Reconstruction (descriptive geometry)

1: true lengths: Borsig-Palais (Wikimedia File: 1881 Palais Borsig.jpg)

In descriptive geometry, reconstruction is understood as a collection of methods to determine true lengths or entire floor plans and elevations including the location of the photographer (eye point) on the basis of a photo ( central projection ) under certain assumptions .

If one only wants to determine true dimensions, it is sufficient to construct the corresponding measuring points for the corresponding directions . Measurement points are vanishing points, from which the distances to be measured are projected onto a straight track (straight line in the illustration) (see below). If you want to reconstruct the ground plan and elevation and the position of the eye point, you use a reversal of the architect's arrangement with which you normally construct central projections.

In the first picture (Borsig-Palais) it was assumed that a) the front edge of the building is in its true length, b) the main point lies in the middle of the picture horizon. In uncut photographs, the main point is always in the center of the image. In this case it is assumed that a piece of the road has been cut off at the lower part of the image. If you do not want to make this assumption, you need further information about the building (e.g. proportions of window edges) in order to determine the main point. It is a technical assumption that the front edge of the building is in the panel. This results in all dimensions except for a fixed factor (scaling). This scaling factor can be determined from the actual length of any edge. So you shouldn't take the word “true” too literally with true length . But on an architect's blueprint, it is not the actual dimensions that can be seen, but the dimensions reduced with a scaling factor (e.g. 1: 100).

Measuring points and true lengths

2: true length of a line in the standing plane (floor plan)

3: true length of a line in the standing plane (perspective)

4: true length, distances parallel to the illustration

5: true length, width, height of a house

For the following considerations, we assume the so-called standard arrangement:

1. The panel is vertical. ${\displaystyle \pi }$
2. The main point , the horizon , the baseline and the distance are known. ${\displaystyle H}$ ${\displaystyle h}$ ${\displaystyle s}$ ${\displaystyle d}$

The horizon usually results from the vanishing points of two mutually perpendicular horizontal directions (see photo). In the plan form a right triangle, i. H. lies on the Thaleskreis . The baseline s is always parallel to the horizon h. A variation in the baseline produces similar reconstructions, i.e. H. they are the same except for a scaling factor. ${\displaystyle h}$ ${\displaystyle F_{1},F_{2}}$ ${\displaystyle O',F'_{1},F'_{2}}$ ${\displaystyle O'}$ ${\displaystyle F'_{1},F'_{2}}$

With the two-panel projection (parallel projection), the line, the true length of which is to be determined, is rotated parallel to one of the crack panels and the real length is then read off in the other crack panel (see true length ). In the case of central projection, it is not sufficient to rotate it parallel to the image board, since in the case of central projection the length of a line parallel to the image board is changed. Unless the rotated route is already in the picture table.

If the route lies in the standing plane (floor plan) and is not parallel to the table: Imagine the route around the track point (the straight line through ) rotated with a vertical axis of rotation in the table on the base line (see figure). Since a rotation is difficult to represent in the perspective image, one thinks up a parallel projection that does the same thing. The associated vanishing point is called the measuring point . It is the same for everyone on parallel routes. In the case of a horizontal line, as assumed here, lies on the horizon and is determined by rotating um in the image table. Justification of this construction: The triangle is isosceles. Because of the existing parallelism (see picture) it is also isosceles, i.e. H. the distances and are the same length. But since we do not assume that the floor plan is known, we have to determine the measuring point in the perspective image (photo) (see image). ${\displaystyle AB}$ ${\displaystyle S_{g}}$ ${\displaystyle g}$ ${\displaystyle A,B}$ ${\displaystyle s}$ ${\displaystyle M_{g}}$ ${\displaystyle AB}$ ${\displaystyle M_{g}}$ ${\displaystyle O'}$ ${\displaystyle F'_{g}}$ ${\displaystyle S'_{g},B',{\tilde {B}}'}$ ${\displaystyle F'_{g},O',M'_{g}}$ ${\displaystyle F'_{g}O'}$ ${\displaystyle F'_{g}M'_{g}}$ ${\displaystyle M_{g}}$

If the line (in the standing plane) is parallel to the table, you can choose any point on the horizon as the measuring point.

Summary: Determination of the true length of a line in the standing plane. ${\displaystyle AB}$

• If the route is parallel to the image panel, you can to choose arbitrarily. ${\displaystyle M_{g}}$ ${\displaystyle h}$
• If the route is not parallel to the illustration.

