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

[Thatcher U. <tu...@tu...>]

From: David Kornmann <da...@ik...>
>
> > Hmmm ... surely what you want is to find the point on the line closest
to
> > the center of the sphere (easy to do) and then if that is less than R
away
> > from the center you have definite intersection, and proceed to compute
the
> > points at which the line intersects the sphere, and otherwise, you
> > conveniently have a point which is in line with the nearest tangential
point
> > of the sphere, so you can easily find that tangential point.
>
> Well, computing the point when the ray intersect with the sphere is not a
big
> deal.
> The problem comes when the ray does not intersect:
>
>  I am not looking for the closest surface point on the sphere then, but a
point
> that
> can be described as follow:
>
> - Take the plane defined with the ray and the center of the sphere.
> - Rotate the ray around its origin and along the plane until it is tangent
to
> the sphere.
> - Compute the tangetial point (easy).
>
> That's what seems to me complicated and which needs to be very accurate.

The description is a little confusing, but if I understand correctly, this
is not hard to solve.  Lets label the points:

R = ray origin
C = center of circle
T = tangent point you're interested in

If I catch your meaning, you want C, T, and the ray to all be coplanar.

If you make a triangle out of R, C and T, you have a right triangle, with
the right angle at T.  You can compute the length of the hypotenuse, it's
just |RC|, and you know the length of CT because it's equal to the circle's
radius.  So you can compute the angle CRT, it's just arcsin(|CT| / |RC|).
The angle RCT is 90 - CRT.  There's some plane which contains all these
points; compute its normal: n = normalize((C-R) x ray_direction).  To find
T, take the vector CR, rotate about n by angle RCT, and make its length
equal to the sphere radius.

What's the purpose of this?

--
Thatcher Ulrich
http://tulrich.com