Lesson 6 – Perspective of points that belong to the straight line r




  1. Consider a plane π°, passing through the centre of projection O and parallel to the picture plane p: we call this plane front plane.
  2. Now we can fix these notable positions:
    T – the point T that the line r has in common with the picture plane;
    P – any points like P, which are located beyond the picture plane;
    I – the point I, which is at infinity;
    Q – any points, like Q, which are located between the picture and the front plane;
    A – the point A that the line r has in common with the front plane p°.
  3. The above mentioned points have the following perspectives on the line r’:
    T has a perspective T’ coincident with T that is the trace of r, as we know;
    P has a perspective P’;
    I, point at infinity, has a perspective I’ that is the vanishing point of the straight line r, but it is also the vanishing point of any line parallel to r;
    Q has a perspective Q’;
    A has a perspective A’ that is the point at infinity of the line r’: in fact, the straight line projecting the point A belongs to the front plane and it is therefore parallel to the straight line r’; this means that they meet at infinity.


You might consider the correspondence between the straight line r and its perspective r’ as a good demonstration of the power of perspective, which allows to imagine and to view infinitely distant entities.


The lines r’ and s’, incident in I’r, are the perspectives of two parallel lines r and s;

The lines u’ and v’, parallel, are perspective of two lines incident in a point A that belongs to the front plane.


If a circle has no points on the front plane, its perspective is an ellipse.

If a circle is tangent to the front plane, its perspective is a parabola, since one of its points has his perspective at infinity.

If a circle passes through the front plane, its perspective is an hyperbole, since two of his points have their perspectives at infinity.