[John]

Say you just finished drawing your latest muscle babe, and you want to include her measurements in a story about her.  How do you do this?  It's not easy to go into the drawing and wrap a tape measure around her arm or leg, so there needs to an alternative.  Here's one that should do the trick.

This method involves geometry, namely elliptical circumference, so it may be good to be familiar with the subject before trying this approach.  It would also help to have a spreadsheet editor to enter the values into (notepad + calculator will work, but it's harder to organize).

What you do is, first, assign a height to this amazon.  Next decide which direction/angle the body part you want to measure is facing the "camera".  Lastly, you take the on-screen measurement of the part, with the on-screen total body height, and figure the elliptical circumference with the ratios and formulas given below.

1. You need to set the height, all measurements will depend on this number.  This sets the overall scale of the amazon, and the rest is a matter of ratio and formula.

2. Second, is the very import matter of deciding the orientation of the body part.  A straight-ahead front, back or side view of the limb is easiest, but for in-between angles, you can approximate how far between the straight-ahead directions your angle is, and using ratios given below to help calculate the depth and width. 

Also, it would help to explain which is "depth" and which is which is "width" for each part.  Let's assume a body is standing straight, facing forward, with their arms perpendicular to the ground, palms facing their body.  "Width" would be the distance left-to-right, across any body part, and "depth" is distance front-to-back.  For example, a bicep's depth is it's "peak" height.

I approximated the following ratios off of a few big female bodybuilders, like Tina Lockwood, Gina Davis, and Conny Brandt.  Ratios are given in r=width/depth,  I used a sample set of 4 women for each part.

— Bicep —
Low: 0.85
High: 0.92
Average: 0.88
Std dev: 0.03
Recommended range: 0.85 <-> 0.95
Comments: "Off-season" (ie softer) arms tend to be wider, but the depth is always significantly greater than the width.

— Thigh
Low: 0.73
High: 0.81
Average: 0.77
Std Dev: 0.04
Recommended range: 0.7 <-> 0.85
Comments: Upper legs are surprisingly deep.  The front may look more impressive and "bumpy", but the side is sometimes close to 40% bigger.  The ratio depends largely on hamstring development.  It doesn't seem to change much from ripped to soft condition.

— Calf —
Low: 0.94
High: 1.13
Average: 1.03
Std dev: 0.08
Recommended range: 0.95 <-> 1.15
Comments: Some calves grow more sideways than backwards.  Choose the ratio that seems more fitted to your creation's shape.  I'd imagine softness has little effect on the ratios. 

3. Now comes measuring and calculating of circumference.  Measuring should be easy if you have an imaging program with a pixel measuring tool (like GIMP).  If not, you can do a rectangular selection, between the two points you are measuring.  All you have to do is read the rectangle's width and height dimensions from the status bar (or wherever it is displayed), and use the Pythagorean equation of d=sqrt(x^2+y^2) to get the pixel distance between points.

The pixel height of the character should be easy to get if the character is standing (or lying) relatively straight.  Otherwise, you will have to add head, neck, torso, pelvis, femur, and shin distances together to get an approximate pixel height.  After getting the pixel height, divide it by the real height you gave the character in step 1.

Once you have the height, measure the diameter of the limb you want to measure, approximately where you would put the measuring tape, if she were taken off the canvas into life.  You want a straight line distance, even if the tape would be curved if projected onto your canvas.  This is a diameter through the limb, not distance around (we're getting to that shortly).

Now you have the diameter of the measured area.  This is your starting point for the final equation.  If the width or depth dimension of the part is pretty much facing the "lens" of your drawing directly, then this should be simpler. If not, no worry.  Say width is 0 degrees and depth is 90 degrees, and your body part is facing somewhere in between.  Well, approximate the angle at which it is facing, between width and depth, and use this formula:

Let A be angle the part is facing the "lens", between 0 (width) and 90 (depth).
Let r be the width/depth ratio for the chosen body part.
Let m be the measured distance.
t will be a value between r and 1.
d will be the body part's depth.
w will be the body part's width.

t=r+(A/90)*(1-r)
d=m/t
w=d*r

Now comes the actual elliptical circumference calculation. With the width and height found, getting a set of elliptical radii is a simple matter of dividing both the width and height by two:

a=w/2
b=d/2

Now there are many ways to calculate elliptical circumference, with varying degrees of precision.  I will only cover the most simple method, as we're just going for a loose estimate here (besides it works quite well when the radii are nearly equal).  That method would be:

C=pi*(a+b)

where C is the approximated elliptical circumference. 

Now that you have this formula, you can claim your 6-foot muscle mistress has 20" biceps with a greater degree of accuracy 😀

[CDR]

Awesome! I'll give it a try with my 3d gals (Where I can make some measures to confirmate the algoritm outputs)
;D

[Reason]

Wow, thanks very much for this. Excellent analysis and description. One day I want to start a Muscle Girl Art Wiki where it would be great to have info like this.

