Reflection (descriptive geometry)

Central projection room with wall mirror

Frontal perspective of a room with a mirror by Johann Erdmann Hummel , around 1820

The reflection of a plane (mirror) is a method of Descriptive Geometry to drawings more realistic and more attractive. While the reflection of an object (e.g. house) on a horizontal plane (e.g. water surface) or a vertical plane (e.g. wall mirror) can be constructed relatively easily with parallel projection, this is much more demanding with central projection was mainly used in painting. The construction of a reflection is even more difficult with an inclined (neither vertical nor horizontal) mirror.

Mirroring and mirror image

Mirroring: principle

The law of reflection in physics states: When a ray of light is reflected (starting from an object) on a reflective plane, the ray is deflected in such a way that:

Angle of incidence = angle of reflection.

One can therefore assume: The reflected beam caught by the eye comes from the point on the perpendicular of the point to the mirror plane at the same distance from the mirror plane but on the other side (see picture). The point is called the point to be mirrored . Analogously one speaks of a mirrored object (house, cuboid, ...). ${\ displaystyle P_ {s}}$ ${\ displaystyle P}$ ${\ displaystyle P_ {s}}$ ${\ displaystyle P}$

For the construction of mirror images in parallel or central projections, this view has the great advantage:

The mirror image of a point is created by the point reflection at the corresponding plumb line (see picture) and is projected like a real object. ${\ displaystyle P_ {s}}$ ${\ displaystyle P}$ ${\ displaystyle P}$ ${\ displaystyle L_ {P}}$

If the floor plan and elevation of an object (house, ...) are available, the reflections can be drawn into the existing cracks according to the above rule and then a picture of the object and mirror image can be constructed in the usual way ( axonometry , incision method , architect arrangement , front perspective ). Often, however, there are already constructed images in parallel or central projection and mirror images should be added later. In this case, possible solutions for common situations (horizontal mirror plane, vertical mirror plane) are given in the literature.

Reflection of a house in a cavalier perspective, horizontal and vertical mirror planes

Mirroring with parallel projection

With parallel projection, the construction of the mirror image of a point (object) is relatively easy. Requirement is:

The mirror plane is defined in the picture by ground and elevation traces and the normal direction of the mirror plane is known in the picture (see example: house).

Example : A house is given in cavalier projection (oblique axonometry ) (see picture).

The horizontal mirror plane (water surface) lies half a unit below the lower edge of the house (blue lines). The normal direction of the mirror plane is the z-direction.
The vertical mirror plane is a plane parallel to the xz plane at a distance of 1/2 unit. The normal here points in the y-direction.

Reflection of a straight line (green) at the stand level (with central projection)

Reflection of a tower in central projection on a water surface

Reflection in central projection

In contrast to parallel projection, with central projection two equally long stretches on a straight line generally do not appear to be of the same length. This makes the construction of mirror images in central perspective images considerably more difficult. Since the mirror image of a point always lies on the perpendicular to the mirror plane (see above) and the direction of the perpendicular is usually known in important examples (horizontal or vertical mirror plane), the mirror image on the image of the perpendicular must be determined with the help of a straight line . For this you have to know how the mirror image of a straight line is determined in the perspective image. In the following consideration, for the sake of simplicity, it is assumed that the mirror plane is the standing plane with the track and the straight line , the horizon. (Track and straight line of flight of a plane are always parallel straight lines in the perspective image!) If the track point and vanishing point of a straight line (in the perspective image) are known, the following applies to the mirrored straight line : ${\ displaystyle P}$ ${\ displaystyle P_ {s}}$ ${\ displaystyle s}$ ${\ displaystyle h}$ ${\ displaystyle g}$ ${\ displaystyle S_ {g}}$ ${\ displaystyle F_ {g}}$ ${\ displaystyle g_ {s}}$

The vanishing point is created (in the perspective picture) by mirroring the vanishing point on the straight line of flight of the mirror plane. (see picture) ${\ displaystyle F_ {gs}}$ ${\ displaystyle F_ {g}}$ ${\ displaystyle h}$
The track point is created (in the perspective picture) by mirroring the track point on the straight line of the mirror plane. (see picture) ${\ displaystyle S_ {gs}}$ ${\ displaystyle S_ {g}}$ ${\ displaystyle s}$

This rule also applies to other mirror planes.

The following examples assume that the drawing board is vertical . This can be recognized by the fact that the main point is on the horizon . ${\ displaystyle H}$ ${\ displaystyle h}$

Example: tower on a river

The mirror plane is horizontal here and thus has the horizon as a straight line of flight. The straight track is . Since horizontal depth lines (vanishing point = main point) have been selected as auxiliary lines for tower points, their common vanishing point remains fixed when mirroring . The track points are mirrored on (see picture). The top of the tower is in the picture panel and is therefore mirrored directly. The brown straight line is the edge of the bank at the level of the standing level. The bank is a plain sloping towards the water. Since the bank edge is horizontal, its vanishing point remains fixed in the reflection. Only the track point needs to be moved. ${\ displaystyle h}$ ${\ displaystyle s_ {w}}$ ${\ displaystyle h}$ ${\ displaystyle s_ {w}}$ ${\ displaystyle F_ {w}}$

Reflection of a vertical line on a wall mirror with central projection

Room with door and wall mirror, template (above) and solution (below)

Example: a room with a wall mirror

When mirroring on a perpendicular mirror, you can take advantage of that

A line parallel to the mirror and the mirrored line form a rectangle in which the diagonals on the mirror plane (intersection of the rectangle with the mirror) intersect and halve this middle line (see picture).
In the case of a central projection, the center point of a line that is parallel to the image table merges into the center point of the image line. (This is not the case with routes that are not parallel to the illustration!)

This property was used in the construction of the mirror image of the door in the perspective image (see picture). The two vanishing points of the horizontal room edges are shown in the template. The right vanishing point is also the vanishing point of the normal of the mirror plane. To construct the mirroring of the vertical door edge , first draw the straight line connecting the vanishing point and determine the point as the center of the two points (images of the plumb points on the mirror). The straight line intersects the straight line at the point . ${\ displaystyle F_ {n}}$ ${\ displaystyle {\ overline {P}} \; {\ overline {Q}}}$ ${\ displaystyle F_ {n}}$ ${\ displaystyle {\ overline {M}}}$ ${\ displaystyle {\ overline {L}} _ {P}, {\ overline {L}} _ {Q}}$ ${\ displaystyle {\ overline {Q}} \; {\ overline {M}}}$ ${\ displaystyle {\ overline {P}} F_ {n}}$ ${\ displaystyle {\ overline {P}} _ {s}}$

literature

Rudolf Fucke, Konrad Kirch, Heinz Nickel: Descriptive Geometry. Fachbuch-Verlag, Leipzig 1998, ISBN 3-446-00778-4 , p. 241.
A. Pumann: Descriptive Geometry. Part 2, Pumann, Coburg 1998, ISBN 3-9800531-1-3 , p. 107.
F. Rehbock: Descriptive Geometry. Springer-Verlag, Berlin / Göttingen / Heidelberg 1969, ISBN 3-540-04557-0 , p. 184.

Web links

Descriptive geometry for architects. (PDF; 1.5 MB). Script (Uni Darmstadt), p. 103.