PROJECTION OF THE SPACE TO THE PLANE

Basics of the projection to the plane

Special mapping of the space to the arbitrarily chosen plane (image plane) is denoted as projection.
Two different basic types of projection to the plane can be distinguished, central and parallel.

Central projection

Let S Î E³ be a real point - the centre of projection, not located in the chosen plane p - the image plane.
View of an arbitrary point A Î E³, A S
is the point A_S Î p, which is the intersection point
of the line s^A = AS - projecting line passing through the point A and the centre of projection S
and the image plane, AS = s^A Ç p (obr. 1.29).

Distance of the centre of projection from the image plane is the characteristic distance of the central projection

| S p | = d.

There is attached no image to the centre of projectio, it has no view.

Parallel projection

Let s be a given direction - direction of the projection, a pencil of lines in the space E³ passing through the same ideal point in the direction s. Let the direction s intersects the chosen arbitrary plane - image plane p in the real point.
View of an arbitrary point A Î E³ is the point A_P Î p,
which is the intersection point of the image plane
and the line s^A, A Î s^A in the direction s, projecting line passing through the point A, AP = s^A Ç p (obr. 1.30).

Parallel projection is a special case of the central projection, when the centre S is an ideal point of the space E³.

Central and parallel projections are bijective mappings of the space E³ on the image plane.
All points in the image plane are invariant in both projections.
All points on any projecting line are mapped to the same point in the image plane, the intersection point.

View of a geometric figure U is a figure in the image plane,
which can be constructed from the views of all points on the figure U.
View of a line, which is not a projecting line, is a line (obr. 1.31).
View of a projecting line is a point.
Invariant intersection point of a line and the image plane is the line trace.
Set of all projecting lines of points located on one line forms the line projecting plane.
View of a line is the intersection line of the line projecting plane and the image plane.

View of a plane, which is not a projecting plane, is the entire image plane.
View of a projecting plane is a line.
Intersection line of a plane and the image plane is the plane trace.
The plane trace is the set of all line traces on all lines located in the plane.
Any line located in the plane, and sharing a common ideal point with the plane trace,
is called principal line in the plane (it has the ideal line trace).
View of a principal line in the plane is a line parallel to the plane trace (obr. 1.32).
Any line in the plane perpendicular to the plane trace (and all principal lines in the same direction)
is called slope line in the plane.
Acute angle j, which the slope line in the plane forms to the image plane,
is the angle of the plane to the image plane, we speak about the slope of the plane to the image plane.

Parallel (central) projection is an affine (collinear) mapping of the space to the plane.

Properties of the parallel projection:

1. View of the figure U, which is located in the plane parallel to the image plane, is the figure congruent to the figure U.

2. Parallelism (common ideal point) is the invariant property of the parallel projection.
Parallel lines not in the direction of projectionare mapped to the parallel views.
Parallel planes have parallel plane traces and views of principal lines.

3. Ratio of three points on the line is invariant, l(ABC) = l(A₁ B₁ C₁).
Centre of a figure is mapped to the centre of the figure view (obr. 1.33).

Orthogonal projection is a parallel projection, while the projecting lines in the direction s are perpendicular to the image plane.
In addition to the properties 1. - 3. of the general parallel projection, special properties are valid for the orthogonal projection and orthographic views of geometric figures under this projection:

4. Let AB be a line segment on the line a, forming the angle j to the image plane p.
Length of the orthographic view A₁B₁ satisfies the following:

| A₁B₁ | = | AB | cos j

The length of the orthographic view of a line segment is therefore
lesser then the length of the original for j Î ( 0°, 90° ) (obr. 1.33, | AB | > | A₁B₁ |),
for j = 90° it is zero (A₁ = B₁),
or it is equal to the length of the original, for j = 0° (obr. 1.34, | A₁ V₁ | = | AV | ).

5. The right angle is mapped as the right angle,
if at least one arm of the angle is parallel to the image plane
and none of the arms is perpendicular to the image plane (obr. 1.34).

Slope line s in the plane a, which is not perpendicular to image plane,
is mapped to the line s₁ perpendicular to the plane trace p^a (and views of the principal lines in the plane).
Orthographic view k₁ of the line k perpendicular to the plane a is perpendicular to the plane trace p^a
(and views of the principal lines in the plane) (obr. 1.35).

The idea to map three-dimensional objects to the plane originated in the practical needs of the human activities and is very important in many branches. For the technician, it is not enough to map the object only, but he needs also to find out some features of the mapped object right from the views, or to reconstruct it entirely and unambiguosly.
This problem can be solved using different types of projection methods.

Projection method is a bijective mapping of the points in the space onto the plane.
The mostly used types of projection methods are:

altitudinal projection
orthogonal projection to two orthogonal image planes - Monge method
orthogonal and oblique axonometric method
central projection and linear perspective
stereoscopic projection (with two centres)

The choice of the proper projection method depends on the usage of the created views.
In art, architecture and civil engineering, the commonly used methods are linear perspective, Monge method,
axonometry, or altitudinal projection.
In machine engineering, Monge and axonometric methods are used.

The creative work of engineers - constructors of new machines, equipments and machine parts,
but also the work of designers and technicians, who design and construct this new objects on the base
of the technical drawings, is unthinkable without a good knowledge of projection methods.
New methods in constructions and computer aided design even stressed the importance and accuracy
of the spatial abilities and reconstruction capabilities of the three-dimensional objects
designed and visualized by projecting in some of the projection methods
and plot at the screen of the computer or using any other computer graphics device (plotter, drawing machine).