix} a_1 \\ a_2 \\ a_3 \ end {pmatrix} + \ lambda \ begin {pmatrix} b_1-a_1 \\ b_2-a_2 \\ b_3-a_3 \ end {pmatrix} + \ mu \ begin {pmatrix} c_1-a_1 \\ c_2-a_2 \\ c_3-a_3 \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9aeb94ec2fc92e96ba55449794ea6eb330d93d2)
The two parameters and can take all real numbers as values and the given points and must not lie on a straight line.
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
![FROM](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
Normal equation of the plane through the point with the normal vector in vector notation:
![\ vec {n} = \ begin {pmatrix} n_1 \\ n_2 \\ n_3 \ end {pmatrix} \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/e29cfae94c9c5c9dd237931850513c4adaa731f4)
-
or.
Coordinate equation
-
with not all equal to 0.![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d)
Conversion of the forms into one another
- Parametric form in normal form:
![\ vec {n} = \ vec {u} \ times \ vec {v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e36017bf7516276e888940eeace892b5dea66641)
- Normal form and coordinate equation:
- The normal form is the same as the coordinate equation, just written a little differently. Explicit: and .
![a = n_ {1}, b = n_ {2}, c = n_ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f443591d0f082dfd6035de5c9d125335088a21a7)
![d = n_1a_1 + n_2a_2 + n_3a_3](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf4992ec3fb5c096ff711a313f01360254e72e5)
- From the parametric form to the coordinate equation:
-
defines three equations; solve one of them and another and plug this into the remaining equation.![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![\ mu](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161)
- From the coordinate equation to the parametric form:
- Either one finds three non-collinear points in the plane by trial and error and puts them in the three-point form of the parametric equation. Alternatively, the following algorithmic approach works: Since not all are equal to 0 (say ), the equation can coordinate to a coordinate dissolve and this coordinate is a function of the other two: . Now there are three non-collinear points in the plane by successively , and uses. I.e. one sets explicitly
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d)
![c \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/be27396bd0e62003728d08329a8767eee94409e2)
![x_3 (x_1, x_2) = \ tfrac {1} {c} \ left (d - ax_1 - bx_2 \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/26011b1918d2490c67ea5b94a0b108b30e3db374)
![(x_1, x_2) = (0.0)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6488a85553df9adf0e6e5e2db5eae6d700232a68)
![(x_1, x_2) = (1.0)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6291a463114e8697579089dea16f2c905a59f6)
![(x_1, x_2) = (0.1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c64ea92b15e68c89e0b5f47c95e575e5770c3584)
-
, and![\ vec {B} = \ begin {pmatrix} 1 \\ 0 \\ x_3 (1,0) \ end {pmatrix}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e3096e98ff3804c0f8fc79e9cc830ae0e897290)
- into the three-point form of the parametric equation.
distances
Distance between the points
![\ left \ vert \ overrightarrow {AB} \ right \ vert = \ left | \ vec {B} - \ vec {A} \ right | = \ sqrt {(b_1-a_1) ^ 2 + (b_2-a_2) ^ 2 + (b_3-a_3) ^ 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39d92c5d849d32559779adb51b258391aeb0a09c)
Distance of the point from the straight line![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
in parametric form :
![\ vec {X} = \ vec {A} + \ lambda \ vec {u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d42a315688468a9b76f98f6a15574ce9f0d44cc9)
![d (P, g) = \ frac {| (\ vec {P} - \ vec {A}) \ times \ vec {u} |} {| \ vec {u} |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3daa7e7bdb5f31abbd75cb7cc5b71874738f55c7)
Distance of the point from the plane![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
with the normal equation (see Hessian normal form ):
![n_1 x_1 + n_2 x_2 + n_3 x_3 + n_0 = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/225df1561eb94b8c3c3bd18388262f6da9535dc6)
