Basic elements of the Euclidean plane will be denoted in the following way:
points – capital Latin letters (A, B, C, D,...)
lines – lower-case Latin letters (a, b, c, d,...)
planes – capital Greek letters (Α, Β, Γ, Δ,...).
## Projection of a PointLet two perpendicular planes Π_{1} and Π_{2} be given in space intersecting at a line x.
The plane Π _{1} is horizontal and is called the 1st projection plane or horizontal projection plane or ground plane.
The plane Π _{2} is vertical and is called the 2nd projection plane or vertical projection plane or frontal plane.
The line of intersection x is called the ground line or reference line.
Let T be an arbitrary point in space. Orthogonal projection of the point T onto the plane Π _{1} is called the 1st projection or horizontal projection of the point T, and is denoted by T'.
_{1}Orthogonal projection of the point T onto the plane Π _{2} is called the 2nd projection or vertical projection of the point T, and is denoted by T''.
We now rotate the plane Π _{1} around the intersection line x counterclockwise by 90^{o}.In this rotation the point T' _{1} transforms to the point T' on the plane Π_{2}.
The point T' ∈ Π_{2} is also called the 1st projection or horizontal projection of the point T.
The correspondence T —> (T',T'') is called Monge's method of projection or the two picture protocol.
The straight line T'T'' ∈ Π_{2} is perpendicular to the intersection line x, and is called line of recall of the point T.
In space, we introduce a left rectangular (Cartesian) coordinate system such that the x-axis coincides with the intersection line of the planes Π_{1} and Π_{2}. Now, every point T in space is determined by its coordinates T(x,y,z), for which the following applies:
d(T,Π and _{1}) = |z|d(T,Π.
_{2}) = |y|Furthermore z = 0, then T lies in the plane Π_{1} (T ∈ Π),
_{1}y = 0, then T lies in the plane Π_{2} (T ∈ Π).
_{2}Both projection planes Π _{1} and Π_{2} divide the space into 2 half-spaces, and together into 4 quadrants.Half-spaces are referred to as upper and lower (above and under the plane Π_{1}), front and back (in front of and behind the plane Π_{2}).
Quadrants are referred to as: I (upper front), II (upper back), III (lower back), IV (lower front).
The half-space or quadrant of T(x,y,z) depends on the sign of the y and z coordinates (see table).
From the horizontal and vertical projections of a point it can be determined to which quadrant it belongs. In the figure below are some examples of points in different quadrants and their orthogonal projections (A ∈ I, B ∈ II, C ∈ III, D ∈ IV). ## Points in special positionsIf a point lies in one of the projection planes, then it will coincide with its projection onto that plane, while the other projection will lie on the x-axis._{1}, then T' = T and T'' ∈ x, i.e. its vertical projection lies on the x-axis.
_{2}, then T'' = T and T' ∈ x, i.e. its horizontal projection lies on the x-axis.
On the following figure projections of the following points are shown: A,B ∈ Π _{1}, C,D ∈ Π_{2} and E ∈ x.
Another two planes in space for which the horizontal and vertical projections are in a special position are: Σ – symmetry plane, which bisects the I^{st} and III^{rd} quadrant - the horizontal and vertical projections of every point are symmetrical around the x-axis.
Κ – coincidence plane, which bisects the II^{nd} and IV^{th} quadrant - the horizontal and vertical projections of every point coincide.
On the figure below orthogonal projections of the two points A,B ∈ Σ are shown. On the figure below orthogonal projections of the two points A,B ∈ Κ are shown. Created by Sonja Gorjanc 3DGeomTeh - Developing project of the University of Zagreb. |