🇨🇩 Geometry

Descriptive geometry: point, line, plane

⏱ 40 minOfficial programme: CD-H2-GEO-03

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🎯 In this lesson, you will learn to

Represent point, line, plane in Monge double projection.

The lesson

Introduction

To make a metal part in a workshop in Likasi, the technician does not work from a photo: he reads a technical drawing where the object is drawn seen from above and seen from the front. These two views are orthogonal projections. Descriptive geometry (Monge’s method) teaches you to represent exactly a point, a line or a plane of space on a flat sheet, and then to recover their true dimensions.

Explanation

Projection planes and the drawing

We use two perpendicular planes: the horizontal plane $\pi$ and the frontal plane $\pi'$ (vertical). They meet along the ground line (or hinge line), written ch.
A point $A$ in space has a horizontal projection $a$ (on $\pi$ ) and a frontal projection $a'$ (on $\pi'$ ).
Then $\pi$ is turned about the ground line until it lies on $\pi'$ : this gives the drawing (épure). The front part of $\pi$ comes below the ground line.
On the drawing, $a$ and $a'$ lie on the same perpendicular to the ground line: the projection line (ligne de rappel).
The height $c$ of $A$ (above $\pi$ ) is read between $a'$ and the ground line; its depth $e$ (distance in front of $\pi'$ ) is read between $a$ and the ground line.

Explanation

The four dihedra

The two projection planes divide space into four dihedra (quadrants):

Dihedron	Position of $A$	$a'$	$a$
1st	above $\pi$ , in front of $\pi'$ ( $c>0$ , $e>0$ )	above ch	below ch
2nd	above $\pi$ , behind $\pi'$ ( $c>0$ , $e<0$ )	above	above
3rd	below $\pi$ , behind $\pi'$ ( $c<0$ , $e<0$ )	below	above
4th	below $\pi$ , in front of $\pi'$ ( $c<0$ , $e>0$ )	below	below

Special cases: a point of $\pi$ has its frontal projection $a'$ on the ground line ( $c=0$ ); a point of $\pi'$ has its horizontal projection $a$ on the ground line ( $e=0$ ).

Since the ground line, the front-back direction and the vertical are pairwise perpendicular, the distance from $A$ to $B$ is $AB=\sqrt{(\Delta x)^2+(\Delta e)^2+(\Delta c)^2}$ .

Explanation

Lines: projections, special lines and traces

A line $D$ in space is represented by its two projections: $d$ (horizontal) and $d'$ (frontal).

Horizontal line (parallel to $\pi$ ): $d'$ parallel to ch; $d$ shows the true length.
Frontal line (parallel to $\pi'$ ): $d$ parallel to ch; $d'$ shows the true length.
Line parallel to the ground line: $d$ and $d'$ parallel to ch.
Vertical line: $d$ reduced to a point, $d'$ perpendicular to ch.
Line perpendicular to $\pi'$ (droite de bout): $d'$ reduced to a point, $d$ perpendicular to ch.
Profile line: $d$ and $d'$ on the same perpendicular to ch.

The horizontal trace $h$ is the point of $D$ with height $0$ ; the frontal trace $f$ is the point of $D$ with depth $0$ .

Explanation

Planes: traces and special planes

A plane not parallel to the ground line is often given by its traces: its horizontal trace (intersection with $\pi$ ) and its frontal trace (intersection with $\pi'$ ), which meet on the ground line. It can also be given by three non-collinear points or by two intersecting lines.

Plane	Traces on the drawing
Horizontal plane (parallel to $\pi$ )	frontal trace parallel to ch, no horizontal trace
Frontal plane (parallel to $\pi'$ )	horizontal trace parallel to ch, no frontal trace
Vertical plane (perpendicular to $\pi$ )	frontal trace perpendicular to ch, horizontal trace in any direction
Plane perpendicular to $\pi'$ (plan de bout)	horizontal trace perpendicular to ch, frontal trace in any direction
Profile plane (perpendicular to ch)	both traces on the same perpendicular to ch

Worked example

A point is located by $(x, e, c)$ : abscissa $x$ along the ground line, depth $e$ and height $c$ , in cm.

