Profile Projection of a PointLet plane Π_{3} be determined by axes y and z.In the left coordinate system O(x,y,z) the plane Π_{3} is called the 3rd projection plane or the profile projection plane. Orthogonal projection of a point T on the plane Π_{3} is called the 3rd projection or the profile projection of the point T and is denoted with T'''_{3}. The plane Π_{3} is rotated clockwise around zaxis for 90^{o} (left or negative rotation) into the drawing plane. In this rotation the point T'''_{3} is mapped into the point T''' on the plane Π_{2}. The point T''' ∈ Π_{2} is also called the 3rd projection or the left profile projection of a point T. The right profile projection T''' ∈ Π_{2} is gained in the case of the right or positive rotation (counterclockwise) around the zaxis for 90^{o}. In the following, we will only observe the left profile projection and therefore it will be simply be referred to as the profile projection. 

Projection of a point in the drawing plane. 
The plane Π_{3} divides the space into two halfspaces — left and right.
View for the left profile projection is the view from the right side.
The planes Π_{1}, Π_{2} and Π_{3} divide the space into eight octants.
Point T(x,y,z) belongs to a certain octant depending on the sign of the coordinates x, y and z (see table).


The line segment A^{o} B^{o} in the plane Π_{3}, for which is valid d (A^{o}, B^{o}) = d (A,B), is constructed by the rotation of the trapezoid AA'''B'''B around the line A'''B''' for 90^{o}. This procedure is analogous to the procedure earlier explained for determining the true size using rotation into plane Π_{1} or Π_{2}, wherein the length of the parallel edges of the trapezoid are determined by the xcoordinates of the points A and B. If the sign of the xcoordinates of points A and B are different (one point belongs to the left and other ro the right halfspace) then the rotated positions we have two triangles instead of a trapezoid. 

Created by Sonja Gorjanc 3DGeomTeh  Developing project of the University of Zagreb.
Translated by Helena Halas and Iva Kodrnja.