Sure vortigen.
Normally a vector can be put into x,y,z (like this see picture :
3d axis) so if you have any point in 3D, you can define it as (x,y,z) or for example (0,0,1) is 1 unit above the plan along the Z axe, whereas (1,0,0) would be 1 unit on the X axe and nothing else, (1,1,1) would be a vector where there is one unit on each axe.
very obviously if you project a vector (x,y,z) onto the plan x,y , you jsut eliminate the third coordinate (x,y,0).
There is anotehr way to see those vector, and this is usually the way we see them in film or photo , in a spheric coordinate where my theta , phi come from :
spheric coordinate so we have length+theta+phi I will represent with bracket [r,theta,phi]. There is naturally a way to go from 1 coordiante to the other
x=r \, \sin\theta \, \cos\varphi \quad
y=r \, \sin\theta \, \sin\varphi \quad
z=r \, \cos\theta \quad
Visually the forshortening effect is so , we see in the photo only a projection of the various vector onto a plan, so taking the Z axis to be perpandicular to the photo film, we see 3D vector projected onto x,y with variopus theta and phi (theta and phi changing is the movement). In other word , if I take a stick , and photography it tumbling, the stick length is always the same but theta and rho change, and going to the projection over x,y you imemdiately see the x,y change.
Please note that the demonstration will not take into account perspective and distance change which add to the final result.
Remark : in my previous post I reverted the role of theta and phi but it does not change anything.
For a hand it is a bit more complicated , but imagine all finger phalange are a single vector. So the first phalange for example go from origin (0,0,0) to P1 (1,0,0) so the vector is (1,0,0) from origin and is horizontal , the second one is inclined in plan x,y and go from (1,0,0) to (2,1,0) (so the vectro is (1,1,0) if placed at origin) and the last one is vertical in the same plan and go from (2,1,0) to (2,2,0) (so vector is (0,1,0) if at origin, remember a vector is final point minus origin point).
Like this :
first phalange is made of 1 , second is diagonal made of 2 and last is horizontal made of 3.
Code:
------------> Y
|1
|1
| 2
V 2
X 33
remark my phalange 2 is much longer than my phalange 1 and 3.
Basically now, going back to [r,theta,phi] for all vector, and rotating along theta and phi to have the angle changed, and projecting AGAIN in x,y, you immediately see that vector length is changed, as well as the apparent angle between them.
Before (I put angle in degree as it is easier than radian):
vector 1 : (1,0,0) = [1,90,0]
vector 2 : (1,1,0) = [SQRT(2),90,45]
vector 3 : (0,1,0) = [1,90,90]
Now we turn by 30° vertically and 30° horizontally (theta=theta+30 and phi=phi+30)
New vector is spheric coordinate :
vector 1 : [1,120,30] = (SQRT(3)/2*1/2,SQRT(3)/2*SQRT(3),-1/2) =(SQRT(3)/4,3/4,-1/2)
vector 2 : [SQRT(2),120,75] = (SQRT(2)*SQRT(3)/2*0.258,SQRT(2)*SQRT(3)/2*0.965,-SQRT(2)/2)
vector 3 : [1,120,120] = (-SQRT(3)/2*1/2,SQRT(3)/2*SQRT(3/2),-1/2) =(-SQRT(3)/4,3/4,-1/2)
projecting into the x,y plan (getting ride of z) +numeric approx
proj vector 1 : = (0.43, 0.75,0)
proj vector 2 : = (0.63,1.18,0)
proj vector 3 : = (-0.43,0.75,0)
I don't have a picture to attach, sorry.
Now adding all together to get our hand back :
X2 Y2
0,43 0,75
1,06 1,93
0,63 2,68
Just plot it in scatter mode in open office or open document.
Comapre to the original
X3 Y3
1 0
1 1
0 1
You haven't 90° anymore but a more bent angle.
Please note that if one started from the more bent one, and turned by -30 theta and -30 phi, you would actually get a less bent hand.
This naturally NO proof that spekator did not cheat, it is a mathematical demonstration that projected angle and length change with turning in all position.
click me for a picture
Top the "3" phalange of a finger. Bottom, we rotated the whole by theta=30 and phi=30.
PS: other rotation would have made the finger apparent angle even more "unbent" (imagine the "finger" inclination going the other way around, I guess this is a rotation of about 75 theta, and 75 phi).
ETA: and before sweaty protest, the whole point is on the doll you have no reference, but looking at the reflection it is obvious the angle theta and phi change are great, the same as with a SWINGING arm the angle changes change greatly. Plus elevation, plus plus plus...
QED