3D Java中的平面线交叉

Question

This post is a reply to: 3D Ray-Quad intersection test in java Because I can't yet comment.

My question is, how did they get: A point M belongs to this plane iff it satisfies this equation: n . ( M - S1 ) = 0

How is the (dotProduct(n, (M - S1)) == 0) suppose to tell us that the ray intersects the quad?

Answer 1

In my opinion Wikipedia answers this question quite well: Wiki

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

Specifically, let r0 be the position vector of some point P0 = (x0, y0, z0), and let n = (a, b, c) be a nonzero vector. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points r such that

With other words you have a normal vector of a plane and a vector created by a point of the plane and the point you are checking if it is on the plane. The dot-product tells you something about the angle between the two vector. Therefore if the vector is parallel to the plane, the point has to be on the plane.