Reflection

Reflection is one of the non deformative transformation that any solid object can be subjected to. It involves determining the mirror image of the object accross a point or a line in 2D and accross a point, a line or a plane or in 3D. In this transformation any changes applied to a coordinate point can be directly applied to the equations of objects as well.

Reflections in 2D

Reflections in 2D can happen in the following ways:

  1. Reflection with respect to coordinate axes or the origin: These are the simplest form of reflection that can happen in 2D. Following summerises these.
    Reflection TypeEquation FormMatrix Form
    Accross x axis \(x' = x\)
    \(y' = -y \)
    \(\begin{bmatrix} 1 & 0 \\0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} 1 & 0 \\0 & -1 \end{bmatrix} = \begin{bmatrix} x' & y' \end{bmatrix} \)
    Accross y axis \(x' = -x\)
    \(y' = y \)
    \(\begin{bmatrix} -1 & 0 \\0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} -1 & 0 \\0 & 1 \end{bmatrix} = \begin{bmatrix} x' & y' \end{bmatrix} \)
    Accross origin (0,0)
    (Similar to rotation by 180° accross origin (0,0))
    \(x' = -x\)
    \(y' = -y \)
    \(\begin{bmatrix} -1 & 0 \\0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} -1 & 0 \\0 & -1 \end{bmatrix} = \begin{bmatrix} x' & y' \end{bmatrix} \)
  2. Reflection with respect to lines \(x=constant\) or \(y=constant\) or arbitrary point \( (o_x,o_y) \): Following summerises these.
    Reflection TypeEquation FormMatrix Form
    Accross line x=c
    (c=constant)
    \(x' = 2c - x \)
    \(y' = y \)
    \(\begin{bmatrix} - 1 & 0 & 2c \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & 1 \end{bmatrix} \begin{bmatrix} -1 & 0 & 0\\0 & 1 & 0 \\ 2c & 0 & 1 \end{bmatrix} = \begin{bmatrix} x' & y' & 1\end{bmatrix} \)
    Accross line y=c
    (c=constant)
    \(x' = x\)
    \(y' = 2c-y \)
    \(\begin{bmatrix} 1 & 0 & 0 \\0 & -1 & 2c \\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\0 & -1 & 0 \\ 0 & 2c & 1 \end{bmatrix} = \begin{bmatrix} x' & y' & 1 \end{bmatrix} \)
    Accross point \( (o_x,o_y) \)
    (Similar to rotation by 180° accross point \( (o_x,o_y) \))
    \(x' = 2o_x - x\)
    \(y' = 2o_y - y \)
    \(\begin{bmatrix} -1 & 0 & 2o_x \\0 & -1 & 2o_y \\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & 1 \end{bmatrix} \begin{bmatrix} -1 & 0 & 0\\0 & -1 & 0 \\ 2o_x & 2o_y & 1 \end{bmatrix} = \begin{bmatrix} x' & y' & 1 \end{bmatrix} \)
    Please note that when \( c, o_x,o_y \) are 0 these equations are similar to reflection with respect to coordinate axes or the origin
  3. Reflection with respect to line \(y=mx\) making and angle \(\theta\) with positive direction of x axis:
    Reflection Based OnEquation FormMatrix Form
    Angle \(\theta\) \(x' = x cos 2\theta + y sin 2\theta \)
    \(y' = x sin 2\theta - y cos 2\theta \)
    \(\begin{bmatrix} cos 2\theta & sin 2\theta \\sin 2\theta & -cos 2\theta\end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} x' \\ y'\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y\end{bmatrix} \begin{bmatrix} cos 2\theta & sin 2\theta \\sin 2\theta & -cos 2\theta\end{bmatrix} = \begin{bmatrix} x' & y'\end{bmatrix} \)
    Slope m (m=\(\tan\theta\)) \(x' = x (\frac{1-m^2}{1+m^2}) + y (\frac{2m}{1+m^2}) \)
    \(y' = x (\frac{2m}{1+m^2}) - y (\frac{1-m^2}{1+m^2}) \)
