Tables show typical formulas related to analytic geometry such as distance between two points.

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Line segment#

Name	Formula	Legend
Distance between two points (line segment length)	Show source $\|AB\|=\sqrt{\left(B_x-A_x\right)^{2}+\left(B_y-A_y\right)^{2}}$	$\|AB\|$ - the distance between points (the length of the segment beginning at point A and ending at point B), $A_x$ , $A_y$ - coordinates of the first point, $B_x$ , $B_y$ - coordinates of the second point.
The midpoint of the line segment: x-coordinate	Show source $C_x=\frac{A_x+B_x}{2}$	$C_x$ - the x-coordinate of the midpoint, $A_x$ - x-coordinate of the first point, $B_x$ - x-coordinate of the second point.
The midpoint of the line segment: y-coordinate	Show source $C_y=\frac{A_y+B_y}{2}$	$C_y$ - the y-coordinate of the midpoint, $A_y$ - y-coordinate of the first point, $B_y$ - y-coordinate of the second point.

Line segment in three-dimensional space#

Name	Formula	Legend
Distance between two points in three-dimensional space (line segment length)	Show source $\|AB\|=\sqrt{\left(B_x-A_x\right)^{2}+\left(B_y-A_y\right)^{2}+\left(B_z-A_z\right)^{2}}$	$\|AB\|$ - the distance between points (the length of the segment beginning at point A and ending at point B), $A_x$ , $A_y$ , $A_z$ - coordinates of the first point, $B_x$ , $B_y$ , $B_z$ - coordinates of the second point.
The midpoint of the line segment in three-dimensional space: x-coordinate	Show source $C_x=\frac{A_x+B_x}{2}$	$C_x$ - the x-coordinate of the midpoint, $A_x$ - x-coordinate of the first point, $B_x$ - x-coordinate of the second point.
The midpoint of the line segment in three-dimensional space: y-coordinate	Show source $C_y=\frac{A_y+B_y}{2}$	$C_y$ - the y-coordinate of the midpoint, $A_y$ - y-coordinate of the first point, $B_y$ - y-coordinate of the second point.
The midpoint of the line segment in three-dimensional space: z-coordinate	Show source $C_z=\frac{A_z+B_z}{2}$	$C_z$ - the z-coordinate of the midpoint, $A_z$ - z-coordinate of the first point, $B_z$ - z-coordinate of the second point.

Symmetry with respect to the x-axis#

Name	Formula	Legend
The symmetric point in respect to the x-axis: x coordinate	Show source $x^{\prime}=-x$	$x^{\prime}$ - the x-coordinate of the point image, $x$ - x-coordinate of the original point.
The symmetric point in respect to the x-axis: y coordinate	Show source $y^{\prime}=y$	$y^{\prime}$ - the y-coordinate of the point image, $y$ - y-coordinate of the original point.

Symmetry with respect to the y-axis#

Name	Formula	Legend
The symmetric point in respect to the y-axis: x coordinate	Show source $x^{\prime}=x$	$x^{\prime}$ - the x-coordinate of the point image, $x$ - x-coordinate of the original point.
The symmetric point in respect to the y-axis: y coordinate	Show source $y^{\prime}=-y$	$y^{\prime}$ - the y-coordinate of the point image, $y$ - y-coordinate of the original point.

Symmetry with the respect to the origin#

Name	Formula	Legend
The symmetric point in respect to the origin: x coordinate	Show source $x^{\prime}=-x$	$x^{\prime}$ - the x-coordinate of the point image, $x$ - x-coordinate of the original point.
The symmetric point in respect to the origin: y coordinate	Show source $y^{\prime}=-y$	$y^{\prime}$ - the y-coordinate of the point image, $y$ - y-coordinate of the original point.

2D translation (two-dimensional space)#

Name	Formula	Legend
Translate point by vector: x coordinate	Show source $x^{\prime}=x+V_x$	$x^{\prime}$ - the x-coordinate of the point image, $x$ - x-coordinate of the original point, $V_x$ - the x-coordinate of the vector.
Translate point by vector: y coordinate	Show source $y^{\prime}=y+V_y$	$y^{\prime}$ - the y-coordinate of the point image, $y$ - y-coordinate of the original point, $V_y$ - the y-coordinate of the vector.

