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

NameFormulaLegend
Distance between two points (line segment length)Show sourceAB=(BxAx)2+(ByAy)2 |AB|=\sqrt{{\left( B_x- A_x\right)}^{2}+{\left( B_y- A_y\right)}^{2}}
  • AB|AB| - the distance between points (the length of the segment beginning at point A and ending at point B),
  • AxA_x, AyA_y - coordinates of the first point,
  • BxB_x, ByB_y - coordinates of the second point.
The midpoint of the line segment: x-coordinateShow sourceCx=Ax+Bx2 C_x=\frac{ A_x+ B_x}{2}
  • CxC_x - the x-coordinate of the midpoint,
  • AxA_x - x-coordinate of the first point,
  • BxB_x - x-coordinate of the second point.
The midpoint of the line segment: y-coordinateShow sourceCy=Ay+By2 C_y=\frac{ A_y+ B_y}{2}
  • CyC_y - the y-coordinate of the midpoint,
  • AyA_y - y-coordinate of the first point,
  • ByB_y - y-coordinate of the second point.

Line segment in three-dimensional space

NameFormulaLegend
Distance between two points in three-dimensional space (line segment length)Show sourceAB=(BxAx)2+(ByAy)2+(BzAz)2 |AB|=\sqrt{{\left( B_x- A_x\right)}^{2}+{\left( B_y- A_y\right)}^{2}+{\left( B_z- A_z\right)}^{2}}
  • AB|AB| - the distance between points (the length of the segment beginning at point A and ending at point B),
  • AxA_x, AyA_y, AzA_z - coordinates of the first point,
  • BxB_x, ByB_y, BzB_z - coordinates of the second point.
The midpoint of the line segment in three-dimensional space: x-coordinateShow sourceCx=Ax+Bx2 C_x=\frac{ A_x+ B_x}{2}
  • CxC_x - the x-coordinate of the midpoint,
  • AxA_x - x-coordinate of the first point,
  • BxB_x - x-coordinate of the second point.
The midpoint of the line segment in three-dimensional space: y-coordinateShow sourceCy=Ay+By2 C_y=\frac{ A_y+ B_y}{2}
  • CyC_y - the y-coordinate of the midpoint,
  • AyA_y - y-coordinate of the first point,
  • ByB_y - y-coordinate of the second point.
The midpoint of the line segment in three-dimensional space: z-coordinateShow sourceCz=Az+Bz2 C_z=\frac{ A_z+ B_z}{2}
  • CzC_z - the z-coordinate of the midpoint,
  • AzA_z - z-coordinate of the first point,
  • BzB_z - z-coordinate of the second point.

Symmetry with respect to the x-axis

NameFormulaLegend
The symmetric point in respect to the x-axis: x coordinateShow sourcex=x x^{\prime}=- x
  • xx^{\prime} - the x-coordinate of the point image,
  • xx - x-coordinate of the original point.
The symmetric point in respect to the x-axis: y coordinateShow sourcey=y y^{\prime}= y
  • yy^{\prime} - the y-coordinate of the point image,
  • yy - y-coordinate of the original point.

Symmetry with respect to the y-axis

NameFormulaLegend
The symmetric point in respect to the y-axis: x coordinateShow sourcex=x x^{\prime}= x
  • xx^{\prime} - the x-coordinate of the point image,
  • xx - x-coordinate of the original point.
The symmetric point in respect to the y-axis: y coordinateShow sourcey=y y^{\prime}=- y
  • yy^{\prime} - the y-coordinate of the point image,
  • yy - y-coordinate of the original point.

Symmetry with the respect to the origin

NameFormulaLegend
The symmetric point in respect to the origin: x coordinateShow sourcex=x x^{\prime}=- x
  • xx^{\prime} - the x-coordinate of the point image,
  • xx - x-coordinate of the original point.
The symmetric point in respect to the origin: y coordinateShow sourcey=y y^{\prime}=- y
  • yy^{\prime} - the y-coordinate of the point image,
  • yy - y-coordinate of the original point.

2D translation (two-dimensional space)

NameFormulaLegend
Translate point by vector: x coordinateShow sourcex=x+Vx x^{\prime}= x+ V_x
  • xx^{\prime} - the x-coordinate of the point image,
  • xx - x-coordinate of the original point,
  • VxV_x - the x-coordinate of the vector.
Translate point by vector: y coordinateShow sourcey=y+Vy y^{\prime}= y+ V_y
  • yy^{\prime} - the y-coordinate of the point image,
  • yy - y-coordinate of the original point,
  • VyV_y - the y-coordinate of the vector.

