Geometric Concepts for Geometric Design, W.Boehm & H.Prautzsch, 1994

Perspective Drawings

Moving the object

Any perspective projection can be obtained by first moving the object, than projecting it, and finally translating the image (see [Boehm94, fig.6.2, p.51]).

1. Vanishing point v : Parallel lines (in Euclidean space) have the same vanishing point v (in Perspective space).
   
2. Vanishing Line L : Set of vanishing points of all lines which are parallel to a plane (in Euclidean space); this is the vanishing line of that plane. Horizontal lines have vanishing points which define the Horizon H : set of vanishing points of all lines parallel to an horizontal plane.
   
3. Trace : The intersection of a plane P with the image plane IP; it is parallel to the vanishing line of that same plane P.
   
4. Angle conservation : The vanishing points of 2 straight lines are seen from the eye under the same angle these lines form in (Euclidean) space.
   
5. Intersection : The vanishing lines of 2 planes intersect at the vanishing point of the intersection of the planes.
   
6. Infinity : The vanishing point of a line parallel to the image plane lies at infinity.

Let the eye (camera lens) be the origin of the reference coordinate frame [0 0 0]. Let also the image system (image plane) be parallel to x- and z-axes so that its origin is at [0 f  0] . The number f  is the eye distance (focal length). It controls the size of the image (on the image plane).

Neutral plane: Plane y = 0 for which all points have images (in the image plane) at infinity, except the eye whose image is not defined (in the image plane); however the eye lies in the neutral plane.

Principal direction : Normal to the neutral and image planes.

Principal ray : Principal direction from the eye.

Principal point h : Vanishing point of the principal ray (in the image plane).

• h = a¹ l¹ + a² l² + a³ l³ ,    where [ l¹ , l² , l³ ] are the barycentric coordinates of h w/r to a¹ , a² and a³, the (3) vanishing points in the i direction .
  
• Let £ be some direction which forms an angle of 45º with the principal ray. Then, the distance between the vanishing point of £ and h is equal to the eye distance f .

Vanishing Plane Þ : Plane defined by the three vanishing points. It is parallel to the image plane.

Parallel Projection

The three vanishing points are at infinity. The mapping matrix represents a parallel projection.

1-Point Perspective

Only one vanishing point is finite, say a² (see fig. 7.3 on p.63; only 1 direction of the "house" or cuboid has perspectiveness).

Then, h = a² , and a¹ and a³ form an orthonormal system parallel to the image plane.

2-Point Perspective

Two vanishing points are finite, say a¹ and a² (see fig. 7.4 on p.64; 2 directions of the "house" or cuboid have perspectiveness).

Then, h = a¹ l¹ + a² l² , and lies on the horizon, H, defined by a¹ and a² . Furthermore, h is at a distance l¹ from a² and distance l² from a¹ .

The eye distance is given by:   f = | a² - a¹ | × sqrt(l¹ × l²) ,
and a³ is perpendicular to a² - a¹ and proportional to f  in length.

3-Point Perspective

All three vanishing points are finite (see fig. 7.5 on p.65; all 3 directions of the "house" or cuboid have perspectiveness).

Then, h is the orthocenter of the triangle defined by a¹ , a² and a³ . The barycentric coordinates of h are in proportion to the respective internal angles of the that triangle, i.e.:
l¹ : l² : l³ = tg(ß¹) : tg(ß²) : tg(ß³) .

The eye distance is given by:   f = | a² - a¹ | × sqrt(l¹ × l²) / sqrt(l¹ + l²) .