Deformed Bullet

Date: 04/11/2002 at 12:03:14
From: Allison Milam
Subject: Circle/Ellipse Diameter

If you initially have a circle and it gets smushed into an ellipse, 
how do you determine what the diameter of the initial circle was, 
knowing the major and minor diameter of the ellipse?

Date: 04/11/2002 at 12:54:44
From: Doctor Peterson
Subject: Re: Circle/Ellipse Diameter

Hi, Allison.

That depends on how it was "smushed"! What was preserved - does it 
still have the same area (as if it is made of clay or some 
uncompressible material), or the same perimeter (as if it is contained 
in an unstretchable membrane, but the inside can compress), or what?

Answering that question will probably make it clear how to answer your 
question. You may need to use our Geometric Formulas page to see how 
to carry out the plan:

   Ellipse
   http://mathforum.org/dr.math/faq/formulas/faq.ellipse.html 

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/11/2002 at 13:25:07
From: Allison Milam
Subject: Circle/Ellipse Diameter

I am a forensic scientist in the field of firearm and toolmark 
examination. We use the diameter of the base of a bullet as a way of 
figuring out caliber. Therefore, when we get a damaged bullet where 
the base has changed from a circle to an ellipse we would like to be 
able to determine what the initial diameter was.

Date: 04/11/2002 at 13:33:54
From: Doctor Peterson
Subject: Re: Circle/Ellipse Diameter

Hi, Allison.

You can probably assume that the area remains the same, although I 
could easily imagine metal deforming in such a way that the area might 
change. With that assumption, you just have to find the area of the 
ellipse

    K = pi A/2 B/2

where A and B are the major and minor axes (diameters), as opposed to 
the usual a and b, the semi-axes (radii).

Since a circle of diameter D with the same area would require

    K = pi (D/2)^2

we have

    pi D^2/4 = pi AB/4

    D^2 = AB

    D = sqrt(AB)

So just take the square root of the product of A and B (called the 
geometric mean) as your presumed original diameter.

I would recommend testing this on some deformed bullets whose caliber 
you know, in order to check our assumptions.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/11/2002 at 13:49:04
From: Allison Milam
Subject: Circle/Ellipse Diameter

Thank you very much for your time.  We are going to manually deform 
some known caliber bullets with a hammer and try out these formulas.  
If it works I will write an article to share the information with 
others in the field.  Again, thank you for your help.