THE CONE

development of the fuselage

The fun of drawing a card board scale model is found in flattening 3D objects to 2D paper particles. As long as the outline mainly consist of straight lines it's just a matter of measuring and scaling but the moment you encounter curvatures a bit more thinking will be required.

In this respect the fuselage of most planes is a combination of cylinders and 
cones. 

Since the cone is an easy to flatten geometrical shape. A rectangle from the length and circonference of the cylinder.
Although not very complicated to calculate the cone is less easy.

I think this can be explained in just twenty steps of which the last five are used to show the incredible advantage of using the DraftSight 2D vector-programme where you still can work on free of charge!

STEPS to build A CONE

STEPS into GEOMETRY

R radius bottom circle and r = radius top circle

h is the height from top circle to the intersecting opposing tangents
and H is the = distance between top and bottom circle

Ru the distance from the intersection of both opposite tangents 
to the bottom circle and meanwhile the radius of the produced cone (!)

ESSENTIAL FORMULAE

the before mentioned radius can be calculated (or checked!) by
means of the square root of the sum of squares of both R and H+h

the top angle is necessary to calculate either the
length of the produced arc or the length of its chord
and, because its length has to correspond with the bottom's
 circonference, it's calculated by the quotient of bottom circle
and the circle of the produced radius times a circle (360º)

the last 'trick' you probably need when your construction is an analog
one will be the trigonometric calculation of the chord length where
 the onboard windows calculator in  scientific mode can do the job.

THE EASY (digital-no-calculate) WAY

Draw your cone-variables in DraftSight and connect/produce the connections

between both endpoints of the parallel diameters. The intersection is autom-aticly found when you use the chamfer instruction.

you have drawn both circles and the big Ru one as well and using the
properties window all circumferences are available values straight forward

he division of both values times 360 supplies the top angle v

this value is the between both radii Ru, cutting off an arc with a
length equal to the circumference of the bottom circle.

it is worthwile to check the above mentioned statement because
you cannot make calculation-errors, but sometimes you're
just one click away from a wrong result...

the same exercise can be done with the top circle values

here you are, your development

BUT WHAT IF . . . 

... upper and bottom surface are mutually 'shifted' ... 

The opposing radii are significant different resulting in two different arcs.

To cut a long story short I wrote a programme in Q(uick)B(asic)64 based on the idea that an arc can be approximated by a series very short chords.

Therefore both surfaces have to be divided into an equal number of sections, each one to become the side of a quadrangle.

This method, also based on the assumption that both surfaces were symmetric and could be divided into two equal parts, was an acceptable solution.

This calculation however showed a few naughty restrictions as displayed below.

4 tests - from left to right
1) egual top and bottom circles 2) different top and bottom circles
3) different and mixed straight lines and curves
4) different and only straight lines

non-regular shapes and possible consequences for the programme's capacity:
from left to the right
1) different lengths & non-circular 2) different lengths & reverse curved
3) different lengths & straight lines 4) different lengths & samehights
& straights lines 5) as 4 but exactly horizontal lines 6) as 4 but shapes higher
and/or lower than the cutting-line

The restrictions of the programme are evident now. 

a) both exact horizontal as vertical straight parts are not accepted;

b) shapes that are over or/and under the extends of the centerline are not

    properly developed.

_________________________________________________________________________________

THE PROGRAMME

For those interested the programme is dowloadable over here.
Mind that it is just the raw code and has to be copied as a .bas file the can be opened in QB64. As soon as you run the program from there QB64 automatically creates the .exe file.

The following steps need to be taken in order to create a development :

Preparation:

1. download and install DraftSight and use it free-of-charge ;
2. create the layers links and rechts;
3. create the map C:/DXF.
4. alter the DraftSight drawing settings (type 'units') :

•  angle setting Grads
•  base angle 100 clockwise
•  length type decimal 

Preparing the DXF:

1. activate layer links (in the layer manager on the layer toolbar);
2. divide the left surface in max. 50 sections (Draw - Point - By Segments);
3. activate layer rechts;
4. divide the right surface equally;
5. if you like to have the developments in a specific layer then activate it now;
6. type 'wblock' (without paranthesis) on the command line and enter;
7. in the Save File window select 'browse';
8. select the Save as bar and select R2013 ASCII Drawing (*.dxf) 1st dxf from top;
9. give your DXF a name and make sure that the adress-bar shows C:>DXF;
10. click OK and see 'selection is empty' - select Yes and enter;
11. select now all the points + both vertical axes;
12. CHECK: if in steps 2 and 4 you chose for 40 segments then the total in the command line is 40 now too;
13. click Save and run cone.exe;
14. Behind 'Naam puntenbestand ?' type the name of your DXF (without DXF);
15. on the message 'c:\Dxf\UITSLAG.SCR opgeslagen' 'Druk toets' press any key;
16. return to your drawing and select Run Script from the Tools pull-down;
17. select uitslag.scr in C:\DXF and click Open.

This should have done the job.

If not feel free to mail me at post@papermodel.nl

Sources of error :

1. the axis is not a vertical straight line (e.g. a polyline or arc !);
2. the surface is not exactly connected to the end of the axis;
3. the number of selected elements (sub. 11 above) is not the same as constructed under sub. 2 and 4 above;
4. the shape of a surface exceeds the y-values of the axis;
5. the connection between two points on the surface is an exact vertical or horizontal straight line;
6. points on the left surface are not in the layer links or these on the right are not in the layer rechts;
7. the map C:\DXF doesn't exist or the block (wblock command) is saved in a different map;
8. you gave your block a name inclusive the extension .DXF.