![d (P, \ epsilon) = \ frac {\ left | n_1 p_1 + n_2 p_2 + n_3 p_3 + n_0 \ right |} {\ sqrt {{n_1} ^ 2 + {n_2} ^ 2 + {n_3} ^ 2} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b30896ba23c66e286408d1a972625f7b42c2ab)
Distance of the point from the plane![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
in parametric form :
![\ vec {X} = \ vec {A} + \ lambda \ vec {u} + \ mu \ vec {v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c155f2a266c823c641b500ab701ac617a10ec80a)
![d (P, \ epsilon) = \ frac {| (\ vec {P} - \ vec {A}) \ cdot (\ vec {u} \ times \ vec {v}) |} {| \ vec {u} \ times \ vec {v} |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a2bd799b68f17fb038383d7a10103ca63424ddc)
Distance of the parallel planes and with the normal equations or :
![\epsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)
![\epsilon'](https://wikimedia.org/api/rest_v1/media/math/render/svg/78f2a815df98759c8f75e757592569d942250171)
![n_1 x_1 + n_2 x_2 + n_3 x_3 + n_0 = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/225df1561eb94b8c3c3bd18388262f6da9535dc6)
![n_1 x_1 + n_2 x_2 + n_3 x_3 + n_0 '= 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/19d758546833172fe9c9e950bbb8a94e06daf00b)
![d (\ epsilon, \ epsilon ') = \ frac {\ left | n_0 - n_0 '\ right |} {\ sqrt {{n_1} ^ 2 + {n_2} ^ 2 + {n_3} ^ 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59a837485e573f78f2e510f6eaee5691019b1942)
Projections
Orthogonal projection of a point onto a straight line in parametric form :
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![\ vec {X} = \ vec {A} + \ lambda \ vec {u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d42a315688468a9b76f98f6a15574ce9f0d44cc9)
![\ vec {P} _g = \ vec {A} + \ frac {(\ vec {B} - \ vec {A}) \ cdot \ vec {u}} {\ vec {u} \ cdot \ vec {u} } \, \ vec {u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f16d8ba26efd16c626be45fcef8b1f7a90a393a5)
Orthogonal projection of a point onto a plane in normal form :
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![\epsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)
![(\ vec {X} - \ vec {A}) \ cdot \ vec {n} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e76a36666dcbd61201052ff82d85d8b266ff63)
![\ vec {P} _ \ epsilon = \ vec {B} - \ frac {(\ vec {B} - \ vec {A}) \ cdot \ vec {n}} {\ vec {n} \ cdot \ vec { n}} \, \ vec {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90dcafa225a56c5a087590c0b2e1669d80d55805)
Parallel projection in the direction of a point onto a plane in normal form :
![{\ vec {v}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![\epsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)
![(\ vec {X} - \ vec {A}) \ cdot \ vec {n} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e76a36666dcbd61201052ff82d85d8b266ff63)
![\ vec {P} _ {\ epsilon, \ vec {v}} = \ vec {B} - \ frac {(\ vec {B} - \ vec {A}) \ cdot \ vec {n}} {\ vec {v} \ cdot \ vec {n}} \, \ vec {v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19c06b80e137b2c37cb9c3368bd0c5c013b89109)
angle
Intersection angle (smaller angle) between two straight lines
with the direction vectors and :
![{\ vec {u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43)
![\ vec {v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753)
![\ cos \ epsilon = \ frac {\ left | \ vec {u} \ cdot \ vec {v} \ right |} {\ left | \ vec {u} \ right | \ left | \ vec {v} \ right |} = \ frac {\ left | u_1 v_1 + u_2 v_2 + u_3 v_3 \ right |} {\ sqrt {{u_1} ^ 2 + {u_2} ^ 2 + {u_3} ^ 2} \ sqrt {{v_1} ^ 2 + {v_2} ^ 2 + { v_3} ^ 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a20059495f1937a1b94d79c6abcc8f52253d2f3b)
Intersection angle between a plane
with the normal vector and a straight line with the direction vector :
![\ vec {u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43)
![\ sin \ epsilon = \ frac {\ left | \ vec {n} \ cdot \ vec {u} \ right |} {\ left | \ vec {n} \ right | \ left | \ vec {u} \ right |} = \ frac {\ left | n_1 u_1 + n_2 u_2 + n_3 u_3 \ right |} {\ sqrt {{n_1} ^ 2 + {n_2} ^ 2 + {n_3} ^ 2} \ sqrt {{u_1} ^ 2 + {u_2} ^ 2 + { u_3} ^ 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/101249e01ee445a0de39195cedcceadbc6d0143b)
Intersection angle between two planes
with the normal vectors and :
