Go to Top

Multi-geodesic lines (1)

The most common method of creating patterns on a membrane involves drawing geodesic lines on the membrane.  Justification of the layout of the geodesic lines is beyond the scope of this text .

Although the geodesic lines can be drawn freely to form patterns of any shape and position, it is common that the route of the geodesic lines follow some models determined:

  1. Geodesic lines more or less parallel to create patterns of elongated rectangular shape to use the fabric with maximum efficiency in fabrication of the membrane.
Patrón paralelo

Parallel pattern

  1. Radial geodesic lines where the center or meeting point coincides at top of the membrane
Patrón radial

Radial pattern

In the first case, how to draw these roughly parallel geodesic lines is something that every user usually does in a particular way depending on different objectives: avoiding waste material, more aesthetic orientations, etc.

WinTess speeds up dramatically this process using the menu: Multi-geodetic .
This offers four possibilities: Trans-auxiliar, Trans-geodesic, Between 2 geodesics and Radial .

The first two options are virtually identical except that an auxiliary line can go from anywhere to anywhere, even to areas where there is no membrane.  Trans-geodesic is always on the membrane which can be a limitation. The user can check through practice as each of these options have their advantages and disadvantages.

First we must look at the number of sections forming the auxiliary or geodesic lines which will be used for Multi-geodesic. Bear in mind that it is the user who chooses how many sections the line will have at the time of generating. WinTess draws a geodesic at each point of the line, at the beginning, the end and at each intermediate point.

How does WinTess do this? As it has been said, an auxiliary or geodesic line is actually a poly-line formed by segments. There are two segments and a plane perpendicular to the tangent at each interior node. The tangent vector is the vectorial sum of each segment. This cuts the perimeter of the membrane at different locations. If it is cuts with two points we find the most typical and simplest case. WinTess draws a geodesic line from end to end. Conversely, if it cuts the perimeter of the membrane in more points then the issue becomes more complicated. If you cut the perimeter in an odd number of points, no trace geodesic cannot be drawn as it is not possible to know which is the remaining point. Conversely, if you cut the perimeter in an even number of points, it can be plotted using different geodesic breakpoints pairs.

Let’s take an example. Suppose a membrane in the form of a hyperbolic paraboloid. We draw a horizontal auxiliary line on the plan view connecting the highest two opposite ends.

malla-1

Planes perpendicular to the line will be obviously vertical planes, all parallel to each other. These planes cut the membrane at two points and therefore they will create more or less parallel geodesic lines (it is important to understand that the geodesic created with these two endpoints need not be contained in the plane, but almost it is)

Línea auxiliar horizontal

Horizontal auxiliary line

These geodesic lines separated by an equal distance on plan will produce patterns of different width due to the curvature of the paraboloid itself.

malla-3

We see that pattern 1 (located on the inclined upper part) is much wider than the pattern 3 (found in the flat middle).

We will repeat the same exercise with a geodesic line located on the membrane, like generatrix line.

Línea geodésica generatriz

Generating a geodesic line

We use Trans-geodesic to create geodesic lines on geodesic line 1. We get something like this.

malla-5

In this case we can see how the four patterns have substantially the same width as the distance of the sections coincides with the membrane, not in the previous case.

malla-6

 

Precaution

As mentioned above, the method of multi-geodesic uses end points of the new geodesic lines found at the edge of the membrane. If we are with a membrane formed by different parts, we can not perform the function of multi-geodesic directly.
doble

In previous drawing membrane we have a bow bent in the middle and two distinct parts by the arc. This arc was generated with an auxiliary line. Therefore it would seem that we could use this auxiliary line to generate the multi-geodesic lines. If we did that we would get the following:

doble-2
Our geodesic lines go from side to the other side, but they are wrong: they don’t even touch the upper arch. They pass under.
What we must do is break down the membrane in two simple membranes and find the pattern in each.

doble-3The experience and continued use of this new command of WinTess will indicate which one is the ideal way to generate multi-geodetic. What is clear that due to the speed in generating them, it does not cost much to go back and change the way you do in order to obtain geodesic for more uniform patterns according to our needs.

Ramon Sastre
April 2014

, ,