Analytical geometry formula collection

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${\displaystyle {\sqrt[{n}]{x}}}$

This article is a collection of formulas for the topic of analytic geometry . Math symbols are used, which are explained in the article List of Math Symbols .

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 . ${\displaystyle x_{1}}$ ${\displaystyle x}$ ${\displaystyle x_{2}}$ ${\displaystyle y}$ ${\displaystyle x_{3}}$ ${\displaystyle z}$ ${\displaystyle \cdot }$ ${\displaystyle \times }$

Analytical geometry of the Euclidean plane

Designations

In the following the point has the coordinates ; the items in that order ${\displaystyle P}$ ${\displaystyle (p_{1},p_{2})}$ ${\displaystyle A,B,C}$ ${\displaystyle (a_{1},a_{2}),(b_{1},b_{2}),(c_{1},c_{2})}$

Points

Points are described by Cartesian coordinates or position vectors .

Coordinate representation of a point

${\displaystyle P(p_{1}|p_{2})}$ or ${\displaystyle P(p_{1},p_{2})}$

Position vector of the point : ${\displaystyle P(p_{1}|p_{2})}$

${\displaystyle {\vec {P}}={\begin{pmatrix}p_{1}\\p_{2}\end{pmatrix}}}$

Connection vector of two points : ${\displaystyle A,B}$

${\displaystyle {\overrightarrow {AB}}={\vec {B}}-{\vec {A}}={\begin{pmatrix}b_{1}-a_{1}\\b_{2}-a_{2}\end{pmatrix}}}$

Center of the route (as position vector): ${\displaystyle AB}$

${\displaystyle {\vec {M}}={\tfrac {1}{2}}\left({\vec {A}}+{\vec {B}}\right)={\tfrac {1}{2}}{\begin{pmatrix}a_{1}+b_{1}\\a_{2}+b_{2}\end{pmatrix}}}$

Division point : The point that divides the line in proportion : ${\displaystyle AB}$ ${\displaystyle \lambda }$

${\displaystyle {\vec {T}}={\frac {1}{1+\lambda }}\left({\vec {A}}+\lambda {\vec {B}}\right)={\frac {1}{1+\lambda }}{\begin{pmatrix}a_{1}+\lambda b_{1}\\a_{2}+\lambda b_{2}\end{pmatrix}}}$

Center of gravity of a triangle : ${\displaystyle ABC}$

${\displaystyle {\vec {S}}={\tfrac {1}{3}}\left({\vec {A}}+{\vec {B}}+{\vec {C}}\right)={\tfrac {1}{3}}{\begin{pmatrix}a_{1}+b_{1}+c_{1}\\a_{2}+b_{2}+c_{2}\end{pmatrix}}}$

Straight lines

Parametric equation of the straight line (point-directional form) through the point with the direction vector : ${\displaystyle A(a_{1}|a_{2})}$ ${\displaystyle {\vec {u}}={\begin{pmatrix}u_{1}\\u_{2}\end{pmatrix}}}$

${\displaystyle {\vec {X}}={\vec {A}}+\lambda {\vec {u}}={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}+\lambda {\begin{pmatrix}u_{1}\\u_{2}\end{pmatrix}}}$

The parameter can take all real numbers as a value and must not be the zero vector. ${\displaystyle \lambda }$ ${\displaystyle {\vec {u}}}$

Parametric equation of the straight line (two-point form) through the points : ${\displaystyle A,B}$

${\displaystyle {\vec {X}}={\vec {A}}+\lambda \left({\vec {B}}-{\vec {A}}\right)={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}+\lambda {\begin{pmatrix}b_{1}-a_{1}\\b_{2}-a_{2}\end{pmatrix}}}$

The parameter can take all real numbers as value and. and must be different. ${\displaystyle \lambda }$ ${\displaystyle A}$ ${\displaystyle B}$

Normal equation of the straight line through the point with the normal vector in vector notation: ${\displaystyle A}$ ${\displaystyle {\vec {n}}={\begin{pmatrix}n_{1}\\n_{2}\end{pmatrix}}}$

${\displaystyle {\vec {n}}\cdot \left({\vec {X}}-{\vec {A}}\right)=0}$ or. ${\displaystyle {\begin{pmatrix}n_{1}\\n_{2}\end{pmatrix}}\cdot {\begin{pmatrix}x_{1}-a_{1}\\x_{2}-a_{2}\end{pmatrix}}=0}$

Coordinate equation, explicit form of the straight line with the slope through the point of the -axis: ${\displaystyle m}$ ${\displaystyle (0|t)}$ ${\displaystyle x_{2}}$

