# line and plane intersection

In 3D, three planes P1, P2 and P3 can intersect (or not) in the following ways: Only two planes are parallel, andthe 3rd plane cuts each in a line[Note: the 2 parallel planes may coincide], 2 parallel lines[planes coincide => 1 line], No two planes are parallel, so pairwise they intersect in 3 lines, Test a point of one line with another line. Line-plane and line-line are not the only intersections in geometry, you will also find line-point intersection as well. Solution: Because the intersection point is common to the line and plane we can substitute the line parametric points into the plane equation to get: 4 (− 1 − 2t) + (1 + t) − 2 = 0. t = − 5/7 = 0.71. Here are some sample "C++" implementations of these algorithms. [1, 1, 2] = 3: A diagram of this is shown on the right. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. How do we find the intersection point of a line and a plane? P (a) line intersects the plane in From MathWorld--A Wolfram Web Resource. Stokes' theorem integration. As it is fundamentally a 2D-package, it doesn't know how to compute the intersection of the line and plane and so doesn't know when to stop drawing the line. https://mathworld.wolfram.com/Line-PlaneIntersection.html, Parallelepiped Stokes' Theorem to evaluate integral. Thus, it is on the line of intersection for the two planes, and the parametric equation of L is: P ( s ) = I + s ( n 1 x n 2 ). Intersection of plane and line.. Walk through homework problems step-by-step from beginning to end. Finally, if the line intersects the plane in a single point, determine this point of intersection. This value can then be plugged back in to (2), (3), and (4) to give the point of intersection . a Plane. Here's the question. Windows. Finding the intersection of an infinite ray with a plane in 3D is an important topic in collision detection. 2. 1 : t1;               // clip to max 1        if (t0 == t1) {                  // intersect is a point            *I0 = S2.P0 +  t0 * v;            return 1;        }        // they overlap in a valid subsegment        *I0 = S2.P0 + t0 * v;        *I1 = S2.P0 + t1 * v;        return 2;    }    // the segments are skew and may intersect in a point    // get the intersect parameter for S1    float     sI = perp(v,w) / D;    if (sI < 0 || sI > 1)                // no intersect with S1        return 0; // get the intersect parameter for S2    float     tI = perp(u,w) / D;    if (tI < 0 || tI > 1)                // no intersect with S2        return 0; *I0 = S1.P0 + sI * u;                // compute S1 intersect point    return 1;}//===================================================================, // inSegment(): determine if a point is inside a segment//    Input:  a point P, and a collinear segment S//    Return: 1 = P is inside S//            0 = P is  not inside SintinSegment( Point P, Segment S){    if (S.P0.x != S.P1.x) {    // S is not  vertical        if (S.P0.x <= P.x && P.x <= S.P1.x)            return 1;        if (S.P0.x >= P.x && P.x >= S.P1.x)            return 1;    }    else {    // S is vertical, so test y  coordinate        if (S.P0.y <= P.y && P.y <= S.P1.y)            return 1;        if (S.P0.y >= P.y && P.y >= S.P1.y)            return 1;    }    return 0;}//===================================================================, // intersect3D_SegmentPlane(): find the 3D intersection of a segment and a plane//    Input:  S = a segment, and Pn = a plane = {Point V0;  Vector n;}//    Output: *I0 = the intersect point (when it exists)//    Return: 0 = disjoint (no intersection)//            1 =  intersection in the unique point *I0//            2 = the  segment lies in the planeintintersect3D_SegmentPlane( Segment S, Plane Pn, Point* I ){    Vector    u = S.P1 - S.P0;    Vector    w = S.P0 - Pn.V0;    float     D = dot(Pn.n, u);    float     N = -dot(Pn.n, w);    if (fabs(D) < SMALL_NUM) {           // segment is parallel to plane        if (N == 0)                      // segment lies in plane            return 2;        else            return 0;                    // no intersection    }    // they are not parallel    // compute intersect param    float sI = N / D;    if (sI < 0 || sI > 1)        return 0;                        // no intersection    *I = S.P0 + sI * u;                  // compute segment intersect point    return 1;}//===================================================================, // intersect3D_2Planes(): find the 3D intersection of two planes//    Input:  two planes Pn1 and Pn2//    Output: *L = the intersection line (when it exists)//    Return: 0 = disjoint (no intersection)//            1 = the two  planes coincide//            2 =  intersection in the unique line *Lintintersect3D_2Planes( Plane Pn1, Plane Pn2, Line* L ){    Vector   u = Pn1.n * Pn2.n;          // cross product    float    ax = (u.x >= 0 ? 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