Reflection

1. Reflections are Non Deformative Transformation. It involves Determining the Mirror Image of the Object across a Point or a Line in 2D and across a Point, a Line or a Plane in 3D. In this transformation any changes applied to a Coordinate Point can be directly applied to the Equations of Objects as well.
2. Reflection in 2D, 3D or Higher Dimension of a Point \(C\) having Position Vector \(\vec{C}\) across any Point \(P\) having Position Vector \(\vec{P}\):
   
   \( \vec{C_R}= 2\vec{P} -\vec{C}\)
   
   where \(\vec{C_R}\)=Position Vector of Reflected Point \(C_R\). The Point \(P\) is called the Point of Projection and it is Mid-Point of the Points \(C\) and \(C_R\).
   
   \( \vec{P}= \frac{\vec{C}+\vec{C_R}}{2}\)
3. Reflection in 2D, 3D or Higher Dimensions of a Point \(C\) having Position Vector \(\vec{C}\) across any Line having Direction Ratio \(\vec{A}\) containing a Point \(B\) having Position Vector \(\vec{B}\):
   
   
   
   \( \vec{C_R}=\vec{C} - 2{\vec{CB}}_\perp =\vec{C} - 2 (\vec{CB} - {\vec{CB}}_{||}) =\vec{C} - 2 (\vec{CB} - \frac{(\vec{A} \cdot \vec{CB})\vec{A}}{{|\vec{A}|}^2})\)   ...(In 2D, 3D and Higher Dimensions)
   \( \vec{C_R}=\vec{C} - 2\frac{\vec{A} \times (\vec{CB}\times \vec{A})}{{|\vec{A}|}^2} =\vec{C} - 2\frac{(\vec{A} \times \vec{CB})\times \vec{A}}{{|\vec{A}|}^2} \)   ...(In 2D and 3D only)
   
   where \(\vec{CB}=\vec{C}-\vec{B}\), \(\vec{C_R}\)=Position Vector of Reflected Point \(C_R\)
   
   Please Check the Derivation of the Formula.
4. Reflection of a Point \(C\) having Position Vector \(\vec{C}\) across any Line in 2D (or across any Plane in 3D or across any Hyper-Plane in Higher Dimensions) having Direction Ratio of Normal \(\vec{A}\) containing a Point \(B\) having Position Vector \(\vec{B}\):
   
   
   
   \( \vec{C_R}= \vec{C} - 2\vec{CB}_{||}= \vec{C} - 2 (\frac{(\vec{A} \cdot \vec{CB}) \vec{A} }{{|\vec{A}|}^2})\)
   
   where \(\vec{CB}=\vec{C}-\vec{B}\), \(\vec{C_R}\)=Position Vector of Reflected Point \(C_R\)
   
   Please Check the Derivation of the Formula.
5. Reflection of a Point having Coordinates (\(x_c,y_c\)) in 2D across Line \(ax + by + c=0\):

   Equation Form	Matrix Form
   \(x_r = x_c - 2D \hat{a} \) \(y_r = y_c - 2D \hat{b} \)	\( \begin{bmatrix} x_r \\ y_r \end{bmatrix} = \begin{bmatrix} x_c \\ y_c \end{bmatrix} - 2D \begin{bmatrix} \hat{a} \\ \hat{b}\end{bmatrix} \)
   (\(\hat{a},\hat{b}\))=Unit Vectors corresponding to coefficients a and b of Line \(D\)= Signed Distance from Point to Line=\(\frac{ax_c+by_c+c}{\sqrt{a^2+b^2}}\) \((x_r,y_r)\)=Coordinates of the Reflected Point
   Please Note that this formula is the derived from formula given in Point 4

6. Reflection of a Point having Coordinates (\(x_c,y_c,z_c\)) in 3D across Plane \(ax + by + cz +d=0\):

   Equation Form	Matrix Form
   \(x_r = x_c - 2D \hat{a} \) \(y_r = y_c - 2D \hat{b} \) \(z_r = z_c - 2D \hat{c} \)	\( \begin{bmatrix} x_r \\ y_r \\ z_r \end{bmatrix} = \begin{bmatrix} x_c \\ y_c \\ z_c \end{bmatrix} - 2D \begin{bmatrix} \hat{a} \\ \hat{b} \\ \hat{c} \end{bmatrix} \)
   (\(\hat{a},\hat{b},\hat{c}\))=Unit Vectors corresponding to coefficients a, b and c of Plane \(D\)= Signed Distance from Point to Plane=\(\frac{ax_c+by_c + cz_c + d}{\sqrt{a^2+b^2+c^2}}\) \((x_r,y_r,z_r)\)=Coordinates of the Reflected Point
   Please Note that this formula is the derived from formula given in Point 4

7. Reflection in 2D across Coordinate Axes or across Origin:

   Reflection Type	Equation Form	Matrix Form
   Across x axis	\(x' = x\) \(y' = -y \)	\(\begin{bmatrix} 1 & 0 \\0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix} \)
   Across y axis	\(x' = -x\) \(y' = y \)	\(\begin{bmatrix} -1 & 0 \\0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}\)
   Across Origin	\(x' = -x\) \(y' = -y \)	\(\begin{bmatrix} -1 & 0 \\0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}\)

