Amaz0ns Forum

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 :D

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

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?

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. 

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.

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.

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. 

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

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 36\28 = 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...  ???

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.

It's simple - Fett started the ball rolling, now MG has taken it a stage futher.  As a geeky scientist (albeit a non-artistic scientist who's also a mathematical dunce) I love this thread. ::) ;D

I'm sorry I missed your posts about this, Fett.  I would have given you some credit if I had known about your posts on this subject.  I haven't been here that long, and haven't read the board very often since I joined.  BTW, Thanks for the better elliptical circumference equation.  It should help me out when I get to parts like chest where the aspect ratio is far from 1:1.

lol Exactly the same calculations i have been using for years ^^ funny to see i am not the only one who calculates like this.

You could simplify that to:
sqrt(2*(a²+b²))*pi*ratio

sqrt((a²+b²)/2)*2
=sqrt(a²+b²)*sqrt(1/2)*2
=sqrt(a²+b²)*sqrt(1/2)*sqrt(4)
=sqrt(a²+b²)*sqrt(2)
=sqrt(2*(a²+b²))

No need to apologize! I was just a bit dumbfounded, wondering if I had in fact made up a memory of posting things. :D

So basically the formula Fett gave was the circumference of the circle

r=sqrt(1/2)*sqrt(a^2+b^2)
(circle circumference is 2*r*pi)

A way to look at this is:
1. Draw a circle
2. Draw a perfect square over that circle, with the bottom left corner at the center of the circle, the bottom right corner on the rim of the circle at 0 degrees, and the top left corner at the 90 degree rim point.
3. Draw a line from the bottom left corner of the square to the top right corner.  This line intersects with the circle rim at the 45 degree point.
4. Scale the graphic to the aspect ratio of the ellipse.

After doing that, the 0 degree line should be equal to a, and the 90 degree line equal to b.  That would make the intersection point drawn in step 3 to be of the length sqrt(1/2)*sqrt(a^2+b^2).  So now draw yet another circle over this set of graphics, with that line (from the center to the ellipse intersection point) as the radius.

You can see the similarity in circumference of both the ellipse and the last circle.

Speaking of height ratios, I sampled a group of 10 FBBs for their part length proportions.  Below is a text table of the average percentages for each part relative to the total body height.

	Average
Head2Collar	18.1%
Collar2Pevis	35.9%
Pelvis2Heel	46.0%
Pelvis2Knee	19.8%
Knee2Heel	26.2%
Head2Pelv	54.0%
Shldr2Elbow	17.5%
Elbow2Wrist	15.8%
Wrist2Ftip	9.0%

Here's the complete chart in image form:
(http://i163.photobucket.com/albums/t299/MuskelGrothe/bodyMeasureChart.gif)

hi !
assuming  the ratio between a and b is often similar and in that way the round of the form is a factor of a dimension ( square : 4.a    - circle : 3.14.a ) ...

 ... you can say for the bicep form , the round is 2.6 x a, then apply the ratio.

Wow, more excellent info. Thanks MuskelGrothe.

Just wondering what you mean by Head, Collar, and Pelvis. Is Head the top of the head or bottom of the head? The same applies for Pelvis and Knee. Also, is Collar the clavicle or manubrium (yes, I have a book on anatomy in front of me).

Also, I was wondering if I could trouble you for the average head size as a percentage relative to the total body height. That way I can convert your percentages to number of heads, which I find easier to work with.

Thanks again for all the work you've done on this.

I'm not very good with anatomical terms, but I'll try to explain the points I was using:
"Head" = crown (highest point standing up)
"Collar" = clavicle (collar bone), at the center groove/hole.  (Below the adam's apple)
"Pelvis" = lowest point of pelvis/crotch/butt standing up.  The bottom of the "V".
"Knee' = center of knee cap.
"Heel" = base where one would stand.
"Elbow" = middle of the elbow, at the tip of the bone
"Wrist" = anywhere along the wristline, but usually the side with the shortest distance from the elbow point on the pic.
"Ftip" = tip of middle finger (usually projected, as most poses aren't open-handed.)

Thanks very much for clarifying.

The bottom of the head to base of the collar is usually quoted as being roughly a third of a head. So according to your measurements, a head should be about 13.5% relative to the height of the body, on average. Does this agree with your measurements?

And in case you were interested, that central hole at the base of the neck is the manubrium.

