Mathematical tables: typical analytic geometry related formulas
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.