Also, just wondering if you tried out your C=pi*(a+b) formula on your 4 reference women, as all of their measurements are known. If so, how accurate was it?

[John]

Here are the results of "measuring" the sample pics with the ratios:

Calf
Name TestCrcm
Tina Lockwood (OS) 20.64
Marja Lehtonen 17.97
Nursel Gurler 16.33
Helle Nielsen 17.27

Thigh
Name TestCrcm
Marja Lehtonen 27.38
Gina Davis (OS) 27.81
Conny Brandt 26.09
Helle Nielsen 26.75

Bicep
Name TestCrcm
Gina Davis (OS) 17.98
Tina Lockwood (OS) 18.76
Marja Lehtonen (A) 16.53
Marja Lehtonen (B) 16.58

Both Gina D. and Tina L. were super-bulky off-season conditions.  Marja L. was ripped (when is she not?) and Helle N., Nursel G., and Conny B. were in moderately soft offseason conditions in the photos used. 

[John]

I goofed on listing the ratios, these are the correct ones

— Calf —
Low: 0.94
High: 1.13
Average: 1.03
Std dev: 0.08
Recommended range: 0.95 <-> 1.15

— Thigh —
Low: 0.85
High: 0.92
Average: 0.88
Std dev: 0.03
Recommended range: 0.85 <-> 0.95

— Bicep —
Low: 0.73
High: 0.81
Average: 0.77
Std Dev: 0.04
Recommended range: 0.7 <-> 0.85

I switched biceps and thighs.

[Reason]

Thanks for the correction, although I have to say I'm surprised that the width to length ratio is smaller for biceps than thighs. Often when I see pictures of thighs dead-on they are quite narrow, often more narrow than the calves.

By the way, this principle is also very useful for going the other way, ie if you have a measurement in mind and want to draw the arm or leg with the correct dimensions. The ratios are also a very useful reference for 3D modelers. So thanks once again for this excellent resource.

Oh and by the way, don't spose there's any chance of you posting the reference pictures you used is there. If that involves violating copyright, you could always post where to legally download them instead.

[John]

I did many of the ratios by screen-capping videos on Youtube.  This was easier to get multiple angles from.  I did the first few off of pics on my HD, but changed over to videos after realizing how hard it would be to get multiple angles from the same shot. (many schmoetogs stick to the same angle)

I never saved the screen-caps to disk, and deleted them afterwards.  I had to use pics saved off of the drive for the tested measurement.

Here are some of the videos I used:

Marja Lehtonen (biceps):
http://www.youtube.com/watch?v=s4D6ZsQNYQA

Tina Lockwood (biceps):
http://www.youtube.com/watch?v=UL5yqOPI3sg

I used the pics from the blue demin dress shoot for Tina's calves.  Here's video from that shoot to see her condition:
http://www.youtube.com/watch?v=o_9C61RYMv4

Gina Davis (biceps & thighs):
http://www.youtube.com/watch?v=R3SH_IHzQjo

This is Helle Nielsen's condition in the photos I used of her:
http://www.amg-lite.com/?view=http://www.amgprofiles.com/Helle/Pics/Helle4003.jpg

These were Conny Brandt's two quad shots used:
http://www.cornelia-brandt.com/FreePics/Galeries/FreePics/Bigs/OffSeason05_010.jpg
http://www.cornelia-brandt.com/FreePics/Galeries/FreePics/Bigs/OffSeason05_012.jpg

Nursel Gurler's shots were basically the condition she always seems to be in. 

[David C. Matthews]

I think you've just written probably one of the most useful documents I've ever run across.

Will this method work for women's bustlines as well?  (I should think it would.)  One of the questions I get asked almost as frequently as "What are Tetsuko's measurements?" is "What is Sonya's bust measurement (or bra size)?"  And I don't know what answer to give  ;D

[Fett]

I could've sworn I've said this before… I've been measuring my gals for ages to keep their proportions right – until I realised I could actually just do so in the same way I make sure their legs are the right length (i.e. I don't need to measure them).

All I did is use the circumference for an oval.

First, I work out the ratio by measuring from the top of the head to the bottom of the groin. This is equal to half their height in inches, so you divide her real height in inches and then take your measurement of your picture (which is in centimetres), then you get the ratio. So if the woman is 6' and your picture is 28cm, then 3628 = ratio.

Next, I find half the height (a) and half the width (b), then do this formula:

([(a² + b²)2]root) x 2 x pi x ratio = measurement in inches

I could've sworn I mentioned this AGES ago…  ???

[Reason]

I agree that these techniques has been discussed before on this forum, but probably before MuscleGrothe's time and definitely with not nearly as much detail.

[Author]

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