# Types of Lines in 3D

A total of 50 types of lines can be laid out in a Cartisean Space in 3D divided into 6 Categories. Following are they:

1. Lines that a perpendicular to a coordinate planes. There are 3 types of such lines.
S.No Equation Explanation $$x=D$$$$y=D$$ These lines are perpendicular to XY plane (and consequently parallel to Z axis). $$y=D$$$$z=D$$ These lines are perpendicular to YZ plane (and consequently parallel to X axis). $$x=D$$$$z=D$$ These lines are perpendicular to XZ plane (and consequently parallel to Y axis).
2. Lines that a pass through origin but are not parallel to any plane. There are 4 types of such lines.
S.No Equation Explanation $$\frac{x}{A} = \frac{y}{B} = \frac{z}{C}$$ OR $$\frac{-x}{A} = \frac{-y}{B} = \frac{-z}{C}$$ These lines are aligned in 1st and 7th octant $$\frac{x}{A} = \frac{y}{B} = \frac{-z}{C}$$ OR $$\frac{-x}{A} = \frac{-y}{B} = \frac{z}{C}$$ These lines are aligned in 2nd and 8th octant $$\frac{x}{A} = \frac{-y}{B} = \frac{z}{C}$$ OR $$\frac{-x}{A} = \frac{y}{B} = \frac{-z}{C}$$ These lines are aligned in 3rd and 5th octant $$\frac{-x}{A} = \frac{y}{B} = \frac{z}{C}$$ OR $$\frac{x}{A} = \frac{-y}{B} = \frac{-z}{C}$$ These lines are aligned aligned in 4th and 6th octant
3. Lines that are parallel to or lie on a coordinate plane and pass through the axis perpendicular to that plane. There are 6 types of such lines
S.No Equation Explanation $$\frac{x}{A} = \frac{y}{B}$$ OR $$\frac{-x}{A} = \frac{-y}{B}$$ AND $$z=D$$ These lines lie on or are parallel to XY plane and are perpendicular to and pass through Z axis $$\frac{-x}{A} = \frac{y}{B}$$ OR $$\frac{x}{A} = \frac{-y}{B}$$ AND $$z=D$$ These lines lie on or are parallel to XY plane and are perpendicular to and pass through Z axis $$\frac{y}{B} = \frac{z}{C}$$ OR $$\frac{-y}{B} = \frac{-z}{C}$$ AND $$x=D$$ These lines lie on or are parallel to YZ plane and are perpendicular to and pass through X axis $$\frac{-y}{B} = \frac{z}{C}$$ OR $$\frac{y}{B} = \frac{-z}{C}$$ AND $$x=D$$ These lines lie on or are parallel to YZ plane and are perpendicular to and pass through X axis $$\frac{x}{A} = \frac{z}{C}$$ OR $$\frac{-x}{A} = \frac{-z}{C}$$ AND $$y=D$$ These lines lie on or are parallel to XZ plane and are perpendicular to and pass through Y axis $$\frac{-x}{A} = \frac{z}{C}$$ OR $$\frac{x}{A} = \frac{-z}{C}$$ AND $$y=D$$ These lines lie on or are parallel to XZ plane and are perpendicular to and pass through Y axis
4. Lines that are parallel to or lie on a coordinate plane and are perpendicular to axis perpendicular to that plane but do not pass through it. There are 12 types of such lines 4 for each coordinate plane.
S.No Equation Intercepts Explanation $$\frac{x-x1}{A} = \frac{y-y1}{B}$$ AND $$z=D$$ XZ Plane intercept point =($$\frac{-Ay1+Bx1}{B},0,D$$) YZ Plane intercept point =($$0,\frac{-Bx1+Ay1}{A},D$$) Lines perpendicular to Z axis either lying on or parallel to XY plane, having intercepts on XZ and YZ plane. 4 subtypes of such lines types are possible $$\frac{y-y1}{B} = \frac{z-z1}{C}$$ AND $$x=D$$ XY Plane intercept point =($$D,\frac{-Bz1+Cy1}{C},0$$) XZ Plane intercept point =($$D,0,\frac{-Cy1+Bz1}{B}$$) Lines perpendicular to X axis either lying on or parallel to YZ plane, having intercepts on XY and XZ plane. 4 subtypes of such lines types are possible $$\frac{x-x1}{A} = \frac{z-z1}{C}$$ AND $$y=D$$ XY Plane intercept point =($$\frac{-Az1+Cx1}{C},D,0$$) YZ Plane intercept point =($$0,D,\frac{-Cx1+Az1}{A}$$) Lines perpendicular to Y axis either lying on or parallel to XZ plane, having intercepts on XY and YZ plane. 4 subtypes of such lines types are possible
5. Lines that pass through a coordinate axis and form intercept on a coordinate plane not involving that axis. There are 24 types of such lines 8 for each coordinate axis/plane.
S.No Equation Intercept Explanation $$\frac{x}{A}=\frac{y-y1}{B}=\frac{z}{C}$$ XZ Plane intercept point =($$\frac{-Ay1}{B},0,\frac{-Cy1}{B}$$) These line pass through Y axis and form intercept on XZ plane. 8 subtypes of such lines types are possible, 4 for negative values of y1 and 4 for positive values of y1. $$\frac{x-x1}{A}=\frac{y}{B}=\frac{z}{C}$$ YZ Plane intercept point =($$0,\frac{-Bx1}{A},\frac{-Cx1}{A}$$) These line pass through X axis and form intercept on YZ plane. 8 subtypes of such lines types are possible, 4 for negative values of x1 and 4 for positive values of x1 $$\frac{x}{A}=\frac{y}{B}=\frac{z-z1}{C}$$ XY Plane intercept point =($$\frac{-Az1}{C},\frac{-Bz1}{C},0$$) These line pass through Z axis and form intercept on XY plane. 8 subtypes of such lines types are possible, 4 for negative values of z1 and 4 for positive values of z1
6. Lines that form non zero intercepts on all three coordinate planes
S.No Equation Intercept Explanation $$\frac{x-x1}{A}=\frac{y-y1}{B}=\frac{z-z1}{C}$$ XY Plane intercept point =($$\frac{-Az1+Cx1}{C},\frac{-Bz1+Cy1}{C},0$$) YZ Plane intercept point =($$0,\frac{-Bx1+Ay1}{A},\frac{-Cx1+Az1}{A}$$) XZ Plane intercept point =($$\frac{-Ay1+Bx1}{B},0,\frac{-Cy1+Bz1}{B}$$) These lines make intercept with all three coordinate planes