${\displaystyle \,x_{2}=mx_{1}+t}$

Restriction: The straight line must not be parallel to the axis. ${\displaystyle x_{2}}$

Coordinate equation, intercept form of the straight line through the points (on the -axis) and (on the -axis): ${\displaystyle (s|0)}$ ${\displaystyle x_{1}}$ ${\displaystyle (0|t)}$ ${\displaystyle x_{2}}$

${\displaystyle {\frac {x_{1}}{s}}+{\frac {x_{2}}{t}}=1}$

Restriction: The given points must not coincide with the origin, i. H. it must and apply. ${\displaystyle s\neq 0}$ ${\displaystyle t\neq 0}$

distances

Distance between points : ${\displaystyle A,B}$

${\displaystyle {\overline {AB}}=\left|{\vec {B}}-{\vec {A}}\right|={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}}}}$

Distance of the point from the straight line with the normal equation (see Hessian normal form ): ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle n_{1}x_{1}+n_{2}x_{2}+n_{0}=0}$

${\displaystyle d(P,g)={\frac {\left|n_{1}p_{1}+n_{2}p_{2}+n_{0}\right|}{\sqrt {{n_{1}}^{2}+{n_{2}}^{2}}}}}$

Distance between two parallel straight lines and with the normal equations or : ${\displaystyle g}$ ${\displaystyle g'}$ ${\displaystyle n_{1}x_{1}+n_{2}x_{2}+n_{0}=0}$ ${\displaystyle n_{1}x_{1}+n_{2}x_{2}+n_{0}'=0}$

${\displaystyle d(g,g')={\frac {\left|n_{0}-n_{0}'\right|}{\sqrt {{n_{1}}^{2}+{n_{2}}^{2}}}}}$

Projections

Orthogonal projection of a point onto a straight line in parametric form : ${\displaystyle B}$ ${\displaystyle g}$ ${\displaystyle {\vec {X}}={\vec {A}}+\lambda {\vec {u}}}$

${\displaystyle {\vec {P}}_{g}={\vec {A}}+{\frac {({\vec {B}}-{\vec {A}})\cdot {\vec {u}}}{{\vec {u}}\cdot {\vec {u}}}}\,{\vec {u}}}$

Orthogonal projection of a point onto a straight line in normal form : ${\displaystyle B}$ ${\displaystyle g}$ ${\displaystyle ({\vec {X}}-{\vec {A}})\cdot {\vec {n}}=0}$

${\displaystyle {\vec {P}}_{g}={\vec {B}}-{\frac {({\vec {B}}-{\vec {A}})\cdot {\vec {n}}}{{\vec {n}}\cdot {\vec {n}}}}\,{\vec {n}}}$

Parallel projection in the direction of a point onto a straight line in normal form : ${\displaystyle {\vec {v}}}$ ${\displaystyle B}$ ${\displaystyle g}$ ${\displaystyle ({\vec {X}}-{\vec {A}})\cdot {\vec {n}}=0}$

${\displaystyle {\vec {P}}_{g,{\vec {v}}}={\vec {B}}-{\frac {({\vec {B}}-{\vec {A}})\cdot {\vec {n}}}{{\vec {v}}\cdot {\vec {n}}}}\,{\vec {v}}}$

angle

Intersection angle (smaller angle) between two straight lines ${\displaystyle \epsilon }$ with the direction vectorsand(compare scalar product ): ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$

${\displaystyle \cos \epsilon ={\frac {\left|{\vec {u}}\cdot {\vec {v}}\right|}{\left|{\vec {u}}\right|\left|{\vec {v}}\right|}}={\frac {\left|u_{1}v_{1}+u_{2}v_{2}\right|}{{\sqrt {{u_{1}}^{2}+{u_{2}}^{2}}}{\sqrt {{v_{1}}^{2}+{v_{2}}^{2}}}}}}$

Surfaces

Area of ​​the triangle (see cross product ): ${\displaystyle ABC}$

${\displaystyle {\begin{aligned}F_{ABC}&={\tfrac {1}{2}}\left|{\overrightarrow {AB}}\times {\overrightarrow {AC}}\right|={\tfrac {1}{2}}\left|\left({\vec {B}}-{\vec {A}}\right)\times \left({\vec {C}}-{\vec {A}}\right)\right|\\&={\tfrac {1}{2}}\left|(a_{1}b_{2}-a_{2}b_{1})+(b_{1}c_{2}-b_{2}c_{1})+(c_{1}a_{2}-c_{2}a_{1})\right|\end{aligned}}}$