3D translation (three-dimensional space)#

Name	Formula	Legend
Translate point by vector: x coordinate	Show source $x^{\prime}=x+V_x$	$x^{\prime}$ - the x-coordinate of the point image, $x$ - x-coordinate of the original point, $V_x$ - the x-coordinate of the vector.
Translate point by vector: y coordinate	Show source $y^{\prime}=y+V_y$	$y^{\prime}$ - the y-coordinate of the point image, $y$ - y-coordinate of the original point, $V_y$ - the y-coordinate of the vector.
Translate point by vector: z coordinate	Show source $z^{\prime}=z+V_z$	$z^{\prime}$ - the z-coordinate of the point image, $z$ - z-coordinate of the original point, $V_z$ - the z-coordinate of the vector.

2D rotation (two-dimensional space)#

Name	Formula	Legend
Rotate point around z-axis: x coordinate	Show source $x^{\prime}=x \cdot cos\left(\theta\right)-y \cdot sin\left(\theta\right)$	$x^{\prime}$ - the x-coordinate of the point image, $x$ - x-coordinate of the original point, $y$ - y-coordinate of the original point, $\theta$ - rotation angle.
Rotate point around z-axis: y coordinate	Show source $y^{\prime}=x \cdot sin\left(\theta\right)+y \cdot cos\left(\theta\right)$	$y^{\prime}$ - the y-coordinate of the point image, $x$ - x-coordinate of the original point, $y$ - y-coordinate of the original point, $\theta$ - rotation angle.

3D rotation around z-axis#

Name	Formula	Legend
Rotate point around z-axis: x coordinate	Show source $x^{\prime}=x \cdot cos\left(\theta\right)-y \cdot sin\left(\theta\right)$	$x^{\prime}$ - the x-coordinate of the point image, $x$ - x-coordinate of the original point, $y$ - y-coordinate of the original point, $\theta$ - rotation angle.
Rotate point around z-axis: y coordinate	Show source $y^{\prime}=x \cdot sin\left(\theta\right)+y \cdot cos\left(\theta\right)$	$y^{\prime}$ - the y-coordinate of the point image, $x$ - x-coordinate of the original point, $y$ - y-coordinate of the original point, $\theta$ - rotation angle.
Rotate point around z-axis: z coordinate	Show source $z^{\prime}=z$	$z^{\prime}$ - the z-coordinate of the point image, $z$ - z-coordinate of the original point.

3D rotation around x-axis#

Name	Formula	Legend
Rotate point around x-axis: x coordinate	Show source $x^{\prime}=x$	$x^{\prime}$ - the x-coordinate of the point image, $x$ - x-coordinate of the original point.
Rotate point around x-axis: y coordinate	Show source $y^{\prime}=y \cdot cos\left(\theta\right)-z \cdot sin\left(\theta\right)$	$y^{\prime}$ - the y-coordinate of the point image, $y$ - y-coordinate of the original point, $z$ - z-coordinate of the original point, $\theta$ - rotation angle.
Rotate point around x-axis: z coordinate	Show source $z^{\prime}=z \cdot cos\left(\theta\right)+y \cdot sin\left(\theta\right)$	$z^{\prime}$ - the z-coordinate of the point image, $z$ - z-coordinate of the original point, $y$ - y-coordinate of the original point, $\theta$ - rotation angle.

3D rotation around y-axis#

Name	Formula	Legend
Rotate point around y-axis: x coordinate	Show source $x^{\prime}=x \cdot cos\left(\theta\right)-z \cdot sin\left(\theta\right)$	$x^{\prime}$ - the x-coordinate of the point image, $x$ - x-coordinate of the original point, $z$ - z-coordinate of the original point, $\theta$ - rotation angle.
Rotate point around y-axis: y coordinate	Show source $y^{\prime}=y$	$y^{\prime}$ - the y-coordinate of the point image, $y$ - y-coordinate of the original point.
Rotate point around y-axis: z coordinate	Show source $z^{\prime}=z \cdot cos\left(\theta\right)+x \cdot sin\left(\theta\right)$	$z^{\prime}$ - the z-coordinate of the point image, $z$ - z-coordinate of the original point, $x$ - x-coordinate of the original point, $\theta$ - rotation angle.

Some facts#

Analytic geometry is a branch of mathematics that deals with the study of geometrical shapes by the analytical and algebraic methods.
Analytical geometry is a bridge between classical geometry and algebra.
Analytical geometry methods allow you to replace the problems known from classical geometry for equivalent problems known from algebra, e.g. into a system of equations.
If you're looking for calculator that takes the classic Euclidean geometry into account, check our other calculator: Math tables: geometry.

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