    \(\begin{bmatrix} \frac{1-m^2}{1+m^2} & \frac{2m}{1+m^2} \\\frac{2m}{1+m^2} & -\frac{1-m^2}{1+m^2}\end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} x' \\ y'\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y\end{bmatrix} \begin{bmatrix} \frac{1-m^2}{1+m^2} & \frac{2m}{1+m^2} \\\frac{2m}{1+m^2} & -\frac{1-m^2}{1+m^2}\end{bmatrix} = \begin{bmatrix} x' & y'\end{bmatrix} \)
  4. Reflection with respect to line \(ax + by=0\):
    Reflection Based OnEquation FormMatrix Form
    When a and b are not unit vectors i.e. \( a^2 + b^2 \neq 1\) \(x' = x (\frac{b^2-a^2}{a^2+b^2}) + y (\frac{-2ab}{a^2+b^2}) \)
    \(y' = x (\frac{-2ab}{a^2+b^2}) + y (\frac{a^2-b^2}{a^2+b^2}) \)
    \(\begin{bmatrix} \frac{b^2-a^2}{a^2+b^2} & \frac{-2ab}{a^2+b^2} \\\frac{-2ab}{a^2+b^2} & \frac{a^2-b^2}{a^2+b^2} \end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} x' \\ y'\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y\end{bmatrix} \begin{bmatrix} \frac{b^2-a^2}{a^2+b^2} & \frac{-2ab}{a^2+b^2} \\\frac{-2ab}{a^2+b^2} & \frac{a^2-b^2}{a^2+b^2} \end{bmatrix} = \begin{bmatrix} x' & y'\end{bmatrix} \)
    When a and b are unit vectors i.e. \( a^2 + b^2 =1\) \(x' = x (1-2a^2) + y (-2ab) \)
    \(y' = x (-2ab) + y (1-2b^2) \)
    \(\begin{bmatrix} 1-2a^2 & -2ab \\-2ab & 1-2b^2 \end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} x' \\ y'\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y\end{bmatrix} \begin{bmatrix} 1-2a^2 & -2ab \\-2ab & 1-2b^2 \end{bmatrix} = \begin{bmatrix} x' & y'\end{bmatrix} \)
    Please note that when a and b are unit vectors i.e. \( a^2 + b^2 =1\), then \(1-2a^2 = b^2-a^2\) and \(1-2b^2 = a^2-b^2\)
  5. Reflection with respect to line \(ax=by\):
    Reflection Based OnEquation FormMatrix Form
    When a and b are not unit vectors i.e. \( a^2 + b^2 \neq 1\) \(x' = x (\frac{b^2-a^2}{a^2+b^2}) + y (\frac{2ab}{a^2+b^2}) \)
    \(y' = x (\frac{2ab}{a^2+b^2}) + y (\frac{a^2-b^2}{a^2+b^2}) \)
    \(\begin{bmatrix} \frac{b^2-a^2}{a^2+b^2} & \frac{2ab}{a^2+b^2} \\\frac{2ab}{a^2+b^2} & \frac{a^2-b^2}{a^2+b^2} \end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} x' \\ y'\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y\end{bmatrix} \begin{bmatrix} \frac{b^2-a^2}{a^2+b^2} & \frac{2ab}{a^2+b^2} \\\frac{2ab}{a^2+b^2} & \frac{a^2-b^2}{a^2+b^2} \end{bmatrix} = \begin{bmatrix} x' & y'\end{bmatrix} \)
    When a and b are unit vectors i.e. \( a^2 + b^2 =1\) \(x' = x (1-2a^2) + y (2ab) \)
    \(y' = x (2ab) + y (1-2b^2) \)
    \(\begin{bmatrix} 1-2a^2 & 2ab \\2ab & 1-2b^2 \end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} x' \\ y'\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y\end{bmatrix} \begin{bmatrix} 1-2a^2 & 2ab \\2ab & 1-2b^2 \end{bmatrix} = \begin{bmatrix} x' & y'\end{bmatrix} \)
    Please note that when a and b are unit vectors i.e. \( a^2 + b^2 =1\), then \(1-2a^2 = b^2-a^2\) and \(1-2b^2 = a^2-b^2\)
  6. Reflection with respect to line making and angle \(\theta\) with positive direction of x axis with y intercept=\(c_y\) and x intercept=\(c_x\):
    Reflection Based OnEquation FormMatrix Form
    Shifting origin to \((c_x,0)\) \(x' = (x-c_x) cos 2\theta + y sin 2\theta + c_x \)
    \(y' = (x-c_x) sin 2\theta - y cos 2\theta \)
    \(\begin{bmatrix} cos 2\theta & sin 2\theta & -c_x cos 2\theta + cx\\sin 2\theta & -cos 2\theta & -c_x sin 2\theta\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & 1\end{bmatrix} \begin{bmatrix} cos 2\theta & sin 2\theta & 0\\sin 2\theta & -cos 2\theta & 0\\-c_x cos 2\theta + cx & -c_x sin 2\theta & 1\end{bmatrix} = \begin{bmatrix} x' & y' & 1\end{bmatrix} \)