Area of ​​the polygon not overridden with the corners : ${\displaystyle P_{1}(p_{11}|p_{12}),\dotsc ,P_{n}(p_{n1}|p_{n2})}$

${\displaystyle {\begin{aligned}A={\Big |}{\tfrac {1}{2}}\cdot &\left(p_{11}p_{22}+p_{21}p_{32}+\dotsb +p_{n-1,1}p_{n2}+p_{n1}p_{12}\right.\\&-\left.p_{21}p_{12}-p_{31}p_{22}-\dotsb -p_{n1}p_{n-1,2}-p_{11}p_{n2}\right){\Big |}\end{aligned}}}$

Circles

Equation of the circle in Cartesian coordinates :

• of the unit circle

${\displaystyle {x_{1}}^{2}+{x_{2}}^{2}=1}$

• general: center in , radius ${\displaystyle (c,d)}$ ${\displaystyle r}$

${\displaystyle (x-c)^{2}+(y-d)^{2}=r^{2}\,}$

in parametric form (general):

${\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}r\,\cos t+c\\r\,\sin t+d\end{pmatrix}}}$ With ${\displaystyle 0\leq t\leq 2\pi }$

Equation of the circle through three points ${\displaystyle P_{1}(x_{1},y_{1}),P_{2}(x_{2},y_{2}),P_{3}(x_{3},y_{3})}$

${\displaystyle {\begin{vmatrix}x^{2}+y^{2}&x&y&1\\x_{1}^{2}+y_{1}^{2}&x_{1}&y_{1}&1\\x_{2}^{2}+y_{2}^{2}&x_{2}&y_{2}&1\\x_{3}^{2}+y_{3}^{2}&x_{3}&y_{3}&1\end{vmatrix}}=0}$

Equation of the circle tangent at the point ${\displaystyle B(b_{1}|b_{2})}$

• Unit circle

  ${\displaystyle \,b_{1}x_{1}+b_{2}x_{2}=1}$

• General:

  ${\displaystyle (x-c)(b_{1}-c)+(y-d)(b_{2}-d)=r^{2}\,}$

Intersection of the straight line with the circle ${\displaystyle y=mx+c}$ ${\displaystyle x^{2}+y^{2}=r^{2}}$ :

${\displaystyle x_{1,2}=-{\frac {cm}{1+m^{2}}}\pm {\frac {1}{1+m^{2}}}{\sqrt {r^{2}(1+m^{2})-c^{2}}}}$

${\displaystyle y_{1,2}={\frac {c}{1+m^{2}}}\pm {\frac {m}{1+m^{2}}}{\sqrt {r^{2}(1+m^{2})-c^{2}}}}$

Center of the circle through three points ${\displaystyle {\vec {X}}}$ that are not on a straight line: ${\displaystyle P_{1}(x_{1},y_{1}),P_{2}(x_{2},y_{2}),P_{3}(x_{3},y_{3})}$

${\displaystyle {\vec {X}}=\left({\begin{array}{c}x_{1}\\x_{2}\end{array}}\right)={\frac {1}{2}}\left({\begin{array}{ccc}x_{1}-x_{3}&y_{1}-y_{3}\\x_{2}-x_{3}&y_{2}-y_{3}\end{array}}\right)^{-1}\left({\begin{array}{c}{\vec {P}}_{1}\cdot {\vec {P}}_{1}-{\vec {P}}_{3}\cdot {\vec {P}}_{3}\\{\vec {P}}_{2}\cdot {\vec {P}}_{2}-{\vec {P}}_{3}\cdot {\vec {P}}_{3}\end{array}}\right)}$

