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.