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 this but I can't understand the equations there (they don't seem to be standard algebraic). I would like an equation (no matter how long) that can be interpreted by a standard programming language (I'm using Java).

Answer 1

Here is a Python example which finds the intersection of a line and a plane.

Where the plane can be either a point and a normal, or a 4d vector (normal form), In the examples below (code for both is provided).

Also note that this function calculates a value representing where the point is on the line, (called fac in the code below). You may want to return this too, because values from 0 to 1 intersect the line segment - which may be useful for the caller.

Other details noted in the code-comments.

---

Note: This example uses pure functions, without any dependencies - to make it easy to move to other languages. With a Vector data type and operator overloading, it can be more concise (included in example below).

# intersection function def isect_line_plane_v3(p0, p1, p_co, p_no, epsilon=1e-6):     """     p0, p1: Define the line.     p_co, p_no: define the plane:         p_co Is a point on the plane (plane coordinate).         p_no Is a normal vector defining the plane direction;              (does not need to be normalized).      Return a Vector or None (when the intersection can't be found).     """      u = sub_v3v3(p1, p0)     dot = dot_v3v3(p_no, u)      if abs(dot) > epsilon:         # The factor of the point between p0 -> p1 (0 - 1)         # if 'fac' is between (0 - 1) the point intersects with the segment.         # Otherwise:         #  < 0.0: behind p0.         #  > 1.0: infront of p1.         w = sub_v3v3(p0, p_co)         fac = -dot_v3v3(p_no, w) / dot         u = mul_v3_fl(u, fac)         return add_v3v3(p0, u)      # The segment is parallel to plane.     return None  # ---------------------- # generic math functions  def add_v3v3(v0, v1):     return (         v0[0] + v1[0],         v0[1] + v1[1],         v0[2] + v1[2],     )   def sub_v3v3(v0, v1):     return (         v0[0] - v1[0],         v0[1] - v1[1],         v0[2] - v1[2],     )   def dot_v3v3(v0, v1):     return (         (v0[0] * v1[0]) +         (v0[1] * v1[1]) +         (v0[2] * v1[2])     )   def len_squared_v3(v0):     return dot_v3v3(v0, v0)   def mul_v3_fl(v0, f):     return (         v0[0] * f,         v0[1] * f,         v0[2] * f,     ) 

If the plane is defined as a 4d vector (normal form), we need to find a point on the plane, then calculate the intersection as before (see p_co assignment).

def isect_line_plane_v3_4d(p0, p1, plane, epsilon=1e-6):     u = sub_v3v3(p1, p0)     dot = dot_v3v3(plane, u)      if abs(dot) > epsilon:         # Calculate a point on the plane         # (divide can be omitted for unit hessian-normal form).         p_co = mul_v3_fl(plane, -plane[3] / len_squared_v3(plane))          w = sub_v3v3(p0, p_co)         fac = -dot_v3v3(plane, w) / dot         u = mul_v3_fl(u, fac)         return add_v3v3(p0, u)      return None 

For further reference, this was taken from Blender and adapted to Python. isect_line_plane_v3() in math_geom.c

---

For clarity, here are versions using the mathutils API (which can be modified for other math libraries with operator overloading).

# point-normal plane def isect_line_plane_v3(p0, p1, p_co, p_no, epsilon=1e-6):     u = p1 - p0     dot = p_no * u     if abs(dot) > epsilon:         w = p0 - p_co         fac = -(plane * w) / dot         return p0 + (u * fac)      return None   # normal-form plane def isect_line_plane_v3_4d(p0, p1, plane, epsilon=1e-6):     u = p1 - p0     dot = plane.xyz * u     if abs(dot) > epsilon:         p_co = plane.xyz * (-plane[3] / plane.xyz.length_squared)          w = p0 - p_co         fac = -(plane * w) / dot         return p0 + (u * fac)      return None