3D Line-Plane Intersection

Question

If given a line (represented by either a vector or two points on the line) how do I find the point at which the line intersects a plane? I\'ve found loads of resources on th

Answer 1

Just to expand on ZGorlock's answer, I have done the dot product, plus and scalse of 3D Vectors. The references for these calculations are Dot Product, Add two 3D vectors and Scaling. Note: Vec3D is just a custom class which has points: x, y and z.

/**
 * Determines the point of intersection between a plane defined by a point and a normal vector and a line defined by a point and a direction vector.
 *
 * @param planePoint    A point on the plane.
 * @param planeNormal   The normal vector of the plane.
 * @param linePoint     A point on the line.
 * @param lineDirection The direction vector of the line.
 * @return The point of intersection between the line and the plane, null if the line is parallel to the plane.
 */
public static Vec3D lineIntersection(Vec3D planePoint, Vec3D planeNormal, Vec3D linePoint, Vec3D lineDirection) {
    //ax × bx + ay × by
    int dot = (int) (planeNormal.x * lineDirection.x + planeNormal.y * lineDirection.y);
    if (dot == 0) {
        return null;
    }

    // Ref for dot product calculation: https://www.mathsisfun.com/algebra/vectors-dot-product.html
    int dot2 = (int) (planeNormal.x * planePoint.x + planeNormal.y * planePoint.y);
    int dot3 = (int) (planeNormal.x * linePoint.x + planeNormal.y * linePoint.y);
    int dot4 = (int) (planeNormal.x * lineDirection.x + planeNormal.y * lineDirection.y);

    double t = (dot2 - dot3) / dot4;

    float xs = (float) (lineDirection.x * t);
    float ys = (float) (lineDirection.y * t);
    float zs = (float) (lineDirection.z * t);
    Vec3D lineDirectionScale = new Vec3D( xs, ys, zs);

    float xa = (linePoint.x + lineDirectionScale.x);
    float ya = (linePoint.y + lineDirectionScale.y);
    float za = (linePoint.z + lineDirectionScale.z);

    return new Vec3D(xa, ya, za);
}

Answer 2

This question is old but since there is such a much more convenient solution I figured it might help someone.

Plane and line intersections are quite elegant when expressed in homogeneous coordinates but lets assume you just want the solution:

There is a vector 4x1 p which describes the plane such that p^T*x =0 for any homogeneous point on the plane. Next compute the plucker coordinates for the line L=ab^T - ba^T where a = {point_1; 1}, b={point_2;1}, both 4x1 on the line

compute: x=L*p = {x0,x1,x2,x3}

x_intersect=({x0,x1,x2}/x3) where if x3 is zero there is no intersection in the euclidean sense.