Skew lines and parallel lines in space
Two lines in space can be in three positions:
- Intersecting: they have one common point. They lie in one plane.
- Parallel: they lie in one plane and never meet.
- Skew: they never meet and no plane contains both. Example: the edge on the floor of a room and a non-parallel edge on the ceiling.
Test for skew lines. If line a lies in a plane and line b meets that plane at a point that is not on a, then a and b are skew.
Useful facts. Through a point outside a line there is exactly one line parallel to it. Two lines parallel to a third line are parallel to each other. Through a line there is exactly one plane parallel to a skew line.
Counting in a cube. Take one edge. There are 3 edges parallel to it, 4 edges that meet it and 4 edges that are skew to it (12 − 1 − 3 − 4 = 4).
Angle between skew lines. Slide one line (keeping its direction) until it meets the other. The angle you get is the angle between the skew lines.
Parallel projection and its properties
Choose a plane π (the picture plane) and a direction l that is not parallel to π. From each point A of a figure draw the line parallel to l. It meets π at the image point A′. This is parallel projection. Sunlight on the ground is a good model: the rays are almost parallel.
What is kept:
- A line goes to a line (or to a single point if the line is parallel to l).
- Parallel lines go to parallel lines.
- Points on one line keep their order, and the ratio of segments on a line is kept: if C divides AB as 2 : 3, then C′ divides A′B′ as 2 : 3. In particular, the midpoint of a segment goes to the midpoint of its image.
- A figure lying in a plane parallel to π keeps its exact shape and size.
What is lost: lengths in different directions, areas, and angles. A right angle can become sharp or blunt. Only the ratio of parallel segments (and segments on one line) survives.
Images of plane figures
Because only parallelism and ratios survive, we can say exactly which images are possible:
- Triangle: any triangle can be the image of any triangle. So an equilateral triangle can be drawn as a right triangle or a long thin one.
- Parallelogram, rectangle, rhombus, square: the image is always a parallelogram (opposite sides stay parallel). A square can never be drawn as a trapezoid.
- Trapezoid: the image is a trapezoid, and the ratio of the parallel sides is kept.
- Regular hexagon: a hexagon with opposite sides parallel and three long diagonals through the centre. The ratios 1 : 2 along the diagonals are kept.
- Circle: an ellipse (or a circle if the picture plane is parallel).
Drawing solids. To draw a cube on paper we draw its front face as a parallelogram, copy it shifted for the back face, join the matching corners, and draw hidden edges with a dashed line. A tetrahedron is drawn as a triangle with one point inside or outside and three lines to its corners. All the rules above keep the drawing correct.
Central projection
In central projection all rays start at one point S (a lamp or your eye). The image of A is where the line SA meets the picture plane.
- A line still goes to a line.
- But parallel lines usually go to lines that meet (like railway tracks).
- The ratio of segments is not kept, and the midpoint is not kept either.
- Lines parallel to the picture plane stay parallel.
This is how a camera and the eye see the world, so artists use it for perspective. Parallel projection is used for engineering drawings, because it keeps parallel lines and ratios.
Sections of a tetrahedron and a parallelepiped
A section of a solid by a plane is the polygon where the plane cuts it. Each side of the section lies on one face of the solid, so the number of sides is at most the number of faces.
- Tetrahedron (4 faces): the section is a triangle or a quadrilateral.
- Parallelepiped or cube (6 faces): the section is a triangle, quadrilateral, pentagon or hexagon.
Theorem. If a plane cuts two parallel planes, the cut lines are parallel. So in a parallelepiped, the sides of a section on opposite faces are parallel.
How to construct a section through three given points:
- If two of the points lie in the same face, join them. That is a side of the section.
- If the plane has already cut one face along a line, draw through a point of the opposite face the parallel line.
- If you are stuck, extend a known side until it meets the plane of another face. This point, called a trace point, lies on the section plane and in that face plane. It gives a new side.
- Keep going until the polygon closes.
Special sections. A plane through the midpoints of the three edges at a cube corner gives an equilateral triangle. A plane through the midpoints of six edges of a cube gives a regular hexagon. A plane through the midpoints of four edges AB, BC, CD, DA of a regular tetrahedron gives a square.
Try it: shadows and a bread slice
Shadows. Hold two pencils parallel above a notebook and shine a torch from far away: the shadows are parallel. Now bring the torch close: the shadows spread out. That is parallel against central projection.
Sections. Cut a loaf of bread or a block of clay straight, then at a slant, and look at the cut face. In the 3D, predict the number of sides before you slide the plane.
Key formulas and definitions
- Parallel projection: lines → lines; parallel → parallel; AC : CB = A′C′ : C′B′
- Midpoint of a segment → midpoint of the image
- Image of a triangle: any triangle; of a parallelogram: a parallelogram
- Central projection: parallelism and ratios are not kept (except lines parallel to the picture plane)
- A plane cutting two parallel planes cuts them along parallel lines
- Section sides ≤ number of faces: tetrahedron ≤ 4, parallelepiped ≤ 6
Worked examples
1. A cube has 12 edges. How many are skew to one chosen edge?
Remove the edge itself (1), the 3 parallel edges and the 4 edges that meet it (2 at each end). 12 − 1 − 3 − 4 = 4 skew edges.
2. Point C is on segment AB with AC : CB = 2 : 3. The image of AB under a parallel projection is A′B′ = 10 cm. Find A′C′.
The ratio on a line is kept, so A′C′ : C′B′ = 2 : 3. A′C′ = 10 × 2/5 = 4 cm.
3. Can the image of an equilateral triangle under parallel projection be a right triangle?
Yes. Any triangle can be the image of any triangle, so an equilateral triangle can be drawn as a right triangle.
4. Can the image of a square be a trapezoid with only one pair of parallel sides?
No. Opposite sides of a square are parallel, and parallel lines stay parallel. Both pairs stay parallel, so the image is a parallelogram.
5. A cube has edge 4 cm. A plane passes through the midpoints of the three edges that meet at one corner. What is the section and its side?
Each cut point is 2 cm from the corner along an edge. Two cut points are at distance √(2² + 2²) = 2√2 ≈ 2.83 cm. All three sides are equal, so the section is an equilateral triangle of side 2√2 cm.
6. In a regular tetrahedron ABCD with edge 6 cm, the plane passes through the midpoints M, N, P, Q of AB, BC, CD, DA. Find the section.
MN is a midline of triangle ABC, so MN ∥ AC and MN = 3. QP ∥ AC and QP = 3 in the same way. NP ∥ BD, MQ ∥ BD, both 3 cm. In a regular tetrahedron opposite edges AC and BD are perpendicular, so the angles are 90°. The section is a square of side 3 cm and area 9 cm².
Common mistakes
- Thinking the parallel-projection image of a square must be a square. It is a parallelogram in general.
- Believing parallel lines stay parallel in central projection. They do not (except when parallel to the picture plane).
- Calling two lines skew just because they do not meet. Parallel lines also do not meet: skew lines must also be non-parallel.
- Drawing a section with more sides than the solid has faces. A tetrahedron section can never be a pentagon.