Skew Lines

Recognising Skew Lines and Orthographic Conventions

In two dimensions, any two straight lines either meet at an intersection point or run parallel. In three-dimensional space, a third condition exists: skew lines.

Two spatial lines are skew when they satisfy three conditions:

• They are not parallel to each other.
• They do not intersect at any point in space, even when extended indefinitely.
• They do not lie in a common plane (they are non-coplanar).

In orthographic drawings, two lines often appear to cross over each other in plan or elevation. An apparent crossing is not proof of an intersection. It only means that one line passes in front of, behind, above or below the other along that direction of sight. The crossing is a real intersection only when the crossing point in the plan and the crossing point in the elevation lie on the same vertical projector.

Parallel in space means parallel in plan and in elevation. Parallel in only one view means the lines are not parallel at all.

Standard Labelling Conventions

Keeping your labelling consistent prevents confusion when transferring dimensions across multiple projection planes:

• Plan points are labelled with lower-case letters: $a, b, c, d$.
• Elevation points carry a prime: $a', b', c', d'$.
• First auxiliary points carry a subscript 1: $a_1, b_1, c_1, d_1$.
• Second auxiliary points carry a subscript 2: $a_2, b_2, c_2, d_2$.

The Containing Plane Principle and Method Selection

To control the direction of a connecting line between two skew lines, you establish a containing plane.

Given two skew lines, $AB$ and $CD$:

1. Pick an endpoint on one line, such as point $B$ on line $AB$.
2. Through $B$, draw a construction line $BE$ parallel to line $CD$ in the plan and in the elevation ($be \parallel cd$, and $b'e' \parallel c'd'$). Point $E$ can be placed at any convenient distance along this line so that it stays neatly on the drawing sheet. A line is only parallel to another in space when it is parallel in both views.
3. Because $AB$ and $BE$ meet at $B$, they form plane $ABE$.

Because line $CD$ is parallel to line $BE$ on this plane, $CD$ is parallel to the entire plane $ABE$, maintaining a constant clearance from it.

Projecting the Edge View of Plane ABE

To view plane $ABE$ as a single line (an edge view), you must look directly along a true length line lying on that plane. The standard choice is a horizontal line (strike line):

1. In elevation, draw a horizontal line across plane $ABE$ parallel to the $XY$ line. For example, draw a level line through $e'$ to cut line $a'b'$ at $f'$.
2. Project $f'$ vertically down to line $ab$ in plan to locate $f$. Join $e$ to $f$. Because line $ef$ is horizontal in space, its plan view shows its true length.
3. Draw the first auxiliary ground line $X_1Y_1$ perpendicular to $ef$ in plan. Choose a position for $X_1Y_1$ that directs the projected views into clear space on your sheet.
4. Project all points ($A, B, C, D$, and check point $E$) perpendicular to $X_1Y_1$. Transfer vertical heights directly from the elevation, measuring from the $XY$ line.

In this first auxiliary view:

• Points $a_1, b_1,$ and $e_1$ align into a single line, forming the edge view of plane $ABE$.
• Line $c_1d_1$ projects parallel to line $a_1b_1$.
• The angle between $a_1b_1$ and $X_1Y_1$ is the true inclination ($\alpha$) of the plane to the Horizontal Plane (HP).

Choosing Your Second Auxiliary Ground Line ($X_2Y_2$)

The orientation of $X_2Y_2$ determines what the second auxiliary view reveals:

The Point View Method for Shortest Perpendicular Distance

If a question asks only for the shortest perpendicular distance (with no requirement for a horizontal or sloping connector), the point view method is faster because it does not require constructing a containing plane.

1. First auxiliary (true length of AB): Draw $X_1Y_1$ parallel to line $ab$ in plan. Project points $A, B, C,$ and $D$ perpendicular to $X_1Y_1$. Transfer heights from the elevation, measured vertically from $XY$. Line $a_1b_1$ now appears at its true length.
2. Second auxiliary (point view of AB): Draw $X_2Y_2$ perpendicular to $a_1b_1$. Project all four points perpendicular to $X_2Y_2$. Transfer distances from $X_1Y_1$ as measured in the plan view (the view two steps back). Line $AB$ collapses into a single point: $a_2b_2$.
3. Measure the distance: From the point $a_2b_2$, drop a perpendicular line onto line $c_2d_2$. Because you are looking directly along $AB$, this line is the common perpendicular and appears at true length. Measure it directly in millimetres.
4. Project back to primary views: Project the contact point on $c_2d_2$ back to the first auxiliary view (perpendicular to $X_2Y_2$) to locate it on $c_1d_1$. From that point, draw a line perpendicular to $a_1b_1$ to locate the contact point on $AB$. Project both endpoints back perpendicular to $X_1Y_1$ into plan to land on $ab$ and $cd$, and then vertically up to elevation to land on $a'b'$ and $c'd'$. Firm in the connector.

Constructing the Shortest Horizontal Distance

A level connector between skew lines must remain parallel to the Horizontal Plane. In the first auxiliary view projected from plan, the $X_1Y_1$ line represents the Horizontal Plane; therefore, every horizontal line in space appears parallel to $X_1Y_1$.