Conic sections

Conic section	ellipse	hyperbole	parabola
properties
Definition: set of all points for which ...	the sum of the distances to the focal points is constantly equal to 2a. ${\displaystyle F_{1},F_{2}}$	the difference in the distances between the two focal points is constant 2a.	the distance to a focal point and the guide line l is constant.
Linear eccentricity	${\displaystyle {\sqrt {a^{2}-b^{2}}}}$	${\displaystyle {\sqrt {a^{2}+b^{2}}}}$	-
Coordinates
Cartesian coordinates	${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$	${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}$	${\displaystyle y^{2}=2px\,}$
Axis-parallel position ${\displaystyle M(c,d)}$	${\displaystyle {\frac {(x-c)^{2}}{a^{2}}}+{\frac {(y-d)^{2}}{b^{2}}}=1}$	${\displaystyle {\frac {(x-c)^{2}}{a^{2}}}-{\frac {(y-d)^{2}}{b^{2}}}=1}$	${\displaystyle (y-d)^{2}=2p(x-c)\,}$
Parametric shape	${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}a\,\cos t\\b\,\sin t\end{pmatrix}}}$ With ${\displaystyle 0\leq t\leq 2\pi }$	${\displaystyle x={\frac {a}{\cos(t)}};\;y=\pm b\tan(t)}$ ${\displaystyle x=\pm a\cosh(t);\;y=b\sinh(t)}$
Straight lines
Tangent in ${\displaystyle P_{1}(p_{1},p_{2})}$	${\displaystyle {\frac {xp_{1}}{a^{2}}}+{\frac {yp_{2}}{b^{2}}}=1}$	${\displaystyle {\frac {xp_{1}}{a^{2}}}-{\frac {yp_{2}}{b^{2}}}=1}$	${\displaystyle yp_{2}=p(y+p_{2})\,}$
Normal through ${\displaystyle P_{1}(p_{1},p_{2})}$	${\displaystyle y-p_{2}={\frac {a^{2}p_{2}}{b^{2}p_{1}}}(x-p_{1})}$	${\displaystyle y-p_{2}=-{\frac {a^{2}p_{2}}{b^{2}p_{1}}}(x-p_{1})}$	${\displaystyle y-p_{2}=-{\frac {p_{2}}{p}}(x-p_{1})}$
Intersection with the straight line ${\displaystyle y=mx+C}$	${\displaystyle x_{1,2}=a^{2}m\alpha \pm \beta \cdot {\sqrt {D}}}$ ${\displaystyle y_{1,2}=b^{2}\alpha \pm m\beta \cdot {\sqrt {D}}}$ ${\displaystyle \alpha :={\frac {C}{b^{2}+a^{2}m^{2}}};\beta :={\frac {ab}{b^{2}+a^{2}m^{2}}};}$ ${\displaystyle D:=a^{2}m^{2}+b^{2}-C^{2}\,}$	${\displaystyle x_{1,2}=a^{2}m\alpha \pm \beta \cdot {\sqrt {D}}}$ ${\displaystyle y_{1,2}=b^{2}\alpha \pm m\beta \cdot {\sqrt {D}}}$ ${\displaystyle \alpha :={\frac {C}{b^{2}-a^{2}m^{2}}};\beta :={\frac {ab}{b^{2}-a^{2}m^{2}}}}$ ${\displaystyle D:=b^{2}+c^{2}-a^{2}m^{2}\,}$	${\displaystyle x_{1,2}={\frac {p-Cm}{m^{2}}}\pm {\frac {1}{m^{2}}}\cdot {\sqrt {D}}}$ ${\displaystyle y_{1,2}={\frac {p}{m}}\pm {\frac {1}{m}}\cdot {\sqrt {D}}}$ ${\displaystyle D:=p\cdot (p-2mC)}$
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). ${\displaystyle a,b\in \mathbb {R} ^{+}}$ ${\displaystyle s}$

