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t value:
t = d ( q p org )
d
.
A.3
Closest Point on a Plane
Consider a plane P defined in the standard implicit manner as all points p
that satisfy
p n = d,
where n is a unit vector. Given a point q , we wish to find the point q
,
which is the result of projecting q onto P. Point q
is the closest point to
q on P.
We showed how to compute the distance from a point to a plane in
Section 9.5.4. To compute q
, we simply displace q by this distance, parallel
to n .
Computing the closest
point on a plane
q
= q + (d − q n ) n
Notice that this is the same as Equation (A.1), which computes the closest
point to an implicit line in 2D.
A.4
Closest Point on a Circle or Sphere
Imagine a 2D point q and a cir-
cle with center c and radius r.
(The following discussion also
applies to a sphere in 3D.) We
wish to find q
, which is the
closest point on the circle to q .
Let d be the vector from q
to c . This vector intersects the
circle at q
. Let b be the vec-
tor from q to q
, as shown in
Figure A.3
Finding the closest point on a circle
Figure A.3.
Now, clearly, b = d −r.
Therefore,
d − r
b =
d .
d
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