Two questions regardingthe design of caravan awnings were posed by a company.The company wishes to produce awnings with a pretty appearance. When an awning is attached to a caravan, some wrinkles could appear. We developed some methods to avoid the wrinkles. The problem is restricted to awnings which are made from one piece of cloth

Beckum, F. van

Hof, B.

Margaretha, H.

Susanto, H.

English

Mathematics in Industry Information Service Eprints Archive

4Caravan Awnings: a Geometrical Problem
B. van 't Hof, H. Margaretha, H. Susanto, EP.H. van Beckum
Abstract
Twoquestions regardingthe design of caravan awningswere posed by a com-
pany.The companywishesto produceawningswith a prettyappearance.When
an awning is attached to a caravan, some wrinkles could appear. We devel-
oped somemethodsto avoidthe wrinkles.The problemis restrictedto awnings
which aremade fromone piece of cloth.
Keywords
Developable surfaces, caravan awning, minimal surface, optimal surface de-
sign.
4.1 Introduction
An interesting problem was posed by Gerjak B.V., a company which specializes in
caravan awnings and tents. Mathematicians were asked to help the company to create
an applicable mathematical model for designing good-looking caravan awnings.
A caravan awning is needed to protect campers from unpleasant weather condi-
tions. The customers' expectation is to have good-looking awnings for their caravans.
When it is attached to the front of the caravan, the awning should be taut and visually
pleasing, so no wrinkles should appear. The back part of the awning is attached along
the upper part of the caravan's body by a rail. This means that the back part of the
awning follows the curved shape of the caravan's body, which is not triangular. In
this report, we consider the classical caravan awning, i.e. the one with a triangular
shape at the front. An example is given in Figure 4.1. This type of awning is pre-
ferred by most people. Because of the specific geometry of the awning's frame, i.e.
45
SWI2000 Caravan Awnings: a Geometrical Problem
Figure 4.1: The classical caravan awning, with triangular shape at the front
the triangular shape at the front and the geometry of the upper part of the caravan's
body at the back, it is not easy to design awning's patterns which suit very well to
this kind of frame and the cloth could wrinkle.
The Gerjak company produces both tailor-made and ready-made awnings. A
tailor-made awning is designed to fit onto one caravan only, whereas a ready-made
awning is designed to fit onto more than onc caravan.
This report is organized as follows: Mathematical preliminaries about devel-
opable surfaces are given in Section 4.2. A model [or designing the tailor-made
awning is discussed in Section 4.3, which also includes computer results for the
awning design. In Section 4.4 we suggest an algorithm to make the best-possible
ready-made awning. Finally, conclusions are given in the last section.
4.2 A brief theory of developable surfaces
The following theory is taken from [1). It explains the notion of developable sur-
faces. In short, a developable surface can be laid-down in the two-dimensional plane,
without splitting into parts, stretching, shrinking, or compressing. First of all, we
introduce the concept of ruled surfaces. A ruled surface is a surface that can be
constructed by straight lines and so admits a parameterization of the form
res, t) = yes) + tz(s), (4.1)
where r, y, Z E ]R3; t, s E [0, 11.For each SE [0, 1], the line L., which passes through
yes) and is parallel to z(s) is called the generator, and the curve yes) is called the
directrix of the surface r.
It turns out that every developable surface is also a ruled surface. A developable
surface is a ruled surface that has the additional property, the unit normal is constant
46
Caravan Awnings: a Geometrical Problem SWI2000
Figure 4.2: An example of a generalized cone
along every generator. Thus, the Gaussian curvature of a developable surface is
identically equal to zero. In terms ofy and z, this means that the determinant vanishes:
IY', z, z'] = O. (4.2)
Now we want to use the form (4.1) to construct a developable surface on a given
close frame "0. Since a developable surface is also a ruled surface, there must be a
line segment inside the surface from any point "o(s) to some other point on the curve.
We shall call this point "0 (a(s)). Now the parameterization of the surface takes the
form
res, t) = (1 - t)"o(s) + txo(a(s)),
By substituting this to (4.2) we get
I' 'I do I " Iy ,z, z = s "o(a(s)) - "o(s), "o(a(s)), "o(s) = O. (4.3)
It is obvious that (4.3) is satisfied if a is constant. The mapping a == constant can be
applied on any generalized cone. Therefore, every generalized cone is developable.
See Figure 4.2 for an example of a generalized cone.
In general, it is not possible to construct a developable surface inside a given
awning's frame for which the second term in the product, l"o(a(s)) - "o(s),
xQ(a(s)), x~(s)l, is equal to zero. Therefore, we construct the awning by combin-
ing generalized cones.
4.3 Tailor-made awnings
Examples of developable caravan awnings which are constructed from three general-
ized cones are given in figures 4.3 and 4.4. For every chosen number of generalized
cones, two basic designs can be chosen: the base of the left-most generalized cone
may be placed either at the front or at the back of the awning. The difference is clear
