Cone Math for Sheet Metal Workers first created 12/03/06 - last modified 11/01/13 Page Author: Ty Harness
Right Cones

A right circular cone has its central axis perpendicular to a base circle. Figure 1c's plan view shows a circle perpendicular to the central axis. Any further cut taken square (or perpendicular) to the centre line will also produce a circle.

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Figure 1a) A model of the Saturn 5 Rocket which is made of right cones and cylinders.
b)Sheet metal right cone connected to 2 cylinders.
c) Right Cone with parallel sections to the base circle.
d) Right cone with oblique sections to the base circle.


Note figure 1d is also is a right cone but with 2 oblique slices which is not the same as an oblique cone. This time the section is known as an ellipse. The ellipse may form a joint as in the case of the right segmental bend or right cone tapered bends. I've written more on right tapered bends here.

Conet2.01 use a technique devised by the famous Greek geometer Apollonius to find the ellipse. Greek scholar Thomas L Heath published a 'Treatise on Conic Sections [3]' which is an English translation of Apollonius' work on conics. I've used the same notation as Heath in figure 2 to show the construction of proposition 27(or I56). ** Apollonius starts with a known ellipse diameter/parameter ratio and shows you how to find a right cone with the given ellipse section. I've reversed the construction procedure here to find the ellipse and parameter from a given cone.

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Figure 2 - Finding the ellipse from a known diameter


How to graphically find the parameter from an axial section, OHK of a cone:

HK is the diameter of the base circle and section HKO is perpendicular to the base. Section AA' is also perpendicular to HKO.

Extend AA' to intersect with the HK plane to find point M.

Draw parallel to AA' from O to intersect the HK plane to find point N.

Circumscribe a circular arc to contain A0A'

Draw parallel to HK through O to meet the AA' plane to find T and also E which intersects the circular arc.

Join A' to E and A to E.

Draw parallel to EA' through Point F which intersect ON and AE to find point P on the AA' plane.

The parameter of the ordinates, pa is equal to PA'

Apollonius' proposition I.13 is pa:AA' = HN.NK : NO^2

That is a rather complicated looking equation. As the section becomes parallel to the base circle the HN, NK and NO^2 must be equal to infinity:infinity, so the parameter to circle ratio becomes 1. You can find more working in Heath[xx] or I've written a little more here where you can see an easier way to find PL.

The diameter,PP' is known as the abscissa and QV the ordinate. Apollonius showed that the ordinate QV^2 = PV.VR. The parameter, PL is needed in order to find the distance VR. In order to graphically find QV I first transfer VR along the line PP' as shown in figure 3. Then I draw a semicircle with a diameter PV+VR and this will find the root of VR.PV. You would then go on to move V along the abscissa and calculate as many points on the curve you desire.

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Figure 3 - Constructing the ellipse curve where PL is the parameter,pa.
Line PP' is AA' in the equation above.




The Conet 2.01 software performs the above construction and then draws on the ellipse section. As a sheet metal worker you will often need to make an elliptical flange, cover or baffle plate. Dickason[2] shows several other methods on how to develop the ellipse graphically on the drawing board using orthographic projection.

Oblique cuts can also form parabolas and hyperbole which are not typical to sheet metal development but they are worth studying and I refer you to the others work references in the footnotes. Please remember the right cone is only a special case and we will move onto the oblique cone.

Volume of a Right Cone

The volume of a full right cone (and oblique cone):
$V = 1/3 A_(base) * height $
where the full derivation can be found here.

The Volume of a frustum cone:
$V = 1/3 *pi*h*(a^2 + a*b + b^2)$
where a is the top radius and b is the base radius



Surface area of a right cone

The surface area of right cone:
$A = pi*r*sqrt(r^2 + h^2)$
where the $sqrt(r^2 + h^2)$ term is only applicable to right cone only - it's the true length. the surface area for a frustum cone is given by:
$A = pi*(a+b)*sqrt(h^2 + (b-a)^2 )$

where the full derivation can been found here.
Oblique Cones (sometimes called a scalene cone)

Figure 3 shows an oblique sheet metal cone with cylinders attached at each end. Typical applications include offset hoppers, offset swan neck pieces, invert level reducers often used in drainage applications, etc. Using an oblique cone can be very useful in sheet metal work because the base plane cut (or section) makes a circle or we shall force it to be so.



