Developable surface

A developable surface called out from intuition in geometry and in differential geometry, cartography and the topology of a two-dimensional surface that can be transformed without internal shape distortion in the Euclidean plane. Without restriction to a finite number of cuts are made, but the items must then be put without further compression or squeezing smooth on one level.

The best known examples are the surfaces of certain three-dimensional bodies such as cubes or cones. The mathematical definition of the inner runs of general metric and curvature and is independent of a possible embedding. However, for the special case of the visual, three-dimensional Euclidean space with induced metric that is there any developable surface is a ruled surface, although ruled surfaces are defined quite differently. The converse is not true, nor the statement of surface embeddings in higher dimensional Euclidean spaces is considered.

A developable ruled surface is also called developable surface.

Definition

An area or more specifically a two -dimensional differentiable manifold is called developable if its Gaussian curvature is zero at every point of the surface, then what exactly happens when one of the two (or both) the principal curvatures is zero.

A Riemannian manifold of dimension whose Riemannian curvature tensor is everywhere zero, is called flat manifold. So one can understand a developable surface as a two -dimensional flat manifold.

Examples

Developable body

The Oloid is one of the very few known body whose entire surface is free of kinks and unwound in one piece. Also, square body, such as prisms, pyramids or polyhedra have developable surfaces, the edges do not affect this, where a geodesic line can run without " kink" on the edge of a prism and has the means of distribution in the plane is rectilinear.

Lateral surfaces

Important developable envelope surfaces are, inter alia, the surfaces of cylinders and cones. The structures located thereon, points, coordinate lines, etc. change their mutual position not if the area is " spread out" in the plane. This property is important for cartography and geodesy, such as conic projections or the pseudo- cylindrical Gauss - Krüger projection.

The feature of shape-persistent settlement applies independently of the cross-section of the original surface, eg for elliptical cylinders.

Approximate rotational body

As can be simple body of revolution such as cylinders or cones as lateral surface handle exactly this is no longer possible for more complex bodies of revolution. In practice, one makes do putting together the body of individual, developable segments which - unlike the shell surfaces - not to the rotation axis, but along the axis of rotation are handled. The greater the number of segments is chosen, the better the assembled body closer to the ideal body of revolution.

A good example of such a body of revolution are approximate onion domes. Basically, with this method can be approximated arbitrary body of revolution - even spheres or ellipsoids - Unwind in segments.

Not developable surfaces

Non developable surfaces are those which are curved in two dimensions, as the ball, which Erdellipsoid or different saddles. Here it is with every image on a plane (Map, optical imaging, etc.) to small or large shape changes, the so-called distortion.

Technical Drawing

In technical drawing is called equivalent to the developable area of ​​the settlement ( an area ). The processing is the diagrammatic representation of the developed surface is required, for example in the production of sheet metal pipes (for example, Plumbing ) for cutting the sheets. ( Flat pattern )

Even if cornered or edged bodies are used in practice rather rare for flat patterns in the training of technical drawing and the one or the other prism or one or the other pyramid is represented unwound to teach the basics of construction of such transactions.

• Settlements

Figure 5: Sheet Metal Processing

Figure 1: Settlement of a sheet metal part

Unsettled frustum

Hexagonal settlement

More

• Length loyalty; ie each sheet goes into a same length. In particular geodesic lines go over into straight lines or line segments.
• The developable surfaces (except the plane) are envelope ( envelopes ) of one-parameter plane groups.