Types of Planes

A total of 33 types of planes can be laid out on a Cartisean Space in 3D divided into 5 Categories. Following are they:

  1. Planes that a perpendicular to a coordinate axis. There are 3 types of such planes.
    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 a pass through origin but do not contain any coordinate axis (and hence their normals are not perpendicular to any coordinate axis) . There are 4 types of such planes.
    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
  3. Planes that a pass through origin but and contain a coordinate axis (and hence their normals perpendicular to that coordinate axis) . There are 6 types of such planes.
    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
  4. Planes that make distinct intercepts on coordinate axes. There are 8 types of such planes, 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 negative. Plane is in 7th 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=\(\frac{D}{C}\)
    All Intercepts are positive. Plane is in 1st 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=\(\frac{D}{C}\)
    Plane is in 8th 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=\(\frac{-D}{C}\)
    Plane is in 3rd 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=\(\frac{-D}{C}\)
    Plane is in 6th 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=\(\frac{-D}{C}\)
    Plane is in 2nd 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=\(\frac{D}{C}\)
    Plane is in 5th 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=\(\frac{D}{C}\)
    Plane is in 4th Octant
  5. Planes that are parallel to a coordinate axis (and hence their normal are perpendicular to that axis). There are 12 types of such planes.
    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