when Figure 4.3 and Figure 4.4 are compared.
47
Caravan Awnings: a Geometrical ProblemSWI2000
This surface has area 8.8718
0.7
0.4
0.6
0.5
0.3
0.2
0.1
o 0
Figure 4.3: An example of a developable awning of type T~(constructed from three
generalized cones)
This surface has area 8.8606
0.7
0.4
~3
2.5
0.6
0.5
0.3
02
0.1
o 0
Figure 4.4: An example of a developable awning of type T3 (constructed from three
generalized cones)
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Caravan Awnings: a Geometrical Problem SWI2000
Figure 4.5: A nondevelopable surface given by z = xy.
This can be used to make a classification of all different designs, which we shall
call TN and T~. A design of type T4' for example, consists of four generalized cones,
and the left-most generalized cone has its base at the front. Analogously, a design
of type T~also consists of four generalized cones, but the left-most generalized cone
has its base at the back.
The two designs of type T3 and T; can only differ in the location of the point
where all the three generalized cones meet. Therefore, the entire space of T3 and
T;-designs may be parameterized by one scalar. In general, the design-classes TN
and Tlv may be parameterized by N - 2 degrees of freedom.
The idea that an arbitrary number of generalized cones may be used for the con-
struction of a developable surface leads to an interesting notion. As an example, we
consider a surface given by z = xy on ° :::::x :::::I and ° :::::y :::::1. This surface is given
in Figure 4.5. This closed frame can be described by the following closed curve in
the three dimensional space:
1
(s, 0, 0) , 0::::: s :::::1,
(1, s - 1, s - 1) ,1::::: s :::::2,
(x(s),y(s),z(s))= (3-s,I,3-s) ,2:::::s:::::3,
(0,4- s, 0) ,3 :::::s :::::4,
in which SE [0,4]. This surface z = xy is ruled, but not developable. Nevertheless,
a developable surface which is constructed from a number of generalized cones can
be fit into the frame. As the number of generalized cones increases, we notice that
• The distance between the developable surface and the surface z = xy becomes
arbitrarily small. This distance is defined as the ]R2 -norm of the difference
between the two surfaces .
• The two dimensional pattern of the developable surface seems to converge to
some limit-shape.
49
SWI2000 Caravan Awnings: a Geometrical Problem
1
Figure 4.6: Approximation of a diagonal line with horizontal and vertical line seg-
ments
Remember that the surface z = xy is non-developable! From this example we see
that somehow, the limit of the series of developable surfaces is not developable.
Two more observations were made by doing numerical investigation:
• The area of the limit-shape is 4% larger than that of the surface z = xy.
• Adjacent triangles in the series of developable surfaces make angles which do
not vanish in the limit.
The limit-surface is non-differentiable. The extra area of the surface is hidden by
folding it back and forth many times. A simple analogy is shown in Figure 4.6. A
diagonal line of length ./2 can be approximated by a number of horizontal and verti-
calline segments. However, the total length of these segments will never converge to
the length of the diagonal, because the ratio between this total length and the length
of the diagonal line is always equal to ./2.
Experimentally, using cardboard and paper for the close frame of the surface
z = xy, it was confirmed that the constructed developable surface gives a wrinkly
impression due to the excess area which had to be hidden by folding.
These ideas led to the following conclusions:
1. If a surface is larger than necessary, it gives a wrinkly appearance.
2. The smallest possible surface is probably optimal (i.e. the prettiest).
Some minimization procedures may be used to find the smallest possible awning
pattern for each of the pattern classes which is defined in this section. In figures 4.3
and 4.4, we show some resulting designs for the caravan awning. In figures 4.7
and 4.8, we show the corresponding patterns of the awnings in figures 4.3 and 4.4,
respectively.
50
Caravan Awnings: a Geometrical Problem SWI2000
_1ooL---~----~----L---~----~-o 50 lOO 150 200 250
____-'--- L---L -'
300 350 400 450 500
Figure 4.7: The 2D pattern of the awning in Figure 4.3
300
Figure 4.8: The 2D pattern of the awning in Figure 4.4
51
Caravan Awnings: a Geometrical ProblemSWI2000
4.4 Ready-made awnings
In the previous section, we presented a method to construct an awning which fits
nicely to one caravan. In this section, we present some methods to design a ready-
made awning. The objective in designing a ready-made awning is to construct the
best awning for some selections of caravans. Since it is not possible to make an
awning that fits perfectly on each of those caravans, we only design an awning which
fits fairly well.
There are two possible approaches to design ready-made awnings:
1. First of all, we choose one caravan as a representative of the entire selection
of caravans. After that we design an awning which fits perfectly well on that
chosen caravan. This idea is presented in Subsection 4.4.1.
2. Minimizing the average excess area of the awning's cloth, when it is attached
to each of the caravans. This idea is presented in Subsection 4.4.2.
The first approach turns out to be much simpler than the second one. However,
the first approach does not in any way guarantee that the solution will be good (and