    Shifting origin to \((0,c_y)\) \(x' = x cos 2\theta + (y-c_y) sin 2\theta\)
    \(y' = x sin 2\theta - (y-c_y) cos 2\theta + c_y \)
    \(\begin{bmatrix} cos 2\theta & sin 2\theta & -c_y sin 2\theta\\sin 2\theta & -cos 2\theta & c_y cos 2\theta + c_y\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & 1\end{bmatrix} \begin{bmatrix} cos 2\theta & sin 2\theta & 0\\sin 2\theta & -cos 2\theta & 0\\-c_y sin 2\theta & c_y cos 2\theta + c_y & 1\end{bmatrix} = \begin{bmatrix} x' & y' & 1\end{bmatrix} \)
  7. Reflection with respect to line making and angle \(\theta\) with positive direction of x axis at a distance c from the origin:
    Equation FormMatrix Form
    \(x' = x cos 2\theta + y sin 2\theta - 2c sin\theta \)
    \(y' = x sin 2\theta - y cos 2\theta + 2c cos\theta \)
    \(\begin{bmatrix} cos 2\theta & sin 2\theta & -2c sin \theta\\sin 2\theta & -cos 2\theta & 2c cos \theta\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & 1\end{bmatrix} \begin{bmatrix} cos 2\theta & sin 2\theta & 0\\sin 2\theta & -cos 2\theta & 0\\ -2c sin \theta & 2c cos \theta & 1\end{bmatrix} = \begin{bmatrix} x' & y' & 1\end{bmatrix} \)
  8. Reflection with respect to line \(ax+by+c=0\) when a and b are unit vectors i.e. \( a^2 + b^2 =1\):
    Equation FormMatrix Form
    \(x' = x (1-2a^2) + y (-2ab) - 2ac \)
    \(y' = x (-2ab) + y (1-2b^2) - 2bc\)
    \(\begin{bmatrix} 1-2a^2 & -2ab & -2ac\\-2ab & 1-2b^2 & -2bc \\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & 1\end{bmatrix} \begin{bmatrix} 1-2a^2 & -2ab & 0\\-2ab & 1-2b^2 & 0\\ -2ac & -2bc & 1\end{bmatrix} = \begin{bmatrix} x' & y' & 1\end{bmatrix} \)
    Please note that since a and b are unit vectors i.e. \( a^2 + b^2 =1\), therefore \(1-2a^2 = b^2-a^2\) and \(1-2b^2 = a^2-b^2\)
  9. Reflection with respect to line \(by=ax + c\) when a and b are unit vectors i.e. \( a^2 + b^2 =1\):
    Equation FormMatrix Form
    \(x' = x (1-2a^2) + y (2ab) - 2ac \)
    \(y' = x (2ab) + y (1-2b^2) + 2bc\)
    \(\begin{bmatrix} 1-2a^2 & 2ab & -2ac\\2ab & 1-2b^2 & 2bc \\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & 1\end{bmatrix} \begin{bmatrix} 1-2a^2 & 2ab & 0\\2ab & 1-2b^2 & 0\\ -2ac & 2bc & 1\end{bmatrix} = \begin{bmatrix} x' & y' & 1\end{bmatrix} \)
    Please note that since a and b are unit vectors i.e. \( a^2 + b^2 =1\), therefore \(1-2a^2 = b^2-a^2\) and \(1-2b^2 = a^2-b^2\)
  10. Reflection with respect to any line \(ax + by + c=0\):
    Equation FormMatrix Form
    \(x' = x - 2D \hat{a} \)
    \(y' = y - 2D \hat{b} \)
    \( \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix} - 2D \begin{bmatrix} \hat{a} \\ \hat{b}\end{bmatrix} \)
    Please note that \(\hat{a},\hat{b}\) are unit vectors corresponding to coefficients a and b and D is the distance from point to line.
  11. Reflection of a point C with respect to any line having direction ratio \(\vec{A}\) making a vector \(\vec{B}\) with any point on the line:
    When \(\vec{B} \perp \vec{A}\) \( \vec{P}= \vec{C} - 2 \vec{B}\)
    When \(\vec{B}\) not \(\perp \vec{A}\) \( \vec{P}= \vec{C} - 2 (\frac{\vec{A} \times (\vec{B} \times \vec{A})}{\vert \vec{A} \vert ^2})\)
    Please note that \(\vec{C}\) is the position vector of point C and \(\vec{P}\) is position vector of reflected point.