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Figure 4 - A sheet metal oblique cone


Therefore if you take another cut parallel to that base circle, such as OP in figure 5a, then the section is also remarkably a circle. This is Apollonius' [I4] proposition. Anyone familiar with similar triangles from their school days will see it's true.

There is one other slice not parallel to the base circle that produces a circular section known as the subcontrary section and Apollonius was also credited with the discovery. Simply flip the cone round 180 degrees about it's axis* and align at the angle D and the base angle D+A becomes the subcontrary angle. Any section parallel to the angle D+A (MN in figure 4) will produce a circular section. This is Apollonius' [I5] proposition.
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Figure 5a - Subcontrary Angle D+A b)Oblique Cone and Sphere section

Figure 5b shows if a perpendicular line from the base circle to the cone apex is drawn this will form the central axis/diameter of a sphere. Where the cone and sphere intersect will be also be a subcontrary section. A worked mechanics problem and neat application of the subcontrary section.

With the subcontrary section being circular it's frequently used in sheet metal work as the segments of an oblique tapered (or plain) bend. If you want to tackle the mathematics then I refer you to Prof. J. McCoy's work below.

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Figure 6 - Circular section EF parallel to the base circle BC


Using ConeT2.01 and clicking the subcontrary check box the section angle is directly set to the subcontrary angle GH. By default the software rotates the section about the central axis, see figure 6. I've used the same letter notation as Dickason [2] in figure 7 which shows if you want the section diameter to be the same as the reported diameter TD the section must be rotated about the SD axis*. The section MN just emphasizes that the SD axis is the bisected axis, please note its section is elliptical and not circular. The AD axis is the bisected angle of the scalene triangle BAC which then means SG = SF and SH = SE thus GH has to be the same diameter as EF.

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Figure 7 - Using Conet2.01 with the subcontrary section


A section that is neither parallel to the base circle or subcontrary will produce either an ellipse, parabola or a branch of a hyperbola [I9]. Figure 8 show the ellipse section.

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Figure 8 - An oblique cone with a section that is not parallel to the base circle


The ellipse section can be utilized for tapered segmental bends in sheet metal work. I've written more here. Please note you can apply the above Apollonius construction method to an oblique cone to find the diameter to parameter ratio and thus the ellipse.

The above examples are considered special cases because where the section plane intersects the base circle plane the intersection is perpendicular to the axial triangle and the diameter. The axial diameter is known as the a principal diameter and the angle of ordinates are perpendicular to the diameter.



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Figure 9 - Conjugate Diameters and Axes


Figure 9 shows what happens when the ellipse axes are not perpendicular. The ordinates are parallel to the intersection formed between the section and base circle planes. It is important to still select an axial triangle at 90 deg. to the intersection, but you can see the angle of ordinates is no longer 90 deg. in the above cone. The figure also shows the ordinates which are common to the circular section and the ellipse. Those coincident ordinates,QV which were used to great effect in finding the parameter earlier. The conjugate diameter will be parallel to the ordinates but is in the centre of the diameter. Remember the diameters must split the ellipse (or circle) in half to be true diameters and not just chords.

The curve is found exactly the same way as the example above where QV^2 = PV.VR, see figure 10.

  • Again start by drawing diameter to scale and the parameter line PL perpendicular to the diameter at either end.
  • Join the opposite end of the diameter to L in this case LP'
  • To find the ordinate,Q and Q' at V then lay off a construction line at the angle of ordinates as predicted by the cone geometry.
  • Draw a line perpendicular to the diameter through V to find R along the line LP'
  • Transfer VR to a construction line perpendicular to angle of ordinates passing through V.
  • Transfer VP to the same line.
  • Draw a circle whose diameter is given by VR+VP on the construction line and where the circle intersects the ordinate passing through V then Q and Q' shall be found which is the square root of PV.VR
  • Draw as many parallel ordinates along the abscissa and repeat until you have enough points to describe the ellipse curve.