3D translation (three-dimensional space)

NameFormulaLegend
Translate point by vector: x coordinateShow sourcex=x+Vx x^{\prime}= x+ V_x
  • xx^{\prime} - the x-coordinate of the point image,
  • xx - x-coordinate of the original point,
  • VxV_x - the x-coordinate of the vector.
Translate point by vector: y coordinateShow sourcey=y+Vy y^{\prime}= y+ V_y
  • yy^{\prime} - the y-coordinate of the point image,
  • yy - y-coordinate of the original point,
  • VyV_y - the y-coordinate of the vector.
Translate point by vector: z coordinateShow sourcez=z+Vz z^{\prime}= z+ V_z
  • zz^{\prime} - the z-coordinate of the point image,
  • zz - z-coordinate of the original point,
  • VzV_z - the z-coordinate of the vector.

2D rotation (two-dimensional space)

NameFormulaLegend
Rotate point around z-axis: x coordinateShow sourcex=x cos(θ)y sin(θ) x^{\prime}= x~\cos\left( \theta\right)- y~\sin\left( \theta\right)
  • xx^{\prime} - the x-coordinate of the point image,
  • xx - x-coordinate of the original point,
  • yy - y-coordinate of the original point,
  • θ\theta - rotation angle.
Rotate point around z-axis: y coordinateShow sourcey=x sin(θ)+y cos(θ) y^{\prime}= x~\sin\left( \theta\right)+ y~\cos\left( \theta\right)
  • yy^{\prime} - the y-coordinate of the point image,
  • xx - x-coordinate of the original point,
  • yy - y-coordinate of the original point,
  • θ\theta - rotation angle.

3D rotation around z-axis

NameFormulaLegend
Rotate point around z-axis: x coordinateShow sourcex=x cos(θ)y sin(θ) x^{\prime}= x~\cos\left( \theta\right)- y~\sin\left( \theta\right)
  • xx^{\prime} - the x-coordinate of the point image,
  • xx - x-coordinate of the original point,
  • yy - y-coordinate of the original point,
  • θ\theta - rotation angle.
Rotate point around z-axis: y coordinateShow sourcey=x sin(θ)+y cos(θ) y^{\prime}= x~\sin\left( \theta\right)+ y~\cos\left( \theta\right)
  • yy^{\prime} - the y-coordinate of the point image,
  • xx - x-coordinate of the original point,
  • yy - y-coordinate of the original point,
  • θ\theta - rotation angle.
Rotate point around z-axis: z coordinateShow sourcez=z z^{\prime}= z
  • zz^{\prime} - the z-coordinate of the point image,
  • zz - z-coordinate of the original point.

3D rotation around x-axis

NameFormulaLegend
Rotate point around x-axis: x coordinateShow sourcex=x x^{\prime}= x
  • xx^{\prime} - the x-coordinate of the point image,
  • xx - x-coordinate of the original point.
Rotate point around x-axis: y coordinateShow sourcey=y cos(θ)z sin(θ) y^{\prime}= y~\cos\left( \theta\right)- z~\sin\left( \theta\right)
  • yy^{\prime} - the y-coordinate of the point image,
  • yy - y-coordinate of the original point,
  • zz - z-coordinate of the original point,
  • θ\theta - rotation angle.
Rotate point around x-axis: z coordinateShow sourcez=z cos(θ)+y sin(θ) z^{\prime}= z~\cos\left( \theta\right)+ y~\sin\left( \theta\right)
  • zz^{\prime} - the z-coordinate of the point image,
  • zz - z-coordinate of the original point,
  • yy - y-coordinate of the original point,
  • θ\theta - rotation angle.

3D rotation around y-axis

NameFormulaLegend
Rotate point around y-axis: x coordinateShow sourcex=x cos(θ)z sin(θ) x^{\prime}= x~\cos\left( \theta\right)- z~\sin\left( \theta\right)
  • xx^{\prime} - the x-coordinate of the point image,
  • xx - x-coordinate of the original point,
  • zz - z-coordinate of the original point,
  • θ\theta - rotation angle.
Rotate point around y-axis: y coordinateShow sourcey=y y^{\prime}= y
  • yy^{\prime} - the y-coordinate of the point image,
  • yy - y-coordinate of the original point.
Rotate point around y-axis: z coordinateShow sourcez=z cos(θ)+x sin(θ) z^{\prime}= z~\cos\left( \theta\right)+ x~\sin\left( \theta\right)
  • zz^{\prime} - the z-coordinate of the point image,
  • zz - z-coordinate of the original point,
  • xx - x-coordinate of the original point,
  • θ\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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