Point of interest, actually- the bone underlying that hole is the manubrium.  That hole is the suprasternal notch, which I've incorrectly called the subclavicle notch ever since being introduced to it as a control point in martial arts training.

Interesting uses of trig... I knew there was a reason to study it in high school. :D

That's about 1/7, which I've gathered is the head/height ratio for most humans.  It does seem this would be higher for shorter people, as heads generally don't change sizes that much between tall and short adults.

I'm a math midget.  I'm not entirely certain I understand exactly how your ratios work.  Is there a chance you could give it to me in other words?

Well for the width-to-depth ratios, let's say you take a limb 10 inches deep. A 0.7 ratio would be 7 inches wide for every 10 inches of depth, so width = 7" and depth = 10".  If the image is taken from somewhere between a width cross-section, and a height cross-section, then the image's cross-section's length would be somewhere between width and depth.  The formula I gave assumes the cross-section's length goes steadily from width (0 degrees) to depth (90 degrees).  So if the rotation is estimated at 22.5 degrees (1/4 of 90 degrees) rotation, the length of the cross-section would be

7+(10-7)*(22.5/90)
=7+3*1/4
=7+0.75
=7.75"

So if you measure about 7.75 scale inches on the photo or art, finding width and depth would be like this

Ratio (r):
r=0.7

Intermediary between r and 1, corresponding to the image's cross-section (t):
t=r+(A/90)*(1-r)
=0.7+(22.5/90)*(1-0.7)
=0.7+0.25*0.3
=0.7+0.075
=0.775

Measurement (m):
m=7.75"

Depth (d):
d=m/t
=7.75/0.775
=10"

Width (w):
w=d*r
=10*0.7
=7"

a & b radii for ellipse:
a=w/2=7/2=3.5
b=d/2=10/2=5

pi Approximation:
pi=3.14

Approximate Circumference (C):
C=sqrt(2*(a^2+b^2))*pi*r
=sqrt(2*(5^2+3.5^2))*3.14*0.7
=sqrt(2*(25+12.25))*2.198
=sqrt(2*37.25)*2.198
=sqrt(77.25)*2.198
=8.789*2.198
=19.318"

----

For the height ratios, it's just % of total height.  You either need to guess what your character's height is, or you can use the real height of the person you are morphing or otherwise basing the art off of.

I think I understand most of the individual things you laid out, but my math deficient mind can't quite put it all together as yet.  I guess it'll make sense later.  Thanks for trying :)

Thanks very much for helping crunch the numbers, and show the terms.  While I don't draw this stuff, it's good to know when I put the physique into words.  :)

Ok, now I'm trying volume and weight.  I ran a test on a very muscular FBB, and got the following percentages.

PART (Shape x (copies)) ..% of volume (all copies)
THIGH (Ellipsoid x 2) ....21.1%
ANKLE (Frustum x 2) ......1.8%
CALF (Ellipsoid x 2) .....9.2%
BUTT (Ellipsoid x 2) .....9.7%
ABDOM (Frustum x 1) ......14.2%
CHEST (Cylinder x 1) .....28.4%
TRAP (Ellipsoid x 0.5) ...0.6%
NECK (Cylinder x 1) ......0.9%
BICEP (Ellipsoid x 2) ....6.0%
FOREARM (Ellipsoid x 2) ..4.7%
DELT (Ellipsoid x 1) .....3.3%

I'm leaving out Head, Feet, and Hands as they don't differ much from person to person in weight or size.  I may need to get a common skeletal weight to figure in for weight.  Do most of these volume proportions look correct, for anyone familiar with the subject?

The bodybuilder in question was 5-6 (66 in), and photographed at about 180 lbs.  Her "muscle volume" came to 4174 cubic inches.  In other words, if she was redistributed into a cube, all of the cube's dimensions would be 16".

I may not have perfected the method (as one could imagine it is pretty hard to get exact off just a picture), but I have put together a spreadsheet to be used for approximating weight off a picture.  Far from an exact science, it at least beats trying to guess the weight just by eyeballing.  Right now it is hosted on my Yahoo! group (you have to sign up for the group).  If anyone wants to host it elsewhere, be my guest.