8. Reflection in 3D across a Coordinate Plane or across a Coordinate Axis (i.e. 2 Coordinate Planes) or across Origin:

   Reflection Type	Equation Form	Matrix Form
   Across xy plane	\(x' = x\) \(y' = y \) \(z' = -z \)	\(\begin{bmatrix} 1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & -1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\)
   Across yz plane	\(x' = -x\) \(y' = y \) \(z' = z \)	\(\begin{bmatrix} -1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\)
   Across zx plane	\(x' = x\) \(y' = -y \) \(z' = z \)	\(\begin{bmatrix} 1 & 0 & 0\\0 & -1 & 0\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\)
   Across xy & yz plane (y Axis)	\(x' = -x\) \(y' = y \) \(z' = -z \)	\(\begin{bmatrix} -1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & -1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\)
   Across yz & zx plane (z Axis)	\(x' = -x\) \(y' = -y \) \(z' = z \)	\(\begin{bmatrix} -1 & 0 & 0\\0 & -1 & 0\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\)
   Across xy & zx plane (x Axis)	\(x' = x\) \(y' = -y \) \(z' = -z \)	\(\begin{bmatrix} 1 & 0 & 0\\0 & -1 & 0\\0 & 0 & -1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\)
   Across Origin (0,0,0)	\(x' = -x\) \(y' = -y \) \(z' = -z \)	\(\begin{bmatrix} -1 & 0 & 0\\0 & -1 & 0\\0 & 0 & -1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}\)

9. Reflection in 2D across Lines \(x=constant\) or \(y=constant\) or Arbitrary Point \( (o_x,o_y) \):

   Reflection Type	Equation Form	Matrix Form
   Across line x=c (c=constant)	\(x' = 2c - x \) \(y' = y \)	\(\begin{bmatrix} - 1 & 0 & 2c \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\)
   Across line y=c (c=constant)	\(x' = x\) \(y' = 2c-y \)	\(\begin{bmatrix} 1 & 0 & 0 \\0 & -1 & 2c \\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\)
   Across Point \((o_x,o_y)\)	\(x' = 2o_x - x\) \(y' = 2o_y - y \)	\(\begin{bmatrix} -1 & 0 & 2o_x \\0 & -1 & 2o_y \\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\)
   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

10. Reflection in 3D across Planes \(x=constant\) or \(y=constant\) or \(z=constant\) or Arbitrary Point \( (o_x,o_y,o_z) \):

    Reflection Type	Equation Form	Matrix Form
    Across plane x=c (c=constant)	\(x' = 2c - x \) \(y' = y \) \(z' = z \)	\(\begin{bmatrix} - 1 & 0 & 0 & 2c \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\)
    Across plane y=c (c=constant)	\(x' = x\) \(y' = 2c-y \) \(z' = z \)	\(\begin{bmatrix} 1 & 0 & 0 & 0\\0 & -1 & 0 & 2c \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\)
    Across plane z=c (c=constant)	\(x' = x\) \(y' = y \) \(z' = 2c-z \)	\(\begin{bmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 2c \\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\)
    Across Point \((o_x,o_y,o_z)\)	\(x' = 2o_x - x\) \(y' = 2o_y - y \) \(z' = 2o_z - z \)	\(\begin{bmatrix} -1 & 0 & 0 & 2o_x \\0 & -1 & 0 & 2o_y \\0 & 0 & -1 & 2o_z \\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\)
    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

11. Reflection in 2D across Lines \(ax + by=0\) or \(ax + by +c =0\) where \(a\) and \(b\) are components of Unit Vector Normal to the Lines:

    Reflection Type	Matrix Form
    Across line \(ax + by=0\)	\(\begin{bmatrix} 1 - 2a^2 & -2ab\\-2ab & 1-2b^2 \end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix} x' \\ y'\end{bmatrix}\)
    Across line \(ax + by + c=0\)	\(\begin{bmatrix} 1 - 2a^2 & -2ab & -2ac\\-2ab & 1-2b^2 & -2bc\\0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ 1\end{bmatrix}\)

12. Reflection in 3D across Planes \(ax + by +cz =0\) or \(ax + by +cz + d=0\) where \(a\), \(b\) and \(c\) are components of Unit Vector Normal to the Planes:

    Reflection Type	Matrix Form
    Across plane \(ax + by + cz=0\)	\(\begin{bmatrix} 1 - 2a^2 & -2ab & -2ac\\-2ab & 1-2b^2 & -2bc \\-2ac & -2bc & 1-2c^2 \end{bmatrix} \begin{bmatrix} x \\ y \\z\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z'\end{bmatrix}\)
    Across plane \(ax + by + cz + d=0\)	\(\begin{bmatrix} 1 - 2a^2 & -2ab & -2ac & -2ad\\-2ab & 1-2b^2 & -2bc & -2bd\\-2ac & -2bc & 1-2c^2 & -2cd \\0 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \\z \\ 1\end{bmatrix} = \begin{bmatrix} x' \\ y' \\ z' \\ 1\end{bmatrix}\)