# 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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