Question

So I was reading over something on this page (http://gamedeveloperjourney.blogspot.com/2009/04/point-plane-collision-detection.html)

The author mentioned

 d = - D3DXVec3Dot(&vP1, &vNormal);

where vP1 is a point on the plane and vNormal is the normal to the plane. I’m curious as to how this gets you the distance from the world origin since the result will always be 0. In addition, just to be clear (since I’m still kind of hazy on the d part of a plane equation), is d in a plane equation the distance from a line through the world origin to the plane’s origin?

Answer 1

In the generic case the distance between a point p and a plane can be computed by

<p - p0, normal>

where <a, b> is the dot product operation

<a, b> = ax*bx + ay*by + az*bz

and where p0 is a point on the plane.

When n is of unity length the dot product between a vector and it is the (signed) length of the projection of the vector on the normal

The formula you are reporting is just the special case when the point p is the origin. In this case

distance = <origin - p0, normal> = - <p0, normal>

This equality is formally wrong because the dot product is about vectors, not points… but still holds numerically. Writing down the explicit formula you get that

(0 - p0.x)*n.x + (0 - p0.y)*n.y + (0 - p0.z)*n.z

is the same as

- (p0.x*n.x + p0.y*n.y + p0.z*n.z)

Indeed a nice way to store a plane is to save the normal n and the value of k = <p0, n> where p0 is any point on the plane (the value of k is independent on which point you choose of the plane).