Development (descriptive geometry)

Developing a truncated cylinder: Example

Developing a cylinder: principle

Development of a truncated cylinder

In geometry, the development of a surface is understood as the expansion of a surface in a plane in such a way that lengths (measured on the original surface) are retained. This is only possible for surfaces whose Gaussian curvature is zero everywhere. Such areas are called developable . They always contain families of straight lines, so they are ruled surfaces . Simple examples are cylinders and cones . In addition to these two types, there are other developable surfaces . Cylinders and cones play a special role in technology and thus also in the representational geometry . Descriptive geometry provides methods to determine the development of cylinders and cones graphically. Since a circuit processed in the processing in each case ( rectified must be) and this is drawing only approximately possible descriptive geometry provides settlement of cylinders and cones only approximately . The graphical approximation of the circumference of a circle by the circumference of a regular 12-sided makes z. B. an error of about 1.14%. The error can of course be reduced by increasing the number of corners. If you accept small invoices, (almost) correct transactions can be generated.

A cylinder ( cone ) is always understood here as the outer surface (i.e. without a base and cover) of a straight circular cylinder (circular cone). A truncated cylinder ( truncated cone ) is a straight circular cylinder (circular cone) cut off at an angle at the top, i. H. the upper edge curve is an ellipse.

Developments find practical meanings through their use as templates. If you draw z. For example, the development of a cylindrical cone (truncated cone) in sheet or paper and cuts out this processing, it can be carried on to produce a model of the winding cylinder cone (frustum).

Development of a straight circular cylinder

Specification: A straight circular cylinder (radius , height ) and a point on it in plan and elevation (see 2nd picture). Wanted: The development of the cylinder and the point . ${\ displaystyle r}$ ${\ displaystyle h}$ ${\ displaystyle P}$
${\ displaystyle P}$

If you cut the cylinder along a straight cylinder line and unwind it in the plane, a rectangle of length and height (of the cylinder) is created. Since the length of the rectangle as a development ( rectification ) of the bottom circle cannot be exactly constructed with a compass and ruler, the circle is approximated by a suitable n-gon (here 12-gon), and this n-gon is wound according to the drawing next to it the elevation. To draw the development of the point in the rectangle, take the distance (green) from the next cylinder line (here straight line 2) in the plan and draw the cylinder line containing the development of in the rectangle. With the arrangement shown in the picture, the height of the point can be taken directly from the elevation. ${\ displaystyle 2 \ pi r}$ ${\ displaystyle h}$ ${\ displaystyle P}$ ${\ displaystyle P '}$ ${\ displaystyle P}$

If you want to develop a truncated cylinder (obliquely cut circular cylinder) as in the third picture, the lower part of the cylinder can be used unchanged. From the upper boundary (ellipse), the ellipse points are developed on the straight lines 0, 1, ..., 12 as described above. The development of these points is particularly easy to construct (see 3rd picture).

An improvement in the handling is obtained through

Measure the cylinder radius and request the correct development of the base circle as a segment of the length and subsequent subdivision (here into 12 equal sections). ${\ displaystyle r}$ ${\ displaystyle 2 \ pi r}$
The horizontal coordinate of the development of a point is calculated from the polar angle of the point in the plan. (When developing the truncated cylinder, the existing division from 1. can be used.) ${\ displaystyle P}$ ${\ displaystyle P '}$

Development of a straight circular cone

Developing a cone: principle

Developing a truncated cone: example

Specification: vertical circular cone (base circle radius , height , surface length ) and a point on it in plan and elevation (see 2nd picture). Wanted: The development of the cone and the point . ${\ displaystyle r}$ ${\ displaystyle h}$ ${\ displaystyle l = {\ sqrt {r ^ {2} + h ^ {2}}}}$ ${\ displaystyle P}$
${\ displaystyle P}$

If one cuts the cone along a straight line and unwinds it in the plane, the result is a circular sector of a circle, the radius of which is equal to the length of a surface line of the cone (see picture). The opening of the circle sector is constructed analogously to the case of the cylinder by an approximation of the bottom circle of the cone by an n-gon (here 12-gon) (see picture). One edge of the (regular) n-gon is removed n times on the circular arc of the development. For the development of the point , draw the outline and elevation of the conical straight line (surface line) on which it lies. The distance of the intersection of this surface line with the base circle to an adjacent surface line of the n-gon is entered on the developed base circle and connected with the center of the circle sector (development of the cone tip). The distance between the point and the apex of the cone is obtained by rotating the line in elevation in parallel (in the picture: purple line). ${\ displaystyle P}$ ${\ displaystyle P}$ ${\ displaystyle P}$ ${\ displaystyle {\ overline {SP}}}$

The development of a truncated cone results (similar to the development of a truncated cylinder) by developing the intersection points of the lid ellipse with the surface lines 0, 1, 2, ..., 12.

There is an improvement in the handling

by measuring the radius of the base circle and applying for the correct development of the base circle as an arc of length (the opening angle of the development is ) and then subdividing it (here into 12 equal sectors). ${\ displaystyle r}$ ${\ displaystyle 2 \ pi r}$ ${\ displaystyle 2 \ pi {\ tfrac {r} {l}}}$
The angle coordinate of the development of a point is obtained by multiplying the angle coordinate of (in the plan) by the factor . ${\ displaystyle P}$ ${\ displaystyle P '}$ ${\ displaystyle {\ tfrac {r} {l}}}$

Another example

Section cone-cylinder with development

In the last example, a straight circular cone and a straight circular cylinder intersect (see picture). The developments of the two surfaces contain the development of the intersection curve (red). (To the sectional curve construction: s. Auxiliary ball method .) By recording the transactions on paper and cutting out the transactions can be carried on winding a model of the cone-cylinder combination prepared.

literature

Fucke, Kirch, Nickel: Descriptive Geometry. Fachbuch-Verlag, Leipzig 1998, ISBN 3-446-00778-4 , p. 71
Graf, Barner: Descriptive Geometry. Quelle & Meyer, Heidelberg 1961, ISBN 3-494-00488-9 . Pp. 125,243
Leopold, C .: Geometric basics of architectural representation. Verlag W. Kohlhammer, Stuttgart 2005, ISBN 3-17-018489-X . P. 162

Web links

Descriptive geometry for architects (PDF; 1.5 MB). Script (Technische Universität Darmstadt), p. 62