The reflection types 3,4 and 5 are different ways of representing reflection accross a line passing through origin which is not parallel or perpendicular to any coordinate axes.

The reflection types 6,7,8 and 9 are different ways of representing reflection accross a line not passing through origin which is not parallel or perpendicular to any coordinate axes.

Reflections in 3D

Reflections in 3D can happen in the following ways:

  1. Reflection with respect to a coordinate plane or a coordinate axis (i.e. 2 coordinate planes) or with respect to origin (0,0,0) (3 coordinate planes): These are the simplest form of reflection that can happen in 3D. Following summerises these.
    Reflection TypeEquation FormMatrix Form
    Accross xy plane \(x' = x\)
    \(y' = y \)
    \(z' = -z \)
    \(\begin{bmatrix} 1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & -1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z\end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -1\end{bmatrix} = \begin{bmatrix} x' & y' & z' \end{bmatrix} \)
    Accross yz plane \(x' = -x\)
    \(y' = y \)
    \(z' = z \)
    \(\begin{bmatrix} -1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} -1 & 0 & 0 \\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} = \begin{bmatrix} x' & y' & z' \end{bmatrix} \)
    Accross zx plane \(x' = x\)
    \(y' = -y \)
    \(z' = z \)
    \(\begin{bmatrix} 1 & 0 & 0\\0 & -1 & 0\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\0 & -1 & 0\\0 & 0 & 1\end{bmatrix} = \begin{bmatrix} x' & y' & z' \end{bmatrix} \)
    Accross xy & yz plane (y Axis) \(x' = -x\)
    \(y' = y \)
    \(z' = -z \)
    \(\begin{bmatrix} -1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & -1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z\end{bmatrix} \begin{bmatrix} -1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -1\end{bmatrix} = \begin{bmatrix} x' & y' & z' \end{bmatrix} \)
    Accross yz & zx plane (z Axis) \(x' = -x\)
    \(y' = -y \)
    \(z' = z \)
    \(\begin{bmatrix} -1 & 0 & 0\\0 & -1 & 0\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} -1 & 0 & 0 \\0 & -1 & 0\\0 & 0 & 1\end{bmatrix} = \begin{bmatrix} x' & y' & z' \end{bmatrix} \)
    Accross xy & zx plane (x Axis) \(x' = x\)
    \(y' = -y \)
    \(z' = -z \)
    \(\begin{bmatrix} 1 & 0 & 0\\0 & -1 & 0\\0 & 0 & -1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\0 & -1 & 0\\0 & 0 & -1\end{bmatrix} = \begin{bmatrix} x' & y' & z' \end{bmatrix} \)
    Accross origin (0,0,0) (all three coordinate planes) \(x' = -x\)
    \(y' = -y \)
    \(z' = -z \)
    \(\begin{bmatrix} -1 & 0 & 0\\0 & -1 & 0\\0 & 0 & -1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} -1 & 0 & 0 \\0 & -1 & 0\\0 & 0 & -1\end{bmatrix} = \begin{bmatrix} x' & y' & z' \end{bmatrix} \)
  2. Reflection with respect to planes \(x=constant\) or \(y=constant\) or \(z=constant\) or arbitrary point \( (o_x,o_y,o_z) \): Following summerises these.
