![]() This formular will be used for 2-plane intersection. And, a negative distance means the point is in opposite side.īefore finding the intersection of 2 planes, we discuss finding the intersection of 3 planes first. For example, if the distance is positive, the point is in the same side where the normal is pointing to. It is useful to determine the direction of the point. Notice this distance is signed can be negative value. For example, the distance from a point (-1, -2, -3) to a plane x + 2y + 2z - 6 = 0 is Note that the distance formula looks like inserting P 2 into the plane equation, then dividing by the length of the normal vector. ![]() The numerator part of the above equation, is expanded įinally, we put it to the previous equation to complete the distance formula The distance D between a plane and a point P 2 becomes The shortest distance from an arbitrary point P 2 to a plane can be calculated by the dot product of two vectors and, projecting the vector to the normal vector of the plane. For example, the distance from the origin for the following plane equation with normal (1, 2, 2) is 2 ĭistance from a Point Distance between Plane and Point Therefore, we can find the distance from the origin by dividing the standard plane equation by the length (norm) of the normal vector (normalizing the plane equation). The equation of the plane can be rewritten with the unit vector and the point on the plane in order to show the distance D is the constant term of the equation If the unit normal vector (a 1, b 1, c 1), then, the point P 1 on the plane becomes ( Da 1, Db 1, Dc 1), where D is the distance from the origin. If the normal vector is normalized (unit length), then the constant term of the plane equation, d becomes the distance from the origin. If we substitute the constant terms to, then the plane equation becomes simpler This dot product of the normal vector and a vector on the plane becomes the equation of the plane. Since the vector and the normal vector are perpendicular each other, the dot product of two vector should be 0. We can define a vector connecting from P 1 to P, which is lying on the plane. Let the normal vector of a plane, and the known point on the plane, P 1. The equation of a plane in 3D space is defined with normal vector (perpendicular to the plane) and a known point on the plane. Example: Intersection Line of 2 Planes (Interactive Demo).
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