definitely not optimal), whereas the second approach does.
4.4.1 The average input approach
In Figure 4.9, we give a description of the general awning's frame by a finite number
of parameters. The representative frame can be obtained by combining these parame-
ters for a number of caravans to a (weighted) average. The weights can be determined
from the relative importance of each of the caravan models, for example: the selling
percentage.
The awning which fits optimally on the representative frame is also expected
to fit nicely on each of the other caravans. However, this design cannot be called
optimal by any meaningful sense of the word. In the following subsection, we will
discuss a different approach, which leads into the construction of an awning which
fits optimally, in some sense of the word. Due to the limitation of time, the authors
were not able to investigate it in great mathematical detail.
4.4.2 The minimal wrinkle approach
In this subsection, we investigate the possibility of constructing the optimal ready-
made awning for a given set of caravans. Each caravan is numbered from 1 to N.
The construction is done by minimizing the average wrinkle on the awning's cloth.
In order to do so, it is clear that we need to measure or to quantify the wrinkle.
First of all, we make the following assumptions.
52
Caravan Awnings: a Geometrical Problem SWI2000
0.8 1.2
0.8,,---,------,-----,---.,..----,---,-------,--,------,------,
o 0.4
Figure 4.9: Parameterization of the awning's frame
• If a flat surface D C R2 is the two-dimensional pattern of a three-dimensional
object E which is nicely attached on a certain close frame, then a larger flat
surface D' ::J D can be forced to fit in that close frame. An experiment was
done by using cardboard and paper. It was not easy to hide the excess area by
folding.
• When the two-dimensional cloth is attached on the close frame, the excess
area, i.e., the area difference between D' and D, appears as wrinkles on the
three-dimensional object. Therefore the quantity 11Dill - 11D 11 is a suitable
measure for wrinkling.
Since the notions of 'a wrinkle' and 'a visually pleasing surface' are not mathe-
matically defined, this assumption can be verified nor falsified. This way of quanti-
fying the wrinkle has the following consequences:
• The shape of the pattern is not considered. Two patterns are defined to be
equally wrinkled if they have the same excess area. Nevertheless in reality,
after we attach them to the close frame, it is possible that one pattern looks
better than the other.
• If one pattern has twice as much excess area as the other pattern, then we say
that the first pattern is twice wrinkled than the other one.
53
Caravan Awnings: a Geometrical ProblemSWI2000
• It does not matter where the wrinkle is located (e.g., at the back, side, middle
or front part of the awning).
The consequences above show the difficulties in defining 'a visually pleasing
surface' mathematically. This notion is abstract and subject to personal feelings.
Another aspect of setting up an awning is the fact that the owner of the awning has
freedom in attaching the awning to the caravan. Although the shape of the caravan
is fixed, he may shift the awning cloth from left to right, adjust the length of the tent
and awning poles and the place where the poles are put. In fact, this action of setting
up the awning/tent determines the parameters Xlcft. hoo, hs and ws, h1cfl, hfront. hright
Wlcft, Wright and the depth of the tent as given in Figure 4.9. These free parameters
may be adjusted in order to minimize the wrinkles.
We summarize the process of setting up a given cloth D on the i-th caravan as
follows: The frame which the owner creates by setting his free parameters define
another pattern D; This ideal pattern must be smaller than the available cloth, D.
In other words D, C D. If this is not the case, the cloth D will not fit in the frame.
The excess area 11DII - 11D, 11measures the wrinkles. We assume that this adjustment
process leads to a unique final result, namely D7. This optimal setup D7 is a function
of the available tent sheet,
D; = D;(D) (4.4)
The basic structure of the minimization procedure for the wrinkle is relatively
simple. The optimal pattern, namely D*, fits optimally on all caravans, 1, ... , N.
This means:
(4.5)
In this formula, V is the space of all possible awning's patterns, and the each weight,
Wi, is due to the relative importance of each caravan.
4.5 Concluding remarks
As a follow-up of the industrial week, the results have been used to answer one of the
question which is asked by the tent/awning manufacturing company. Some examples
of the awning patterns were produced by using computer graphics. These examples
have been used to make a few adjustments to the designs which are actually made by
the company. Further ideas about the optimal shapes for ready-made awnings were
developed, but they were beyond the manufacturer's needs.
54
Caravan Awnings: a Geometrical Problem SWI2000
Acknowledgments
The authors would like to thank A. Lubberdink from Gerjak B.V. for introducing
the problem, Marta Pellicer from Universitat Politecnica de Catalunya and Agus Y.
Gunawan from Technische Universiteit Eindhoven for helping in the nice discussion.
Bibliography
[1] M.P. Do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall,
Inc., 1976.
55
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