![\ vec {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24e08685bde682d6b67c3a22b026d16dbbf8dfcc)
![\ vec {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49569db585c1b6306d5ffd91161775f67235fae0)
![\ cos \ epsilon = \ frac {\ left | \ vec {m} \ cdot \ vec {n} \ right |} {\ left | \ vec {m} \ right | \ left | \ vec {n} \ right |} = \ frac {\ left | m_1 n_1 + m_2 n_2 + m_3 n_3 \ right |} {\ sqrt {{m_1} ^ 2 + {m_2} ^ 2 + {m_3} ^ 2} \ sqrt {{n_1} ^ 2 + {n_2} ^ 2 + { n_3} ^ 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/049a59e94d335e0a93510ba3a770c66114174a7b)
Volumes
Volume of the tetrahedron (compare Spat product ): ( )
![P_0P_1P_2P_3](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4382f7e2fb5ca518c350778d33518135ba4c117)
![\ vec a: = \ overrightarrow {P_0P_1} \, \ \ vec b: = \ overrightarrow {P_0P_2} \, \ \ vec c: = \ overrightarrow {P_0P_3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c06c8640fd9e202c7821b2f824fff832b8d206)
![V = \ Big | \ frac {1} {6} [\ vec a, \ vec b, \ vec c] \ Big | = \ Big | \ frac {1} {6} \ begin {vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \ end {vmatrix} \ Big |](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a40d68981efb70354b03d942e4cd3e43c280114)
Bullets
Cartesian coordinates
- Unit sphere:
![{x_1} ^ 2 + {x_2} ^ 2 + {x_3} ^ 2 = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a70409e09d2c54e3e800fd6f06fab0b68d73a6)
- General: (center: )
![(ABC)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae973a762a92b9cd3eafe7f283890ccfa9b887e8)
![(x_1-a) ^ 2 + (x_2-b) ^ 2 + (x_3-c) ^ 2 = r ^ 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1ef25ef0e0397d3b549ab2e061decb73bd0881)
Parametric form (in the origin)
-
with and![0 \ le \ vartheta \ le \ pi](https://wikimedia.org/api/rest_v1/media/math/render/svg/b879e24278c205c9a490cc539edbe64fc3e79a41)
Center of the sphere through four points
and , which are not in a plane:
![\ vec {A}, \ vec {B}, \ vec {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80ec6d102910002941ff44f103f6f16cb20390be)
![{\ vec {P}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/765f1dd50e122eb3e565c9bfee85de8f74d47f27)
![\ vec {X} = \ left (\ begin {array} {c} x_ {1} \\ x_ {2} \\ x_ {3} \ end {array} \ right) = \ frac {1} {2} \ left (\ begin {array} {ccc} a_ {1} -p_ {1} & a_ {2} -p_ {2} & a_ {3} -p_ {3} \\ b_ {1} -p_ {1 } & b_ {2} -p_ {2} & b_ {3} -p_ {3} \\ c_ {1} -p_ {1} & c_ {2} -p_ {2} & c_ {3} -p_ { 3} \ end {array} \ right) ^ {- 1} \ left (\ begin {array} {c} \ vec {A} \ cdot \ vec {A} - \ vec {P} \ cdot \ vec {P } \\ \ vec {B} \ cdot \ vec {B} - \ vec {P} \ cdot \ vec {P} \\ \ vec {C} \ cdot \ vec {C} - \ vec {P} \ cdot \ vec {P} \ end {array} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e22377929247faa7ff95ea0e91d3919bebc98b20)
Second order surfaces
Ellipsoid with the semi-axes , center at the origin, semi-axes parallel to or axis:
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d)
![x_ {1}, x_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/188263943645114e27e316cc1be787861e5b67be)
![x_ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/766d09a498699be10e276ad49145c921f8cbe335)
![\ frac {{x_1} ^ 2} {a ^ 2} + \ frac {{x_2} ^ 2} {b ^ 2} + \ frac {{x_3} ^ 2} {c ^ 2} = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/85f8779a47accc837f036e79fd33a2c59bdaa664)
Hyperboloid with semi-axes :
![ABC](https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d)
![\ frac {x_1 ^ 2} {a ^ 2} + \ frac {x_2 ^ 2} {b ^ 2} - \ frac {x_3 ^ 2} {c ^ 2} = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe5f82fba8e0d47fc1c3f88dbc6f3c7430053b5)
Paraboloid with vertex at the origin:
![\ frac {x_1 ^ 2} {a ^ 2} \ pm \ frac {x_2 ^ 2} {b ^ 2} -2z = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/e067a0efb1dc2cbb865102b6bf9874fe7a2f5ac7)
Plus provides an elliptical, minus a hyperbolic paraboloid.
Cone with semiaxes of the ellipse, point at the origin:
![from](https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8)
![\ frac {x_1 ^ 2} {a ^ 2} + \ frac {x_2 ^ 2} {b ^ 2} - \ frac {x_3 ^ 2} {c ^ 2} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ddebfd4518187cded4fce4f1f3b374d524a592b)