Re: [Algorithms] Nearest point on plane for 2D IK

[<ro...@do...>]

Jeff Lander  wrote:

>Since we were talking about skeleton animation, I have a question.
>
>For my IK system, I prefer to use actual 2D joints where appropriate =
rather
> then use the general 3D form and limit with DOF restrictions.=20
> I have received some email asking me how to do this with my CCD
> IK routine so I wanted to write up a paragraph on the=20
>little math routine for the magazine. =20
>I think it is pretty obvious to people familiar with vector math,
> but I have found that many developers don't have the background
>and need some help.=20


> I will put this in with my continuing series on why dot product is your=
 friend.
>


>I want to sanity check it to make sure this is a good, fast way to do =
this.=20
> I know that it works but it may not be the best method.
>
>Given a bone object in 3 space, B, and a desired 3D target to attempt to=
 reach, T. =20
>The method is to determine the plane that B is allowed to rotate,=20
>then project T into that plane.  Then I solve for the rotation angle.
>
>To project T onto B's free rotational plane:
>
>N =3D free axis of rotation
>
>V =3D T - B		// Creates vector from base to target

It is not at all clear to me what you mean by B.  Is it a pure vector
(like N) or is it the position vector of a point (like T).  Does it
represent the position of a bone or the relative positions of the two
ends of the bone?=20

This equation only makes sense if B is the position vector of a point.
In that case, it tells where the bone is, but not how it is oriented.
But also, in this case,  "B's free rotational plane" is without
meaning.

>P =3D T - N ( N dot V)	// Project V onto axis and subtract that from the=
 target
>

I think that your criterion for P is that it be perpendicular to N.
This P satisfies that criterion, that is, P is the perpendicular
projection of V on the plane perpendicular to N,  only in the special
case that B is perpendicular to N, which I think is probably not true,
in general, for the systems you want to work with.

The proper vector formula for the perpendicular projection of V on the
plane perpendicular to N is=20

P =3D V - (N dot V)N

and it holds independently of how you construct V.