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Types of Planes

  1. A total of 33 Kinds of Planes can be laid out on a 3D Cartisean Space which can be Classified into following 5 Types
    1. Planes that have Intercept on Only 1 of the Coordinate Axes
    2. Planes that have Non-Zero Intercepts on 2 of the Coordinate Axes
    3. Planes that have Non-Zero Intercepts on All 3 Coordinate Axes
    4. Planes that Pass through Origin and Contain 1 Coordinate Axis
    5. Planes that Pass through Origin but Do Not Contain Any Coordinate Axis
  2. Based on their Cartesian Coordinate Equations, these 5 Types of Planes can be classified into following Subtypes. Please note that All Subtypes of a particular Plane Type have Similar Kind of Cartesian Coordinate Equations.
    1. Planes that have Intercept on Only 1 of the Coordinate Axes and hence are Perpendicular to that Coordinate Axis. These are of following 3 Subtypes
      S.NoEquationInterceptsExplanation
      1. \( x=C \) x Intercept=C
      y Intercept=Undefined
      z Intercept=Undefined      		 These planes are perpendicular to X axis (and consequently parallel to Y and Z axis). The plane \(x=0\) is the YZ plane.
      2. \( y=C \) x Intercept=Undefined
      y Intercept=C
      z Intercept=Undefined      		 These planes are perpendicular to Y axis (and consequently parallel to X and Z axis). The line \(y=0\) is the XZ plane.
      3. \( z=C \) x Intercept=Undefined
      y Intercept=Undefined
      z Intercept=C      		 These planes are perpendicular to Z axis (and consequently parallel to X and Y axis). The line \(z=0\) is the XY plane.
    2. Planes that have Non-Zero Intercepts on 2 of the Coordinate Axes and are Parallel to 3rd Coordinate Axis (and hence their Normals are Perpendicular to the 3rd Axis). These are of following 12 Subtypes
      S.NoEquationInterceptsExplanation
      1. \( Ax+By+D=0 \)
      OR
      \(-Ax-By-D=0\)       		x Intercept=\(\frac{-D}{A}\)
      y Intercept=\(\frac{-D}{B}\)
      z Intercept=Undefined      		 These planes are parallel to Z axis and their normals are perpendicular to it
      2. \( Ax+By-D=0 \)
      OR
      \(-Ax-By+D=0\)       		x Intercept=\(\frac{D}{A}\)
      y Intercept=\(\frac{D}{B}\)
      z Intercept=Undefined      		 These planes are parallel to Z axis and their normals are perpendicular to it
      3. \( Ax-By+D=0 \)
      OR
      \(-Ax+By-D=0\)       		x Intercept=\(\frac{-D}{A}\)
      y Intercept=\(\frac{D}{B}\)
      z Intercept=Undefined      		 These planes are parallel to Z axis and their normals are perpendicular to it
      4. \( -Ax+By+D=0 \)
      OR
      \(Ax-By-D=0\)       		x Intercept=\(\frac{D}{A}\)
      y Intercept=\(\frac{-D}{B}\)
      z Intercept=Undefined      		 These planes are parallel to Z axis and their normals are perpendicular to it
      5. \( By+Cz+D=0 \)
      OR
      \(-By-Cz-D=0\)       		x Intercept=Undefined
      y Intercept=\(\frac{-D}{B}\)
      z Intercept=\(\frac{-D}{C}\)      		 These planes are parallel to X axis and their normals are perpendicular to it
      6. \( By+Cz-D=0 \)
      OR
      \(-By-Cz+D=0\)       		x Intercept=Undefined
      y Intercept=\(\frac{D}{B}\)
      z Intercept=\(\frac{D}{C}\)      		 These planes are parallel to X axis and their normals are perpendicular to it
      7. \( By-Cz+D=0 \)
      OR
      \(-By+Cz-D=0\)       		x Intercept=Undefined
      y Intercept=\(\frac{-D}{B}\)
      z Intercept=\(\frac{D}{C}\)      		 These planes are parallel to X axis and their normals are perpendicular to it
      8. \( -By+Cz+D=0 \)
      OR
      \(By-Cz-D=0\)       		x Intercept=Undefined
      y Intercept=\(\frac{D}{B}\)
      z Intercept=\(\frac{-D}{C}\)      		 These planes are parallel to X axis and their normals are perpendicular to it
      9. \( Ax+Cz+D=0 \)
      OR
      \(-Ax-Cz-D=0\)       		x Intercept=\(\frac{-D}{A}\)
      y Intercept=Undefined
      z Intercept=\(\frac{-D}{C}\)      		 These planes are parallel to Y axis and their normals are perpendicular to it
      10. \( Ax+Cz-D=0 \)
      OR
      \(-Ax-Cz+D=0\)       		x Intercept=\(\frac{D}{A}\)
      y Intercept=Undefined
      z Intercept=\(\frac{D}{C}\)      		 These planes are parallel to Y axis and their normals are perpendicular to it
      11. \( Ax-Cz+D=0 \)
      OR
      \(-Ax+Cz-D=0\)       		x Intercept=\(\frac{-D}{A}\)
      y Intercept=Undefined
      z Intercept=\(\frac{D}{C}\)      		 These planes are parallel to Y axis and their normals are perpendicular to it
      12. \( -Ax+Cz+D=0 \)
      OR
      \(Ax-Cz-D=0\)       		x Intercept=\(\frac{D}{A}\)
      y Intercept=Undefined
