In the screenshot below, the red colored bone is selected and only that bone and the blue parent bone will react on the IK tool, but the green parents will not, because the parent of the blue bone has two child bones.
For an animation of a segmented creature I'm creating, I want to be able to do IK on a chain of bones, but still have branches attached to each of the bones in the chain. Those branches represent the legs, four legs for each segment.
If I can somehow modify the bone position of the unattached branches, I will have to calculate where those branches are, relative to the position of the "parent" bone (it could be called a "virtual parent", because it behave like a parent, but it isn't attached to the branch-bone in a parent-child relationship.
I hope I'm still perfectly clear to you.
Now for the maths. And maths isn't my strong point, alas. So would someone with more recent experience in this matter review my formulas?
___________
Suppose, I have the position of the virtual parent bone P, and the position of the virtual child bone Q. The position of Q will depend on three values:
- position of point P
- angle of the parent bone, alpha
- scale of the parent bone, s
Because Q will rotate around P (due to the changing bone angle of the virtual parent bone in P), I first need to transform the coordinates of Q so, that P is at the origin (0,0). If I know where the position of P is in each frame, I can simple at its coordinates to the coordinates calculated in the transformed coordinates system.
For P = ( xp(t) , yp(t) ) and Q = ( xq(t) , yq(t) )
u(t) = xq(t) - xp(t) [1a]
v(t) = yq(t) - yp(t) [1b]
I can now express the cartesian coordinates as polar coordinates
s(t) is bone scale of parent bone
m is the distance between points P and Q, for a scale equal to one:
m = sqrt ( u(0) * u(0) + v(0) * v(0) ) / s(0)
u(t) = m * s(t) * cos( beta + alpha ) [2a]
v(t) = m * s(t) * sin( beta + alpha ) [2b]
because at t=0, alpha is zero, this applies:
u(0) = m * s(0) * cos( beta )
v(0) = m * s(0) * sin( beta )
or:
cos( beta ) = u(0) / ( m * s(0) ) [3a]
sin( beta ) = v(0) / ( m * s(0) ) [3b]
From this maths source I know that this applies:
cos( beta + alpha ) = cos( beta ) * cos( alpha ) - sin( beta ) * sin( alpha )
sin( beta + alpha ) = sin( beta ) * cos( alpha ) + cos( beta ) * sin( alpha )
If [3a] and [3b] are substituted, this applies:
cos( beta + alpha )
= u(0) * cos( alpha ) / ( m * s(0) )
- v(0) * sin( alpha ) / ( m * s(0) )
sin( beta + alpha )
= v(0) * cos( alpha ) / ( m * s(0) )
+ u(0) * sin( alpha ) / ( m * s(0) )
If [2a] and [2b] are substituted, this applies:
u(t) = ( s(t) / s(0) ) * ( u(0) * cos( alpha ) - v(0) * sin( alpha ) )
v(t) = ( s(t) / s(0) ) * ( v(0) * cos( alpha ) - u(0) * sin( alpha ) )
If [1a] and [1b] are substituted, this applies:
xq(t) = xp(t) + ( s(t) / s(0) ) * ( u(0) * cos( alpha ) - v(0) * sin( alpha ) )
yq(t) = yp(t) + ( s(t) / s(0) ) * ( v(0) * cos( alpha ) - u(0) * sin( alpha ) )
in which:
s(0), u(0) and v(0) are constants
xp(t), yp(t), s(t) and alpha are variables, depending on the values of the bone position, bone angle and bone scale of the virtual parent bone in frame number t
__________
Is this reasoning correct. I need to know, otherwise I can't really continue. Perhaps someone has found a similar solution, or can point me to a web page with a solution, so I can check if I'm correct.
Thanks in advance.