Curve	Domain of definition	analytical function equation		curvature ${\displaystyle \kappa }$	Characterization of their curvature
Straight	${\displaystyle x\in \mathbb {R} }$ ${\displaystyle s\in \mathbb {R} }$	${\displaystyle y(x)=ax}$ ${\displaystyle {\begin{pmatrix}r(s)\\\varphi (s)\end{pmatrix}}={\begin{pmatrix}s\\b\end{pmatrix}}}$	explicitly cartesian explicitly polar parametric	${\displaystyle 0}$	zero
circle	${\displaystyle \varphi \in [0,2\pi ]}$	${\displaystyle r(\varphi )=a}$	explicitly polar	${\displaystyle {\tfrac {1}{a}}}$	constant
equilateral hyperbola	${\displaystyle \varphi \in \left]-{\tfrac {\pi }{4}},{\tfrac {\pi }{4}}\right[}$	${\displaystyle r(\varphi )^{2}={\tfrac {2a^{2}}{\cos(2\varphi )}}}$	implicitly polar	${\displaystyle -{\tfrac {1}{r}}}$	inversely proportional to the signed "distance"
Lemniscates	${\displaystyle \varphi \in \left[-{\tfrac {\pi }{4}},{\tfrac {\pi }{4}}\right]}$	${\displaystyle r(\varphi )^{2}=2a^{2}\cos(2\varphi )}$	implicitly polar	${\displaystyle {\tfrac {3r}{2a^{2}}}}$	proportional to the signed "distance"
Logarithmic spiral	${\displaystyle \varphi \in \mathbb {R} }$	${\displaystyle r(\varphi )=ae^{b\varphi }}$	explicitly polar	${\displaystyle {\tfrac {1}{r{\sqrt {1+b^{2}}}}}}$ ${\displaystyle {\tfrac {1}{bs}}}$	inversely proportional to the distance inversely proportional to the arc length
Clothoid	${\displaystyle s\in \mathbb {R} }$	${\displaystyle {\begin{pmatrix}x(s)\\y(s)\end{pmatrix}}={\begin{pmatrix}C_{b}(s)\\S_{b}(s)\end{pmatrix}}\ }$	Cartesian parametric	${\displaystyle 2bs}$	proportional to their arc length
Catenoids	${\displaystyle x\in \mathbb {R} }$	${\displaystyle y(x)=a\cosh({\tfrac {x}{a}})}$ ${\displaystyle s(x)=a\sinh({\tfrac {x}{a}})}$	explicitly Cartesian	${\displaystyle {\tfrac {a}{y^{2}}}}$ ${\displaystyle {\tfrac {a}{a^{2}+s^{2}}}}$	inversely proportional to the square of their x-axis distance
Circle Volvents	${\displaystyle s\in \mathbb {R} _{0}^{+}}$	${\displaystyle {\begin{pmatrix}r(s)\\\varphi (s)\end{pmatrix}}={\begin{pmatrix}a{\sqrt {1+2s}}\\{\sqrt {2s}}-\arctan {({\sqrt {2s}})}\end{pmatrix}}\ }$	explicitly polar parametric	${\displaystyle {\tfrac {1}{a{\sqrt {2s}}}}}$	inversely proportional to the root of their arc length

Here and denote the Fresnel integrals . ${\displaystyle C_{b}(x)}$ ${\displaystyle S_{b}(x)}$

Analytical geometry of three-dimensional Euclidean space

Designations

In the following, the points have the coordinates in this order . ${\displaystyle X,P,A,B,C}$ ${\displaystyle (x_{1},x_{2},x_{3}),(p_{1},p_{2},p_{3}),(a_{1},a_{2},a_{3}),(b_{1},b_{2},b_{2}),(c_{1},c_{2},c_{3})}$

Points

Points are described by Cartesian coordinates or position vectors .

Coordinate representation

${\displaystyle P(p_{1}|p_{2}|p_{3})}$

Position vector

${\displaystyle {\vec {P}}={\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\end{pmatrix}}}$

Connection vector of two points : ${\displaystyle AB}$

${\displaystyle {\overrightarrow {AB}}={\vec {B}}-{\vec {A}}={\begin{pmatrix}b_{1}-a_{1}\\b_{2}-a_{2}\\b_{3}-a_{3}\end{pmatrix}}}$

Midpoint of the route : ${\displaystyle AB}$

${\displaystyle {\vec {M}}={\tfrac {1}{2}}\left({\vec {A}}+{\vec {B}}\right)={\tfrac {1}{2}}{\begin{pmatrix}a_{1}+b_{1}\\a_{2}+b_{2}\\a_{3}+b_{3}\end{pmatrix}}}$

Division point that divides the distance in proportion : ${\displaystyle AB}$ ${\displaystyle \lambda }$

${\displaystyle {\vec {T}}={\frac {1}{1+\lambda }}\left({\vec {A}}+\lambda {\vec {B}}\right)={\frac {1}{1+\lambda }}{\begin{pmatrix}a_{1}+\lambda b_{1}\\a_{2}+\lambda b_{2}\\a_{3}+\lambda b_{3}\end{pmatrix}}}$

Center of gravity of a triangle with the corners : ${\displaystyle A,B,C}$

${\displaystyle {\vec {S}}={\tfrac {1}{3}}\left({\vec {A}}+{\vec {B}}+{\vec {C}}\right)={\tfrac {1}{3}}{\begin{pmatrix}a_{1}+b_{1}+c_{1}\\a_{2}+b_{2}+c_{2}\\a_{3}+b_{3}+c_{3}\end{pmatrix}}}$

Straight lines

Parametric equation of a straight line (point-directional form) through the point with the direction vector : ${\displaystyle A}$ ${\displaystyle {\vec {u}}={\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}}$

${\displaystyle {\vec {X}}={\vec {A}}+\lambda {\vec {u}}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}+\lambda {\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}}$

The parameter can take all real numbers as a value and must not be the zero vector. ${\displaystyle \lambda }$ ${\displaystyle {\vec {u}}}$

