Re: Distance between point and plane
- From: "Stephen J. Herschkorn" <sjherschko@xxxxxxxxxxxx>
- Date: Fri, 14 Sep 2007 22:18:43 -0400
Simona wrote:
My problem is: i've a 3 points, a plane defined fy this 3 points (P1,
P2, P3). I've other points. I want to calculate the distance between
that points and the plane.
I look for the equation, but i get only the absolute value.
I need to know if the points are under or over my plane. I get the
normal to my plane for that, ok.
If i make the plane with P1, P2, P3 and then i make the plane with
P2,P1,P3 or P3, P2, P1, the coefficient a, b, c and d (plane: ax+by+cz
+d=0) change.
How i can obtain a distance with sign?
The directed distance from the point (u,v,w) to the plane {(x,y,z): a x + b y + c z + d = 0} is
- (a u + b v + c w + d) / (+/-) sqrt(a^2 + b^2 + c^2), where the sign in the denominator is that of the last nonzero coefficient amongst a, b, and c. Here, when c is nonzero, the distance is negative if (u,v,w) lies above the plane (in the sense of the z-axis), positive if (u,v,w) lies below the plane. See Steen and Ballou, Analytic Geometry.
Note that (u, v, w) and (u0, v0, w0) lie on different sides of the plane iff
(a u + b v + c w + d) (a u0 + b v0 + c w0 + d) < 0.
--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan
.
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