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Types of Lines in 3D

  1. A total of 73 Kinds of Lines can be laid out on a 3D Cartisean Space which can be Classified into following 6 Types
    1. Lines that Pass through Origin and are Not Parallel or Perpendicular to Any of the Coordinate Axes or Coordinate Planes.
    2. Lines that are Parallel To or Lie On a Coordinate Plane and Pass through the Coordinate Axis Perpendicular to that Coordinate Plane.
    3. Lines that are Perpendicular to One of the Coordinate Planes and are Parallel to Coordinate Axis Perpendicular to that Coordinate Plane.
    4. Lines that Pass through One of the Coordinate Axes and form Intercept on the Coordinate Plane Perpendicular to that Coordinate Axis.
    5. Lines that Form Intercepts on 2 Coordinate Planes and are Parallel To or Lie On the 3rd Coordinate Plane.
    6. Lines that Form Intercepts on All 3 Coordinate Planes.
  2. Based on their Cartesian Coordinate Equations, these 6 Types of Lines can be classified into following Subtypes. Please note that All Subtypes of a particular Line Type have Similar Kind of Cartesian Coordinate Equations.
    1. Lines that Pass through Origin and are Not Parallel or Perpendicular to Any of the Coordinate Axes or Coordinate Planes. These can be of following 4 Subtypes.
      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
    2. Lines that are Parallel To or Lie On a Coordinate Plane and Pass through the Coordinate Axis Perpendicular to that Coordinate Plane. These can be of following 6 Subtypes.
      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
    3. Lines that are Perpendicular to One of the Coordinate Planes and are Parallel to Coordinate Axis Perpendicular to that Coordinate Plane. These can be of following 3 Subtypes.
      S.NoEquationInterceptExplanation
      1. \( x=D_1 \)
      \( y=D_2 \)      		 XY Plane intercept point =(\(D_1,D_2,0\)) These lines are perpendicular to XY plane (and consequently parallel to Z axis).
      2. \( y=D_1 \)
      \( z=D_2 \)      		 YZ Plane intercept point =(\(0,D_1,D_2\)) These lines are perpendicular to YZ plane (and consequently parallel to X axis).
      3. \( x=D_1 \)
      \( z=D_2 \)      		 XZ Plane intercept point =(\(D_1,0,D_2\)) These lines are perpendicular to XZ plane (and consequently parallel to Y axis).
    4. Lines that Pass through One of the Coordinate Axes and form Intercept on the Coordinate Plane Perpendicular to that Coordinate Axis. There are 24 Subtypes 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 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 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 are possible, 4 for negative values of z1 and 4 for positive values of z1
    5. Lines that Form Intercepts on 2 Coordinate Planes and are Parallel To or Lie On the 3rd Coordinate Plane. There are 12 Subtypes 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
    6. Lines that Form Intercepts on All 3 Coordinate Planes. These can be of 24 Subtypes.
      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 form Intercepts with All 3 Coordinate Planes. Since any Quadrant of a Plane can be Penetrated in 6 Unique Ways and a Plane has 4 Quadrants, 24 such subtypes of Lines are possible
Related Topics and Calculators
Types of Lines in 2D,    Types of Planes
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