Levels

Parametric equation of the plane (point-direction form) through the point with the direction vectors and : ${\displaystyle A}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {X}}={\vec {A}}+\lambda {\vec {u}}+\mu {\vec {v}}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}+\lambda {\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}+\mu {\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}}$

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 ) ${\displaystyle \lambda }$ ${\displaystyle \mu }$ ${\displaystyle {\vec {u}},{\vec {v}}}$ ${\displaystyle {\vec {u}},{\vec {v}}\neq 0}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$

Parametric equation of a plane (three-point form) through the points : ${\displaystyle A,B,C}$

${\displaystyle {\vec {X}}={\vec {A}}+\lambda \left({\vec {B}}-{\vec {A}}\right)+\mu \left({\vec {C}}-{\vec {A}}\right)={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}+\lambda {\begin{pmatrix}b_{1}-a_{1}\\b_{2}-a_{2}\\b_{3}-a_{3}\end{pmatrix}}+\mu {\begin{pmatrix}c_{1}-a_{1}\\c_{2}-a_{2}\\c_{3}-a_{3}\end{pmatrix}}}$

The two parameters and can take all real numbers as values ​​and the given points and must not lie on a straight line. ${\displaystyle \lambda }$ ${\displaystyle \mu }$ ${\displaystyle A,B}$ ${\displaystyle C}$

Normal equation of the plane through the point with the normal vector in vector notation: ${\displaystyle A}$ ${\displaystyle {\vec {n}}={\begin{pmatrix}n_{1}\\n_{2}\\n_{3}\end{pmatrix}}\neq 0}$

${\displaystyle {\vec {n}}\cdot \left({\vec {X}}-{\vec {A}}\right)=0}$ or. ${\displaystyle {\begin{pmatrix}n_{1}\\n_{2}\\n_{3}\end{pmatrix}}\cdot {\begin{pmatrix}x_{1}-a_{1}\\x_{2}-a_{2}\\x_{3}-a_{3}\end{pmatrix}}=0}$

Coordinate equation

${\displaystyle {\begin{pmatrix}a&b&c\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}=ax_{1}+bx_{2}+cx_{3}=d}$with not all equal to 0. ${\displaystyle a,b,c}$

Conversion of the forms into one another

• Parametric form in normal form:

  ${\displaystyle {\vec {n}}={\vec {u}}\times {\vec {v}}}$

• Normal form and coordinate equation:

  The normal form is the same as the coordinate equation, just written a little differently. Explicit: and . ${\displaystyle a=n_{1},b=n_{2},c=n_{3}}$ ${\displaystyle d=n_{1}a_{1}+n_{2}a_{2}+n_{3}a_{3}}$

• From the parametric form to the coordinate equation:

  ${\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\vec {X}}={\vec {A}}+\lambda {\vec {u}}+\mu {\vec {v}}}$defines three equations; solve one of them and another and plug this into the remaining equation. ${\displaystyle \lambda }$ ${\displaystyle \mu }$

• 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 ${\displaystyle a,b,c}$ ${\displaystyle c\neq 0}$ ${\displaystyle x_{3}(x_{1},x_{2})={\tfrac {1}{c}}\left(d-ax_{1}-bx_{2}\right)}$ ${\displaystyle (x_{1},x_{2})=(0,0)}$ ${\displaystyle (x_{1},x_{2})=(1,0)}$ ${\displaystyle (x_{1},x_{2})=(0,1)}$

  ${\displaystyle {\vec {A}}={\begin{pmatrix}0\\0\\x_{3}(0,0)\end{pmatrix}}}$, and ${\displaystyle {\vec {B}}={\begin{pmatrix}1\\0\\x_{3}(1,0)\end{pmatrix}}}$ ${\displaystyle {\vec {C}}={\begin{pmatrix}0\\1\\x_{3}(0,1)\end{pmatrix}}}$

  into the three-point form of the parametric equation.

distances

Distance between the points ${\displaystyle A,B}$

${\displaystyle \left\vert {\overrightarrow {AB}}\right\vert =\left|{\vec {B}}-{\vec {A}}\right|={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}+(b_{3}-a_{3})^{2}}}}$

Distance of the point from the straight line ${\displaystyle P}$ ${\displaystyle g}$ in parametric form : ${\displaystyle {\vec {X}}={\vec {A}}+\lambda {\vec {u}}}$

${\displaystyle d(P,g)={\frac {|({\vec {P}}-{\vec {A}})\times {\vec {u}}|}{|{\vec {u}}|}}}$