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    Figure 10 - Predicting the curve with given conjugate diameters.


    Apollonius goes on to work with the conjugate diameters PP' and DD' and shows if you know the conjugate diameters the parameter can be simply found from proposition I.15:

    PP':QQ' = QQ':PL

    You can see from figure 10 that if the conjugate diameters are known then it's not too difficult to reverse the construction procedure above to predict VR and thus extend P'R to the end of the diameter to predict PL graphically.

    Jan De Witt has written in depth on the drawing of ellipses. Grootendorst[4] has translated De Witt's 'Elementa Curvarum Linearum' work into english and include useful comments and references on the subject.

    You can also find a document below by Prof. J. McCoy which shows how Jan De Witt tackled the subject of drawing an ellipse starting with conjugate diameters and axes. Also Prof. McCoy shows how to find the principal diameters and axes of the ellipse using Apollonius' I.58 proposition. The principal diameters being along the major and minor orthoganal axes.

    Surface Area of an Oblique Cone
    The surface area of an oblique cone is a complicated matter you can find more on the subject here: obliqueconearea.htm

    Oblique Cylinder

    Again a subcontrary section can be found in the oblique cylinder.

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    Figure 9 - Oblique cylinder subcontrary section


    Typical applications might include an oblique pipe where a circular cover is required for inspection or maintenance. Also the oblique segmental bend where the cross section of the tube is elliptical but the joints are circular. More on segmental bends can be found here: lobster back bends

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    Figure 10 - Cuts parallel to the base circle plane and subcontrary plane are circular as shown by the silver discs.




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    Figure 11 - Oblique Cylinder with circular flanges parallel to each other


    References

    (1) Boyer, C.B.,, A History of Mathematics, John Wiley & Sons, 2nd Ed., : (), p147.

    (2) Dickason, A., The Geometry of Sheet Metal Work, Longman Scientific & Technical, 1st, UK: (1967), 234 - 235.

    (3) Heath, T., L.,, Apollonius of Perga Treatise on Conic Sections, Cambridge: At the University Press, 1st, : (1896), .

    (4) Grootendorst, Jan De Witt's Elementa Curvarum Linearum, Liber Primus, Retrieved: (2000), 259-262.
    Foot notes

    **Heath reorders Apollonius' propositions to present his translation book. He references the original propositions in square brackets. For example [I.7] which refers to book 1 proposition 7.
    Acknowledgements

    I've only scraped the surface of Apollonius' work with the first 15 propositions. There's only 385 to go.

    *Special thanks to Prof. John McCoy for sharing his knowledge, enthusiasm and advice on the mathematics of Apollonius. Especially for pointing out that if the section is rotated about the bisected axis SD then the subcontrary section has the same diameter as a section taken parallel to the plane.

    'Modeling an Apollonian Cone' by J.McCoy 22nd Oct. 2012
    'Comments on Apollonius proposition I.58' by J.McCoy 29th Nov. 2012
    'Tutorial on I.58' by J.McCoy 31st Dec. 2012

    Spreadsheets launched via links in the PDF and should be located in the same folder as the pdf file.
    The spreadsheets are designed for educational use using MS Excel(R) and LibreOffice(R) and contain macro code that is the copyright of Prof. J.McCoy.

    Further Study Material:
    Oblique Cone and Sphere Intersection. A problem from Timoshenko's book: 'Advanced Dynamics 1947'.
    Dandelin Spheres of a right Cone



    Further Study Material by others:

    The answers are nearly always in these books:

    An introduction to the ancient and modern geometry of conics, being a geometrical treatise on the conic sections with a collection of problems and historical notes and prolegomena by Taylor, C.

    A Treatise on Conic Sections by Salmon, G.

    They don't write books like those two anymore.

    NJW's excellent lecture series on Apollonius and Greek mathematics

    Brouillon Project

    www.wilbourhall.org

    'What Does a Circle Look Like?' by Mr W Casselman


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