      z Intercept=\(\frac{-D}{C}\)      		 These planes are parallel to Y axis and their normals are perpendicular to it
    3. Planes that have Non-Zero Intercepts on All 3 Coordinate Axes. These are of following 8 Subtypes, 1 for each octant
      S.NoEquationInterceptsExplanation
      1. \( Ax+By+Cz-D=0 \)
      OR
      \(-Ax-By-Cz+D=0\)       		x Intercept=\(\frac{D}{A}\)
      y Intercept=\(\frac{D}{B}\)
      z Intercept=\(\frac{D}{C}\)      		 All Intercepts are positive. The Projection of the Origin on Plane is in the 1st Octant
      2. \( Ax+By-Cz-D=0 \)
      OR
      \(-Ax-By+Cz+D=0\)       		x Intercept=\(\frac{D}{A}\)
      y Intercept=\(\frac{D}{B}\)
      z Intercept=\(\frac{-D}{C}\)      		 The Projection of the Origin on Plane is in the 2nd Octant
      3. \( Ax-By+Cz+D=0 \)
      OR
      \(-Ax+By-Cz-D=0\)       		x Intercept=\(\frac{-D}{A}\)
      y Intercept=\(\frac{D}{B}\)
      z Intercept=\(\frac{-D}{C}\)      		 The Projection of the Origin on Plane is in the 3rd Octant
      4. \( -Ax+By+Cz-D=0 \)
      OR
      \(Ax-By-Cz+D=0\)       		x Intercept=\(\frac{-D}{A}\)
      y Intercept=\(\frac{D}{B}\)
      z Intercept=\(\frac{D}{C}\)      		 The Projection of the Origin on Plane is in the 4th Octant
      5. \( Ax-By+Cz-D=0 \)
      OR
      \(-Ax+By-Cz+D=0\)       		x Intercept=\(\frac{D}{A}\)
      y Intercept=\(\frac{-D}{B}\)
      z Intercept=\(\frac{D}{C}\)      		 The Projection of the Origin on Plane is in the 5th Octant
      6. \( -Ax+By+Cz+D=0 \)
      OR
      \(Ax-By-Cz-D=0\)       		x Intercept=\(\frac{D}{A}\)
      y Intercept=\(\frac{-D}{B}\)
      z Intercept=\(\frac{-D}{C}\)      		 The Projection of the Origin on Plane is in the 6th Octant
      7. \( Ax+By+Cz+D=0 \)

      OR
      \(-Ax-By-Cz-D=0\)       		x Intercept=\(\frac{-D}{A}\)
      y Intercept=\(\frac{-D}{B}\)
      z Intercept=\(\frac{-D}{C}\)      		 All Intercepts are negative. The Projection of the Origin on Plane is in the 7th Octant
      8. \( Ax+By-Cz+D=0 \)
      OR
      \(-Ax-By+Cz-D=0\)       		x Intercept=\(\frac{-D}{A}\)
      y Intercept=\(\frac{-D}{B}\)
      z Intercept=\(\frac{D}{C}\)      		 The Projection of the Origin on Plane is in the 8th Octant
    4. Planes that a Pass through Origin and Contain 1 Coordinate Axis (and hence their Normals are Perpendicular to that Coordinate Axis). These are of following 6 Subtypes
      S.NoEquationInterceptsExplanation
      1. \( Ax+By=0 \)
      OR
      \(-Ax-By=0 \)       		x Intercept=0
      y Intercept=0
      z Intercept=Undefined      		 These planes contain Z axis and their normals are perpendicular to it
      2. \( Ax-By=0 \)
      OR
      \(-Ax+By=0 \)       		x Intercept=0
      y Intercept=0
      z Intercept=Undefined      		 These planes contain Z axis and their normals are perpendicular to it
      3. \( By+Cz=0 \)
      OR
      \(-By-Cz=0 \)       		x Intercept=Undefined
      y Intercept=0
      z Intercept=0      		 These planes contain X axis and their normals are perpendicular to it
      4. \( By-Cz=0 \)
      OR
      \(-By+Cz=0 \)       		x Intercept=Undefined
      y Intercept=0
      z Intercept=0      		 These planes contain X axis and their normals are perpendicular to it
      5. \( Ax+Cz=0 \)
      OR
      \(-Ax-Cz=0 \)       		x Intercept=0
      y Intercept=Undefined
      z Intercept=0      		 These planes contain Y axis and their normals are perpendicular to it
      6. \( Ax-Cz=0 \)
      OR
      \(-Ax+Cz=0 \)       		x Intercept=0
      y Intercept=Undefined
      z Intercept=0      		 These planes contain Y axis and their normals are perpendicular to it
    5. Planes that a Pass through Origin but Do Not Contain Any Coordinate Axis. These are of following 4 Subtypes
      S.NoEquationInterceptsExplanation
      1. \( Ax+By+Cz=0 \)
      OR
      \(-Ax-By-Cz=0\)       		x Intercept=0
      y Intercept=0
      z Intercept=0      		 These planes have normals aligned in 1st and 7th octant
      2. \( Ax+By-Cz=0 \)
      OR
      \(-Ax-By+Cz=0\)       		x Intercept=0
      y Intercept=0
      z Intercept=0      		 These planes have normals aligned in 2nd and 8th octant
      3. \( Ax-By+Cz=0 \)
      OR
      \(-Ax+By-Cz=0\)       		x Intercept=0
      y Intercept=0
      z Intercept=0      		 These planes have normals aligned in 3rd and 5th octant
      4. \( -Ax+By+Cz=0 \)
      OR
      \(Ax-By-Cz=0\)       		x Intercept=0
      y Intercept=0
      z Intercept=0      		 These planes have normals aligned in 4th and 6th octant
Related Topics and Calculators
Types of Lines in 2D,    Types of Lines in 3D
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