Distance of the point from the plane ${\displaystyle P}$ ${\displaystyle \epsilon }$ with the normal equation (see Hessian normal form ): ${\displaystyle n_{1}x_{1}+n_{2}x_{2}+n_{3}x_{3}+n_{0}=0}$

${\displaystyle d(P,\epsilon )={\frac {\left|n_{1}p_{1}+n_{2}p_{2}+n_{3}p_{3}+n_{0}\right|}{\sqrt {{n_{1}}^{2}+{n_{2}}^{2}+{n_{3}}^{2}}}}}$

Distance of the point from the plane ${\displaystyle P}$ ${\displaystyle \epsilon }$ in parametric form : ${\displaystyle {\vec {X}}={\vec {A}}+\lambda {\vec {u}}+\mu {\vec {v}}}$

${\displaystyle d(P,\epsilon )={\frac {|({\vec {P}}-{\vec {A}})\cdot ({\vec {u}}\times {\vec {v}})|}{|{\vec {u}}\times {\vec {v}}|}}}$

Distance of the parallel planes and with the normal equations or : ${\displaystyle \epsilon }$ ${\displaystyle \epsilon '}$ ${\displaystyle n_{1}x_{1}+n_{2}x_{2}+n_{3}x_{3}+n_{0}=0}$ ${\displaystyle n_{1}x_{1}+n_{2}x_{2}+n_{3}x_{3}+n_{0}'=0}$

${\displaystyle d(\epsilon ,\epsilon ')={\frac {\left|n_{0}-n_{0}'\right|}{\sqrt {{n_{1}}^{2}+{n_{2}}^{2}+{n_{3}}^{2}}}}}$

Projections

Orthogonal projection of a point onto a straight line in parametric form : ${\displaystyle B}$ ${\displaystyle g}$ ${\displaystyle {\vec {X}}={\vec {A}}+\lambda {\vec {u}}}$

${\displaystyle {\vec {P}}_{g}={\vec {A}}+{\frac {({\vec {B}}-{\vec {A}})\cdot {\vec {u}}}{{\vec {u}}\cdot {\vec {u}}}}\,{\vec {u}}}$

Orthogonal projection of a point onto a plane in normal form : ${\displaystyle B}$ ${\displaystyle \epsilon }$ ${\displaystyle ({\vec {X}}-{\vec {A}})\cdot {\vec {n}}=0}$

${\displaystyle {\vec {P}}_{\epsilon }={\vec {B}}-{\frac {({\vec {B}}-{\vec {A}})\cdot {\vec {n}}}{{\vec {n}}\cdot {\vec {n}}}}\,{\vec {n}}}$

Parallel projection in the direction of a point onto a plane in normal form : ${\displaystyle {\vec {v}}}$ ${\displaystyle B}$ ${\displaystyle \epsilon }$ ${\displaystyle ({\vec {X}}-{\vec {A}})\cdot {\vec {n}}=0}$

${\displaystyle {\vec {P}}_{\epsilon ,{\vec {v}}}={\vec {B}}-{\frac {({\vec {B}}-{\vec {A}})\cdot {\vec {n}}}{{\vec {v}}\cdot {\vec {n}}}}\,{\vec {v}}}$

angle

Intersection angle (smaller angle) between two straight lines ${\displaystyle \epsilon }$ with the direction vectors and : ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$

${\displaystyle \cos \epsilon ={\frac {\left|{\vec {u}}\cdot {\vec {v}}\right|}{\left|{\vec {u}}\right|\left|{\vec {v}}\right|}}={\frac {\left|u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}\right|}{{\sqrt {{u_{1}}^{2}+{u_{2}}^{2}+{u_{3}}^{2}}}{\sqrt {{v_{1}}^{2}+{v_{2}}^{2}+{v_{3}}^{2}}}}}}$

Intersection angle between a plane ${\displaystyle \epsilon }$ with the normal vector and a straight line with the direction vector : ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {u}}}$

${\displaystyle \sin \epsilon ={\frac {\left|{\vec {n}}\cdot {\vec {u}}\right|}{\left|{\vec {n}}\right|\left|{\vec {u}}\right|}}={\frac {\left|n_{1}u_{1}+n_{2}u_{2}+n_{3}u_{3}\right|}{{\sqrt {{n_{1}}^{2}+{n_{2}}^{2}+{n_{3}}^{2}}}{\sqrt {{u_{1}}^{2}+{u_{2}}^{2}+{u_{3}}^{2}}}}}}$

Intersection angle between two planes ${\displaystyle \epsilon }$ with the normal vectors and : ${\displaystyle {\vec {m}}}$ ${\displaystyle {\vec {n}}}$