1. Draw the vanishing point of the straight line on which the route lies. ${\displaystyle F_{g}}$
2. Draw above or below the principal point at a distance (distance) and turn around at the horizon . This gives the measuring point . ${\displaystyle d}$ ${\displaystyle O'}$ ${\displaystyle O'}$ ${\displaystyle F_{g}}$ ${\displaystyle h}$ ${\displaystyle M_{g}}$

The projection of the path of on the service line provides the true length of the track. ${\displaystyle AB}$ ${\displaystyle M_{g}}$ ${\displaystyle s}$

If the line is vertical , i. H. is parallel to the illustration, you can also choose the measuring point anywhere on the horizon. The measuring point (vanishing point) and the route define a vertical plane that intersects the image table in the associated track. As in the horizontal case, the distance from the measuring point is projected onto the track of the plane (see picture). ${\displaystyle AB}$

The true length of a non-horizontal and non-vertical line is determined with the help of a suitable plane containing the line. The measuring point is then on the distant line of the auxiliary plane. As in the above case, the distance from the measuring point is projected onto the track of the auxiliary plane.

Figure 5 shows the perspective picture of a house including the main point, horizon and base line. The numbers indicate the order of construction steps to determine the true length, width, height and ridge height.

Comment on the Borsig-Palais (see above): In this case, the base line ( ) is not suitable for determining true lengths, since projecting from a measuring point ( ) onto the base line would result in grinding cuts , which would make the construction inaccurate. For this reason, the length and width of the building were constructed on a replacement level with a base line (above the windows). ${\displaystyle s_{1}}$ ${\displaystyle M_{1},M_{2}}$ ${\displaystyle s_{2}}$

Note: In the literature it is described a) how one can determine true lengths with an inclined picture board and b) the requirements of the standard arrangement (see above) can be weakened by further information about the building.

Reconstruction of the floor plan and elevation

6: Reconstruction of the floor plan and elevation of a house, external orientation

The external orientation of a central projection is understood to mean the position of the eye point and the image table relative to the object that is being imaged. The architect's arrangement , which is used for the construction of central projections, is reversed around the external orientation with a known standard arrangement (vertical picture panel, H, h and d are known, see above) . Figure 6 shows the individual reconstruction steps and their order (see numbers):

1. Draw a parallel to the horizon below or above the perspective image . is the floor plan of the picture board (see architect arrangement). ${\displaystyle \pi '}$ ${\displaystyle h}$ ${\displaystyle \pi '}$
2. Transfer the main point and all necessary vanishing points to the floor plan. The ground plan of the eye point lies on the perpendicular to in at a distance , the distance. ${\displaystyle H}$ ${\displaystyle O'}$ ${\displaystyle \pi '}$ ${\displaystyle H'}$ ${\displaystyle d}$
3. Reconstruction of a straight line that lies in the standing plane: Determine (if not already done in 2.) the outline of the vanishing and track point of the straight line . is then a parallel to through . ${\displaystyle F'_{g},S'_{g}}$ ${\displaystyle g}$ ${\displaystyle g'}$ ${\displaystyle O'F'_{g}}$ ${\displaystyle S'_{g}}$
4. Reconstruction of a point which lies in the ground plane: Draw with the aid of the solder on and the layout of the projection beam (straight line ). With the help of the ground plan, another straight line (e.g. contour lines or house edges, ...) is finally obtained . ${\displaystyle P}$ ${\displaystyle {\overline {P}}'}$ ${\displaystyle {\overline {P}}}$ ${\displaystyle \pi '}$ ${\displaystyle O'{\overline {P}}'}$ ${\displaystyle P}$ ${\displaystyle P'}$
5. A point that is not in the standing plane (e.g. a ridge point) can be reconstructed in the same way if its floor plan is known or can be constructed in the perspective image. The height of such a point is obtained from its true length (see above).

See also

true length

literature

• Fucke, Kirch, Nickel: Descriptive Geometry. Fachbuch-Verlag, Leipzig 1998, ISBN 3-446-00778-4 , p. 247
• Cornelie Leopold : Geometric Basics of Architectural Representation. Verlag W. Kohlhammer, Stuttgart 2005, ISBN 3-17-018489-X , p. 242

Web links

• Normal (orthogonal) axonometry with simple examples
• Descriptive geometry for architects (PDF; 1.5 MB). Script (Uni Darmstadt), p. 116