Problem: Draw $A(2, 3, 4)$ and $B(5, -2, 3)$ on the drawing and give their dihedron.

Step 1: For $A$ , draw the projection line 2 cm from the origin. Place $a'$ 4 cm above ch and $a$ 3 cm below. Since $c>0$ and $e>0$ , $A$ is in the 1st dihedron; the distance $aa'$ is $4+3=7$ cm.

Step 2: For $B$ , $e=-2<0$ : $B$ is behind $\pi'$ , so $b$ is 2 cm above ch. With $c=3>0$ , $b'$ is 3 cm above ch.

Answer: $A$ is in the 1st dihedron; $B$ is in the 2nd dihedron, with both projections above the ground line, 1 cm apart.

Worked example

A point is located by $(x, e, c)$ : abscissa $x$ along the ground line, depth $e$ and height $c$ , in cm.

Problem: Find the traces of the line $(AB)$ with $A(0, 1, 3)$ and $B(4, 3, 1)$ .

Step 1: From $A$ to $B$ , $x$ increases by 4, $e$ by 2 and $c$ decreases by 2. A point of the line is $(4t, 1+2t, 3-2t)$ , $t\in\mathbb{R}$ .

Step 2: Horizontal trace ( $c=0$ ): $3-2t=0$ , so $t=1.5$ and $h(6, 4, 0)$ .

Step 3: Frontal trace ( $e=0$ ): $1+2t=0$ , so $t=-0.5$ and $f(-2, 0, 4)$ .

Answer: $h(6, 4, 0)$ and $f(-2, 0, 4)$ . On the drawing, $h'$ lies on ch at abscissa 6, and $f$ lies on ch at abscissa $-2$ .

Worked example

A point is located by $(x, e, c)$ : abscissa $x$ along the ground line, depth $e$ and height $c$ , in cm.

Problem: What is the true length of the segment $[AB]$ with $A(1, 2, 1)$ and $B(5, 2, 4)$ ?

Step 1: $A$ and $B$ have the same depth 2: $ab$ is parallel to ch, so $[AB]$ is a segment of a frontal line.

Step 2: So its frontal projection $a'b'$ is in true length: $\Delta x=4$ and $\Delta c=3$ .

Step 3: $a'b'=\sqrt{4^2+3^2}=\sqrt{25}=5$ .

Answer: $AB=5$ cm, as confirmed by the formula $\sqrt{4^2+0^2+3^2}$ .

Watch out

Watch out: Do not mix up the two projections. The height is read on the frontal projection $a'$ , the depth on the horizontal projection $a$ . For a point in the 1st dihedron, $a'$ is above the ground line and $a$ below: a point whose two projections are on the same side is not in the 1st dihedron. Finally, a projection of a segment is never longer than the segment: $ab$ shows the true length only for a horizontal segment, and $a'b'$ only for a frontal segment.

Key points

Two projection planes $\pi$ (horizontal) and $\pi'$ (frontal), ground line ch; projections $a$ and $a'$ on the same projection line.
Height: distance from $a'$ to ch; depth: distance from $a$ to ch.
1st dihedron: $a'$ above, $a$ below; 2nd: both above; 3rd: $a'$ below, $a$ above; 4th: both below.
Special lines: horizontal, frontal, vertical, perpendicular to $\pi'$ , profile; traces $h$ ( $c=0$ ) and $f$ ( $e=0$ ).
Special planes recognised by their traces: horizontal, frontal, vertical, perpendicular to $\pi'$ , profile.

✍️ Exercises

Answer one exercise at a time. Your answer is checked right away, with an explanation.

Exercise 1 of 8

★ Easy

On a drawing, both projections $a$ and $a'$ of a point $A$ lie above the ground line. In which dihedron is $A$ ?

🎯 In this lesson, you will learn to

The lesson

👋 Introduction

📘 Explanation

📘 Explanation

📘 Explanation

📘 Explanation

✏️ Worked example

✏️ Worked example

✏️ Worked example

⚠️ Watch out

⭐ Key points

✍️ Exercises

Introduction

Explanation

Explanation

Explanation

Explanation

Worked example

Worked example

Worked example

Watch out

Key points