${\displaystyle \cos \epsilon ={\frac {\left|{\vec {m}}\cdot {\vec {n}}\right|}{\left|{\vec {m}}\right|\left|{\vec {n}}\right|}}={\frac {\left|m_{1}n_{1}+m_{2}n_{2}+m_{3}n_{3}\right|}{{\sqrt {{m_{1}}^{2}+{m_{2}}^{2}+{m_{3}}^{2}}}{\sqrt {{n_{1}}^{2}+{n_{2}}^{2}+{n_{3}}^{2}}}}}}$

Volumes

Volume of the tetrahedron (compare Spat product ): ( ) ${\displaystyle P_{0}P_{1}P_{2}P_{3}}$ ${\displaystyle {\vec {a}}:={\overrightarrow {P_{0}P_{1}}}\ ,\ {\vec {b}}:={\overrightarrow {P_{0}P_{2}}}\ ,\ {\vec {c}}:={\overrightarrow {P_{0}P_{3}}}}$

${\displaystyle V={\Big |}{\frac {1}{6}}[{\vec {a}},{\vec {b}},{\vec {c}}]{\Big |}={\Big |}{\frac {1}{6}}{\begin{vmatrix}a_{x}&a_{y}&a_{z}\\b_{x}&b_{y}&b_{z}\\c_{x}&c_{y}&c_{z}\end{vmatrix}}{\Big |}}$

Bullets

Cartesian coordinates

• Unit sphere:

  ${\displaystyle {x_{1}}^{2}+{x_{2}}^{2}+{x_{3}}^{2}=1}$

• General: (center: ) ${\displaystyle (a,b,c)}$

  ${\displaystyle (x_{1}-a)^{2}+(x_{2}-b)^{2}+(x_{3}-c)^{2}=r^{2}}$

Parametric form (in the origin)

${\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}r\,\sin \vartheta \cos \varphi \\r\,\sin \vartheta \sin \varphi \\r\cos \vartheta \end{pmatrix}}}$with and ${\displaystyle 0\leq \vartheta \leq \pi }$ ${\displaystyle 0\leq \varphi \leq 2\pi }$

Center of the sphere through four points ${\displaystyle {\vec {X}}}$ and , which are not in a plane: ${\displaystyle {\vec {A}},{\vec {B}},{\vec {C}}}$ ${\displaystyle {\vec {P}}}$

${\displaystyle {\vec {X}}=\left({\begin{array}{c}x_{1}\\x_{2}\\x_{3}\end{array}}\right)={\frac {1}{2}}\left({\begin{array}{ccc}a_{1}-p_{1}&a_{2}-p_{2}&a_{3}-p_{3}\\b_{1}-p_{1}&b_{2}-p_{2}&b_{3}-p_{3}\\c_{1}-p_{1}&c_{2}-p_{2}&c_{3}-p_{3}\end{array}}\right)^{-1}\left({\begin{array}{c}{\vec {A}}\cdot {\vec {A}}-{\vec {P}}\cdot {\vec {P}}\\{\vec {B}}\cdot {\vec {B}}-{\vec {P}}\cdot {\vec {P}}\\{\vec {C}}\cdot {\vec {C}}-{\vec {P}}\cdot {\vec {P}}\end{array}}\right)}$

Second order surfaces

Ellipsoid with the semi-axes , center at the origin, semi-axes parallel to or axis: ${\displaystyle a,b,c}$ ${\displaystyle x_{1},x_{2}}$ ${\displaystyle x_{3}}$

${\displaystyle {\frac {{x_{1}}^{2}}{a^{2}}}+{\frac {{x_{2}}^{2}}{b^{2}}}+{\frac {{x_{3}}^{2}}{c^{2}}}=1}$

Hyperboloid with semi-axes : ${\displaystyle a,b,c}$

${\displaystyle {\frac {x_{1}^{2}}{a^{2}}}+{\frac {x_{2}^{2}}{b^{2}}}-{\frac {x_{3}^{2}}{c^{2}}}=1}$

Paraboloid with vertex at the origin:

${\displaystyle {\frac {x_{1}^{2}}{a^{2}}}\pm {\frac {x_{2}^{2}}{b^{2}}}-2z=0}$

Plus provides an elliptical, minus a hyperbolic paraboloid.

Cone with semiaxes of the ellipse, point at the origin: ${\displaystyle a,b}$

${\displaystyle {\frac {x_{1}^{2}}{a^{2}}}+{\frac {x_{2}^{2}}{b^{2}}}-{\frac {x_{3}^{2}}{c^{2}}}=0}$