    Reflection TypeEquation FormMatrix Form
    Accross plane x=c
    (c=constant)
    \(x' = 2c - x \)
    \(y' = y \)
    \(z' = z \)
    \(\begin{bmatrix} - 1 & 0 & 0 & 2c \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z & 1 \end{bmatrix} \begin{bmatrix} -1 & 0 & 0 & 0\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 2c & 0 & 0 & 1\end{bmatrix} = \begin{bmatrix} x' & y' & z' & 1\end{bmatrix} \)
    Accross plane y=c
    (c=constant)
    \(x' = x\)
    \(y' = 2c-y \)
    \(z' = z \)
    \(\begin{bmatrix} 1 & 0 & 0 & 0\\0 & -1 & 0 & 2c \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0\\0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 2c & 0 & 1\end{bmatrix} = \begin{bmatrix} x' & y' & z' & 1\end{bmatrix} \)
    Accross plane z=c
    (c=constant)
    \(x' = x\)
    \(y' = y \)
    \(z' = 2c-z \)
    \(\begin{bmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 2c \\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0\\0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 2c & 1\end{bmatrix} = \begin{bmatrix} x' & y' & z' & 1\end{bmatrix} \)
    Accross point \( (o_x,o_y,o_z) \) \(x' = 2o_x - x\)
    \(y' = 2o_y - y \)
    \(z' = 2o_z - z \)
    \(\begin{bmatrix} -1 & 0 & 0 & 2o_x \\0 & -1 & 0 & 2o_y \\0 & 0 & -1 & 2o_z \\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z & 1 \end{bmatrix} \begin{bmatrix} -1 & 0 & 0 & 0\\0 & -1 & 0 & 0 \\0 & 0 & -1 & 0\\ 2o_x & 2o_y & 2o_z & 1 \end{bmatrix} = \begin{bmatrix} x' & y' & z' & 1 \end{bmatrix} \)
    Please note that when \( c, o_x,o_y,o_z \) are 0 these equations are similar to reflection with respect to coordinate planes or the origin
  3. Reflection with respect to plane \(ax + by + cz=0\) (i.e. a plane passing through origin) when a , b and c are unit vectors i.e. \( a^2 + b^2 + c^2 =1\):
    Equation FormMatrix Form
    \(x' = x (1-2a^2) + y (-2ab) + z (-2ac)\)
    \(y' = x (-2ab) + y (1-2b^2) + z(-2bc)\)
    \(z' = x (-2ac) + y (-2bc) + z(1-2c^2)\)
    \(\begin{bmatrix} 1-2a^2 & -2ab & -2ac\\-2ab & 1-2b^2 & -2bc \\-2ac & -2bc & 1-2c^2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z'\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z\end{bmatrix} \begin{bmatrix} 1-2a^2 & -2ab & -2ac\\-2ab & 1-2b^2 & -2bc \\-2ac & -2bc & 1-2c^2 \end{bmatrix} = \begin{bmatrix} x' & y' & z'\end{bmatrix} \)
  4. Reflection with respect to plane \(ax + by + cz +d=0\) (i.e. a plane not passing through origin) when a , b and c are unit vectors i.e. \( a^2 + b^2 + c^2 =1\):
    Equation FormMatrix Form
    \(x' = x (1-2a^2) + y (-2ab) + z (-2ac) - 2ad\)
    \(y' = x (-2ab) + y (1-2b^2) + z(-2bc) - 2bd\)
    \(z' = x (-2ac) + y (-2bc) + z(1-2c^2) - 2cd\)
    \(\begin{bmatrix} 1-2a^2 & -2ab & -2ac & -2ad\\-2ab & 1-2b^2 & -2bc & -2bd \\-2ac & -2bc & 1-2c^2 & -2cd\\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\hspace{.5cm} OR \hspace{.5cm} \begin{bmatrix} x & y & z & 1\end{bmatrix} \begin{bmatrix} 1-2a^2 & -2ab & -2ac & 0 \\-2ab & 1-2b^2 & -2bc & 0\\-2ac & -2bc & 1-2c^2 &0 \\-2ad & -2bd & -2cd & 1\end{bmatrix} = \begin{bmatrix} x' & y' & z' & 1\end{bmatrix} \)
  5. Reflection with respect to any plane \(ax + by + cz +d=0\):
    Equation FormMatrix Form
    \(x' = x - 2D \hat{a} \)
    \(y' = y - 2D \hat{b} \)
    \(z' = z - 2D \hat{c} \)
    \( \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} - 2D \begin{bmatrix} \hat{a} \\ \hat{b} \\ \hat{c} \end{bmatrix} \)
    Please note that \(\hat{a},\hat{b},\hat{c}\) are unit vectors corresponding to coefficients a, b and c and D is the distance from point to plane.
  6. Reflection of a point C with respect to any line having direction ratio \(\vec{A}\) making a vector \(\vec{B}\) with any point on the line:
    When \(\vec{B} \perp \vec{A}\) \( \vec{P}= \vec{C} - 2 \vec{B}\)
    When \(\vec{B}\) not \(\perp \vec{A}\) \( \vec{P}= \vec{C} - 2 (\frac{\vec{A} \times (\vec{B} \times \vec{A})}{\vert \vec{A} \vert ^2})\)
    Please note that \(\vec{C}\) is the position vector of point C and \(\vec{P}\) is position vector of reflected point.