Monge's Method - Perpendicularity
RECALL basic definitions and
theorems on perpendicularity.
Orthogonal projection of perpendicular straight lines
It was mentioned earlier that projecting an angle, measure of its
projection is NOT the same. Therefore:
Orthogonal projections of two perpendicular lines are NOT perpendicular
The right angle (and all other angles) will be projected in true size if its
sides are parallel to the plane of projection.
The following holds for the right angle:
Orthogonal projection of the right angle is in true size (also a right angle) if
one of its sides is parallel to the plane of projection.
Proof of this statement can be seen on the image below.
A line a is parallel to the plane of projection Π
and perpendicular to the line b. We need to prove that
their projections are also perpendicular, i.e. a' ⊥ b'.
A line a is perpendicular to the vertical projecting plane E
that contains the line b. Namely, a is perpendicular to two lines in
this plane: line b and the projection ray z.
(Line a is parallel to the plane Π, while z is perpendicular to Π.)
Since a' is parallel to a, it must be perpendicular to the plane E.
Hence, a' is perpendicular to all lines in the plane E,
as well as to the line b'.
Proof can be written as:
a ⊥ b & a | | Π => a ⊥ E => a' ⊥ E => a' ⊥ b'.
Straight line and a plane perpendicular to each other
Any line n perpendicular to the plane P is
called a normal line of the plane P
and their point of intersection, N = n ∩ P
is called pedal (perpendicular foot) of the line n.
A line n is perpendicular to a
plane P if and only if its horizontal
projection (top view) is perpendicular to the horizontal trace of the plane and its
vertical projection (front view) is perpendicular to the vertical trace of the plane, i.e.
n ⊥ P <=> p' ⊥ r1 & p'' ⊥ r2.
This claim follows from the fact that a line perpendicular to a plane must be
perpendicular to all lines contained in that plane, also to its traces. Since the
line is perpendicular to the horizontal trace r1 (r2) that
belongs to the plane Π1 (Π2),
horizontal (vertical) projection of this angle will be in true size - right
The image below shows this statement and also shows those principal lines of the
plane that intersect the normal line in its pedal point.
Assignment 1: Construct the projections of a line
that passes through the given point and it is perpendicular to the plane given by
Assignment 2: Construct the traces of a plane
that passes through the given point and it is perpendicular to the given line.
Assignment 3: Determine the distance of a given
point and a given plane (plane is given by its traces).
Assignment 4: Determine the distance between a
given point and a given line.
Bisector plane of a line segment
T ∈ Σ<=>d(A,T) = d(B,T)
Set of points that are equally distanced from points A and B belong to the
plane called bisector plane of the line segment AB.
The bisector plane of the line segment AB contains the midpoint
of the line AB and it is perpendicular to the line AB.
Assignment 5: Construct the traces of the
bisector plane of a given line segment.
True length of the normal line
Let n be the normal line of the plane P
and N their intersection point.
be the 1st steepest line of the plane P that passes
through the point N, and p2 the
2nd steepest line that passes through N.
Due to the property of orthogonal projection of the right angle mentioned before (n' ⊥ r1, n'' ⊥ r2),
lines n and p1 will have the same top view (n' = p'1)
and the lines n and p2 will have the same front view (n'' = p''2).
Therefore, the lines n and p1 lie in the same horizontal projecting
plane and the lines n and p2 lie in the same vertical projecting
These properties are shown in the images below.
The normal line and the1st steepest line through the pedal of the
normal lie in the same
horizontal projecting plane.
The normal line rotated into the plane Π1
is perpendicular to the 1st steepest line rotated as well, they intersect at the
projection of the pedal.
The normal line and the 2st steepest line through the pedal of the
normal lie in the same
vertical projecting plane.
The normal line rotated into the plane Π2
is perpendicular to the 2st steepest line rotated as well, they intersect at the
projection of the pedal.
Assignment 6: Lay a line through the point T(1,3,–)
that is perpendicular to the plane P(–3,4,3)
and has length 4.
Mutually perpendicular planes
Two planes are mutually perpendicular if any one of these planes contain at
least one line that is perpendicular to the other plane.
If two planes are mutually perpendicular then every plane contain
infinitely many lines perpendicular to the other plane (pencil of perpendicular
Using Monge's method it is NOT possible to determine whether two planes
with given traces are perpendicular.
That is because perpendicular planes don't have perpendicular traces.
Try to visualize an example.
Assignment 7: Construct the traces of a plane
through a given line that is perpendicular to a given plane.
Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja -
3DGeomTeh - Developing project of the University of Zagreb