Descriptive geometry: intersections and polyhedra

Before building a warehouse in Kisangani, the architect draws the building on a flat sheet. How can an object in space be represented exactly on a plane, and how can lengths, intersections and sections be read from the drawing? That is the purpose of descriptive geometry: the object is projected onto two perpendicular planes and we work on the resulting drawing, called the two-view drawing.

Projections and the two-view drawing

We use a horizontal plane $\pi$ and a frontal (vertical) plane $\pi'$, which meet along the ground line, also called the hinge line ($ch$). A point $A$ in space has:

• a horizontal projection $a$ (on $\pi$) and a frontal projection $a'$ (on $\pi'$);
• a height: its distance to the horizontal plane;
• a depth: its distance to the frontal plane.

To obtain the two-view drawing, $\pi$ is rotated about the hinge line until it lies in the plane of $\pi'$. Then $a$ and $a'$ lie on the same perpendicular to $ch$ (the reference line): for a point in front of $\pi'$ and above $\pi$, $a'$ is above $ch$ at a distance equal to the height, and $a$ is below it at a distance equal to the depth.

Lines, planes and intersections

• The traces of a line are its intersection points with $\pi$ (horizontal trace, height zero) and with $\pi'$ (frontal trace, depth zero).
• A general plane is often given by its traces: its lines of intersection with $\pi$ and with $\pi'$; they meet on the hinge line.
• Intersection of two planes: it is a line. It passes through the common point of the two horizontal traces and through the common point of the two frontal traces.
• Line and plane: a line is parallel to a plane if it is parallel to a line of that plane. Otherwise it pierces the plane at one point, found by cutting the plane with an auxiliary plane containing the line.
• Two lines in space are intersecting, parallel or skew (not coplanar): they intersect if their crossing points in the horizontal view and in the frontal view lie on the same reference line.

Polyhedra and plane sections

A polyhedron is a solid bounded by polygonal faces (cube, prism, pyramid…). For a convex polyhedron, Euler’s relation links the number of vertices $S$, edges $A$ and faces $F$:
$$S-A+F=2.$$
The section of a polyhedron by a plane is a polygon each of whose sides is the segment common to the plane and to a face. Method: find the points where the plane cuts the edges, then join them face by face.

• A plane parallel to the bases of a prism gives a section congruent to the bases.
• A plane parallel to the base of a pyramid, at distance $d$ from the apex (height $h$), gives a polygon similar to the base with ratio $k=\dfrac dh$; its area is $k^2$ times the area of the base.
• A section of a cube has at most $6$ sides, since the cube has $6$ faces.