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.NoEquationExplanation
    1. \( x=D \)
    \( y=D \)
    These lines are perpendicular to XY plane (and consequently parallel to Z axis).
    2. \( y=D \)
    \( z=D \)
    These lines are perpendicular to YZ plane (and consequently parallel to X axis).
    3. \( x=D \)
    \( z=D \)
    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.NoEquationExplanation
    1. \(\frac{x}{A} = \frac{y}{B} = \frac{z}{C}\)
    OR
    \(\frac{-x}{A} = \frac{-y}{B} = \frac{-z}{C}\)
    These lines are aligned in 1st and 7th octant
    2. \(\frac{x}{A} = \frac{y}{B} = \frac{-z}{C}\)
    OR
    \(\frac{-x}{A} = \frac{-y}{B} = \frac{z}{C}\)
    These lines are aligned in 2nd and 8th octant
    3. \(\frac{x}{A} = \frac{-y}{B} = \frac{z}{C}\)
    OR
    \(\frac{-x}{A} = \frac{y}{B} = \frac{-z}{C}\)
    These lines are aligned in 3rd and 5th octant
    4. \(\frac{-x}{A} = \frac{y}{B} = \frac{z}{C}\)
    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.NoEquationExplanation
    1. \(\frac{x}{A} = \frac{y}{B}\)
    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
    2. \(\frac{-x}{A} = \frac{y}{B}\)
    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
    3. \(\frac{y}{B} = \frac{z}{C}\)
    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
    4. \(\frac{-y}{B} = \frac{z}{C}\)
    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
    5. \(\frac{x}{A} = \frac{z}{C}\)
    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
    6. \(\frac{-x}{A} = \frac{z}{C}\)
    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.NoEquationInterceptsExplanation
    1. \(\frac{x-x1}{A} = \frac{y-y1}{B}\)
    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
    2. \(\frac{y-y1}{B} = \frac{z-z1}{C}\)
    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
    3. \(\frac{x-x1}{A} = \frac{z-z1}{C}\)
    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.NoEquationInterceptExplanation
    1. \(\frac{x}{A}=\frac{y-y1}{B}=\frac{z}{C}\) 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.
    2. \(\frac{x-x1}{A}=\frac{y}{B}=\frac{z}{C}\) 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
    3. \(\frac{x}{A}=\frac{y}{B}=\frac{z-z1}{C}\) 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.NoEquationInterceptExplanation
    1. \(\frac{x-x1}{A}=\frac{y-y1}{B}=\frac{z-z1}{C}\) 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