**Theory**

- Consider a plane π
**°**, passing through the centre of projection**O**and parallel to the picture plane**p**: we call this plane*front plane*. - 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****°**. - 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.

**Notes**

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.

**Practice**

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.

**Notes**

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.