1. In the first auxiliary view, lines $a_1b_1$ and $c_1d_1$ run parallel. Draw a line parallel to $X_1Y_1$ between them. Its length is the true length of the shortest horizontal distance.
2. To locate its exact position, set up $X_2Y_2$ perpendicular to $X_1Y_1$. This directs your line of sight straight along the level direction.
3. Project the lines across $X_2Y_2$, transferring distances from $X_1Y_1$ back to the plan view.
4. Lines $a_2b_2$ and $c_2d_2$ intersect. This crossing is the point view of the horizontal connector.
5. Project this intersection point back to the first auxiliary view. It defines the horizontal segment running parallel to $X_1Y_1$ between $AB$ and $CD$.
6. Project both contact points back to plan (perpendicular to $X_1Y_1$) and then vertically up to elevation.

Self-Checks for Drafting Accuracy

Before firming in your solution, verify these two geometric checks:

• In elevation, the connector must be parallel to the $XY$ line.
• In plan, the connector must be perpendicular to the strike line $ef$.

Trigonometric Check

In the first auxiliary view, the shortest perpendicular distance $d$, the shortest horizontal distance $h$, and the plane inclination angle $\alpha$ form a right-angled triangle:

Because $\sin\alpha$ can never be greater than 1, the shortest horizontal distance $h$ is always at least as long as the shortest distance $d$ measured along the common perpendicular. They are equal only when the containing plane is vertical ($\alpha = 90^\circ$). Use this formula strictly to check your measured drawing values, never as a replacement for graphical construction.

True Angle Between Lines (Skew vs Intersecting)

True Angle Between Skew Lines

Although skew lines never intersect, their directions form a spatial angle. In the second auxiliary view of the containing plane method (where $X_2Y_2$ is drawn parallel to the edge view $a_1b_1$):

• Plane $ABE$ is viewed square-on, so both lines $AB$ and $CD$ appear at their true length.
• The angle between $a_2b_2$ and $c_2d_2$ is the true angle between the skew lines.
• Unless the question says otherwise, give the acute angle between the two lines.

True Angle Between Lines That Meet

If two lines share a common vertex (such as $AB$ and $BF$ meeting at $B$), they are coplanar. You do not construct an external containing plane; you find the true shape of the triangle they form.

Method 1 (Auxiliary view):

1. Connect the endpoints to form triangle $ABF$ in plan and elevation.
2. Draw a horizontal line across the triangle in elevation and project it down to plan to get its true length.
3. Draw $X_1Y_1$ perpendicular to this true-length line in plan to project the triangle as an edge view.
4. Draw $X_2Y_2$ parallel to the edge view to reveal the triangle in true shape. Measure the angle at $B$ with a protractor.

Method 2 (Rotation / True lengths):

1. Find the true length of each of the three sides ($AB, BF, FA$) by rotating each line in plan until it is parallel to $XY$, then measuring its length in elevation.
2. Construct the triangle full size using compasses from these three true lengths.
3. Measure the included angle at the shared vertex.

Finishing the Drawing, Dimensioning, and Shared Applications

Completing and Dimensioning the Drawing

The marking scheme awards separate marks for projecting the shortest line back into the plan and the elevation, so a solution that stops in the auxiliary views throws those marks away.

• Label every view: Use the question's lettering throughout ($a, b, a', b', a_1, b_1, a_2, b_2$).
• Indicate dimensions clearly: Measure the line in the view where it appears at true length (the first auxiliary in the containing plane method) and write the dimension beside it (e.g. T.L. = 50 mm).
• Distance to the HP: If asked for the height of a horizontal connector above the Horizontal Plane, measure and dimension it in the elevation vertically above the $XY$ line.
• Lines produced: If the lines in the second auxiliary do not intersect within their drawn lengths, extend them until they cross. The connector attaches to the lines produced; label this clearly.
• Leave construction lines: Keep all projection rays and construction lines light and visible as evidence of method.

Connection to Geologic Geometry

If you take Geologic Geometry in Section C, this exact geometry reappears under different names:

• Two boreholes passing through an ore stratum define two lines in space.
• A level line on the stratum surface is the strike line (identical to the horizontal line $ef$).
• Projecting an auxiliary view perpendicular to the strike line shows the stratum as an edge view; its inclination to $X_1Y_1$ is the dip.
• The perpendicular distance between the parallel headwall and footwall edge views is the true thickness of the stratum.

Key terms

Skew Lines

Straight lines in three-dimensional space that are non-parallel, non-intersecting, and non-coplanar.

Containing Plane

A construction plane drawn through one skew line parallel to the second skew line, causing both lines to appear parallel in an auxiliary edge view.

Common Perpendicular

The unique line segment in space that meets two skew lines simultaneously at right angles; its length is the absolute shortest distance between them.

Shortest Perpendicular Distance

The absolute minimum spatial distance between two skew lines, measured along their common perpendicular.

Shortest Horizontal Distance

The shortest connecting segment between two skew lines that lies completely parallel to the horizontal plane of reference.

Strike Line

A horizontal line constructed on an inclined plane; it runs parallel to XY in elevation and displays its true length in plan.

Edge View

A projection where a plane appears as a single straight line, produced by looking directly along a true length line lying on that plane.

Point View

A projection where a line appears as a single point, produced by looking directly along its true length.

True Angle

The angle between the spatial directions of two lines, measured in a view where both lines appear at their true length.

Lines Produced

The extension of line segments beyond their given endpoints to locate a point of intersection or common connector.