Weight Measurement Aides (http://groups.yahoo.com/group/MuskelGrotheCreations/files/Measurement%20Aids/)

Here is an illustration of the volume zones used.
(http://i163.photobucket.com/albums/t299/MuskelGrothe/WeightZones.gif)

Of course, part of the fun of drawing amazons is guessing the weight.  Even so, I grew tired of just guessing and wanted a better method.  My guess have been pretty accurate for the two fantasy girls I tested out.  I also tested it on a few real FBBs, and the weights seemed to hold up.  With the inexact method, there will still be some guesswork involved, but now you can at least have a "neighborhood" to place your guess in.

Hi MuskelGrothe.  Many thanks for creating this useful tool!  I've been trying to apply it to a commissioned piece of artwork, and run into a couple issues for which I have questions.  ???

First, the example spreadsheet entries for the chest appear to use a width greater than the length (75 pixels versus 45 pixels).  Your diagram shows chest width as the measurement from collarbone to abdomen, while the chest length is the measurement across from side-to-side, so by the diagram, length should be greater than width.  Which is correct, or does it matter?

Second, your ratios width/depth for thighs, biceps, and calves don't give guidance for a normal, non-muscular woman.  I tend to use the ratios closest to 1 in that case.

Third, the weight sheet contains an input for Target weight in pounds.  Isn't that what we're trying to calculate?  So what purpose does that serve?  For now I've been putting in something that approximates 0.036 lb/in^3, erring on the low side for less muscular women and on the high side for more muscular women.

Also, I've put in a row for elliptical circumference using Ramanujan's approximation, which is

pi*[3(a+b) - sqrt((3a+b)(a+3b))]

where a and b are the major and minor radii.  This seems to work out pretty well.  An example formula is

=PI*(1.5*(C8+C9)-0.5*SQRT((3*C8+C9)*(3*C9+C8)))*$B18/$B17

which gives a circumference in inches.

I love this thread.

Heh, sure, what's not to like?  We get to apply our analytical skills to one of our favorite aesthetic subjects!

I messed up on that entry.  But since cylindrical volume simplifies to =(1/4)*pi*w*h*d, it doesn't matter which dimension is which, due to the commutative law.



For volume, I think it the ratios are a little less important, especially for longer bones (limbs). 


Target weight doesn't affect the computed weight at all.  I just included it to help adjust the formula if needed.  The idea was to compare a known weight of an individual with the computed result.  The constant PoundsPerCubicInch and PPCIDev (PoundsPerCubicInch random deviation range) are used with the volume, calculated skeleton weight, and constant adult head weight to get the result.  If you want to change the PoundsPerCubicInch or PPCIDev values, just type them in on the constant page.  I got PoundsPerCubicInch by averaging out a sample set, and PPCIDev was the standard deviation of that set.


I'll put a circumference measurement on the spreadsheet soon.  Biceps, Forearms, Thighs, Calves, Waist, and Chest should be on it. 

Ah, good to know.  It does, however, matter for chest measurements like circumference.  And I don't know what a typical depth to width or depth to height ratio for the chest would be, and I was hoping you'd have some advice there.
 

Great, I'll not worry about it overmuch then.  It seems to me though that for my Lina commission (http://amaz0ns.com/option,com_smf/Itemid,135/topic,7636.0/) I got some rather large circumferences for her normal form,  20" thighs and 18" calves, before her little growth spurt.  Looking things up, that seems on the high side for a normal woman.  But perhaps she really did have thundrous thighs. :D  Or maybe ellipses aren't the greatest model of a cross-sectional leg.


Ah, then I'm guessing it was just the random element of the computation that was fooling me.  Thanks for confirming that.


That would be great!  Thanks again for setting this up!

I posted the updated sheet last night.  Measurements for thigh, calf, bicep, forearm, chest, and waist are calculated.  Others like neck can be added easily by copying and pasting a cell.

For chest, here is an alternate approximation to the one on the sheet:
=HTRat*AVERAGE((2*I7+I9*(PI-2));2*(I7+I9))

The first one
2*I7+I9*(PI-2)
is the circumference of a circle with diameter=depth, plus 2*(height-depth).  I forget what it's called (I'm sure there's a name for it) but it's the shape formed by the rectangle (h-d)x(d) placed between the two halves of the circle diameter=depth. (a form of a rounded rectangle)

The second parameter
2*(I7+I9)
is just the perimeter of the rectangle d x h

For really busty women, it may be good to figure in the boobs & pecs separately.