# 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.No Equation Intercepts Explanation $$x=C$$ x Intercept=Cy Intercept=Undefinedz 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. $$y=C$$ x Intercept=Undefinedy Intercept=Cz 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. $$z=C$$ x Intercept=Undefinedy Intercept=Undefinedz 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.No Equation Intercepts Explanation $$Ax+By+Cz=0$$ OR $$-Ax-By-Cz=0$$ x Intercept=0y Intercept=0z Intercept=0 These planes have normals aligned in 1st and 7th octant $$Ax+By-Cz=0$$ OR $$-Ax-By+Cz=0$$ x Intercept=0y Intercept=0z Intercept=0 These planes have normals aligned in 2nd and 8th octant $$Ax-By+Cz=0$$ OR $$-Ax+By-Cz=0$$ x Intercept=0y Intercept=0z Intercept=0 These planes have normals aligned in 3rd and 5th octant $$-Ax+By+Cz=0$$ OR $$Ax-By-Cz=0$$ x Intercept=0y Intercept=0z 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.No Equation Intercepts Explanation $$Ax+By=0$$ OR $$-Ax-By=0$$ x Intercept=0y Intercept=0z Intercept=Undefined These planes contain Z axis and their normals are perpendicular to it $$Ax-By=0$$ OR $$-Ax+By=0$$ x Intercept=0y Intercept=0z Intercept=Undefined These planes contain Z axis and their normals are perpendicular to it $$By+Cz=0$$ OR $$-By-Cz=0$$ x Intercept=Undefinedy Intercept=0z Intercept=0 These planes contain X axis and their normals are perpendicular to it $$By-Cz=0$$ OR $$-By+Cz=0$$ x Intercept=Undefinedy Intercept=0z Intercept=0 These planes contain X axis and their normals are perpendicular to it $$Ax+Cz=0$$ OR $$-Ax-Cz=0$$ x Intercept=0y Intercept=Undefinedz Intercept=0 These planes contain Y axis and their normals are perpendicular to it $$Ax-Cz=0$$ OR $$-Ax+Cz=0$$ x Intercept=0y Intercept=Undefinedz 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.No Equation Intercepts Explanation $$Ax+By+Cz+D=0$$ 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 $$Ax+By+Cz-D=0$$ 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 $$Ax+By-Cz+D=0$$ 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 $$Ax-By+Cz+D=0$$ 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 $$-Ax+By+Cz+D=0$$ 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 $$Ax+By-Cz-D=0$$ 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 $$Ax-By+Cz-D=0$$ 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 $$-Ax+By+Cz-D=0$$ 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.No Equation Intercepts Explanation $$Ax+By+D=0$$ 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 $$Ax+By-D=0$$ 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 $$Ax-By+D=0$$ 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 $$-Ax+By+D=0$$ 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 $$By+Cz+D=0$$ OR $$-By-Cz-D=0$$ x Intercept=Undefinedy Intercept=$$\frac{-D}{B}$$z Intercept=$$\frac{-D}{C}$$ These planes are parallel to X axis and their normals are perpendicular to it $$By+Cz-D=0$$ OR $$-By-Cz+D=0$$ x Intercept=Undefinedy Intercept=$$\frac{D}{B}$$z Intercept=$$\frac{D}{C}$$ These planes are parallel to X axis and their normals are perpendicular to it $$By-Cz+D=0$$ OR $$-By+Cz-D=0$$ x Intercept=Undefinedy Intercept=$$\frac{-D}{B}$$z Intercept=$$\frac{D}{C}$$ These planes are parallel to X axis and their normals are perpendicular to it $$-By+Cz+D=0$$ OR $$By-Cz-D=0$$ x Intercept=Undefinedy Intercept=$$\frac{D}{B}$$z Intercept=$$\frac{-D}{C}$$ These planes are parallel to X axis and their normals are perpendicular to it $$Ax+Cz+D=0$$ OR $$-Ax-Cz-D=0$$ x Intercept=$$\frac{-D}{A}$$y Intercept=Undefinedz Intercept=$$\frac{-D}{C}$$ These planes are parallel to Y axis and their normals are perpendicular to it $$Ax+Cz-D=0$$ OR $$-Ax-Cz+D=0$$ x Intercept=$$\frac{D}{A}$$y Intercept=Undefinedz Intercept=$$\frac{D}{C}$$ These planes are parallel to Y axis and their normals are perpendicular to it $$Ax-Cz+D=0$$ OR $$-Ax+Cz-D=0$$ x Intercept=$$\frac{-D}{A}$$y Intercept=Undefinedz Intercept=$$\frac{D}{C}$$ These planes are parallel to Y axis and their normals are perpendicular to it $$-Ax+Cz+D=0$$ OR $$Ax-Cz-D=0$$ x Intercept=$$\frac{D}{A}$$y Intercept=Undefinedz Intercept=$$\frac{-D}{C}$$ These planes are parallel to Y axis and their normals are perpendicular to it