#### Theory

The straight lines parallel to the picture plane can have two possible positions:

– they can belong to the picture plane and in this case all their points belong to the picture plane;

– they are out of the picture plane and in this case they have no point in common with the picture plane.

If the line belongs to the picture plane, it is possible to draw and measure it without performing any projecting operation.

If the line is out of the picture plane, it meets the picture plane at infinity and therefore:

– the trace of the straight line is at infinity and we say that it is the direction of the line;

– the vanishing point of the perspective of this line is at infinity, too, and it is the same direction;

– therefore, the perspectives of straight lines that are parallel to the picture plane are parallel to each other.

(1) Now, consider a horizontal line like **d**, which forms an angle of 45° with the picture plane:

– set the point **Td**, that is the trace of the line;

– construct the straight line **d°**, which passes through the centre of projection **O** and that is parallel to **d**;

– since the lines d and **d°** are parallel, they form the same angle of 45° with the picture plane;

– the line **d°** meets the picture plane in **I’d**, which is the vanishing point of the perspective **d’** of **d**;

– as we know, the line (**OO°**) is perpendicular to the picture plane and the triangle (**OO°I’d**) is therefore isosceles;

– thus: the distance (**OO°**) is equal to the distance (**O°I’d**).

(2) Consider, furthermore, a straight line **s** that belongs to a vertical plane **a** which is perpendicular to the picture plane. As you know (see lesson 7 and lesson 5 – Notes) if a line belongs to a plane, it has the trace on the trace of the plane and the vanishing point on the vanishing line of the plane. Therefore, the vanishing point **I’s** of the straight line **s **is the point that the distance circle and the vanishing line **i’****a** have in common.

This means that the straight lines sloping at 45° to the picture plane have the vanishing point on the distance circle. As we will see shortly, these points can be used to measure distances on lines that are perpendicular to the picture plane and, for this reason, they are also called *distance points*.

About the history of the distance points see: Kirsti Andersen, *The Geometry of an Art – The History of the Mathematical Theory of Perspective from Alberti to Monge*, Springer 2007.

#### Practice

(3) Draw two vertical planes on two horizontal parallel lines forming a 45° angle with the picture plane.

– Set, as you like, the points **td** and **te**, which are the traces of the two horizontal lines **d** and **e**;

– since **d** and **e** are horizontal, the vanishing point of their perspective, **I’d**, belongs to the horizon (vanishing line of the horizontal planes) and to the distance circle;

– draw **d’** and **e’**;

– now draw, as you like it, the sides **f’** and **g’** of the two planes, which are parallel to the picture plane and to each other, since the planes are vertical, like the picture plane is.

(4) If you want to draw the perspective **s’** of a line that belongs to a vertical plane **a**, perpendicular to the picture plane:

– set the vanishing point **I’s** on the vanishing line **I’****a** and on the distance circle;

– draw the perspective **s’**.