问题描述
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如果给定一条直线（由矢量或直线上的两个点表示），如何找到直线与平面相交的点？我已经找到了很多资源，但是我不明白那里的方程式（它们似乎不是标准的代数）。我想要一个可由标准编程语言（我使用Java）解释的方程式（无论时间长短）。
解决方法
这是Java中的一种方法，用于查找线和平面之间的交点。没有包含向量方法，但是它们的功能很容易说明。
/**
* 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 Vector lineIntersection(Vector planePoint,Vector planeNormal,Vector linePoint,Vector lineDirection) {
if (planeNormal.dot(lineDirection.normalize()) == 0) {
return null;
}
double t = (planeNormal.dot(planePoint) - planeNormal.dot(linePoint)) / planeNormal.dot(lineDirection.normalize());
return linePoint.plus(lineDirection.normalize().scale(t));
}
Vector
# 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)
else:
# 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,)
p_co
def isect_line_plane_v3_4d(p0,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,u)
else:
return None
isect_line_plane_v3()
# point-normal plane
def isect_line_plane_v3(p0,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)
else:
return None
# normal-form plane
def isect_line_plane_v3_4d(p0,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)
else:
return None
x = 5 - 13t
y = 5 - 11t
z = 5 - 8t
(x,y,z)
x + 2y + 2z = 5
x
y
z
t
t
t
t
y
z
#Based on http://geomalgorithms.com/a05-_intersect-1.html
from __future__ import print_function
import numpy as np
epsilon=1e-6
#Define plane
planeNormal = np.array([0,1])
planePoint = np.array([0,5]) #Any point on the plane
#Define ray
rayDirection = np.array([0,-1,-1])
rayPoint = np.array([0,10]) #Any point along the ray
ndotu = planeNormal.dot(rayDirection)
if abs(ndotu) < epsilon:
print (\"no intersection or line is within plane\")
w = rayPoint - planePoint
si = -planeNormal.dot(w) / ndotu
Psi = w + si * rayDirection + planePoint
print (\"intersection at\",Psi)
# n: normal vector of the Plane
# V0: any point that belongs to the Plane
# P0: end point 1 of the segment P0P1
# P1: end point 2 of the segment P0P1
n = np.array([1.,1.,1.])
V0 = np.array([1.,-5.])
P0 = np.array([-5.,-1.])
P1 = np.array([1.,2.,3.])
w = P0 - V0;
u = P1-P0;
N = -np.dot(n,w);
D = np.dot(n,u)
sI = N / D
I = P0+ sI*u
print I
[-3.90909091 1.18181818 -0.27272727]
/**
* findLinePlaneIntersectionCoords (to avoid requiring unnecessary instantiation)
* Given points p with px py pz and q that define a line,and the plane
* of formula ax+by+cz+d = 0,returns the intersection point or null if none.
*/
function findLinePlaneIntersectionCoords(px,py,pz,qx,qy,qz,a,b,c,d) {
var tDenom = a*(qx-px) + b*(qy-py) + c*(qz-pz);
if (tDenom == 0) return null;
var t = - ( a*px + b*py + c*pz + d ) / tDenom;
return {
x: (px+t*(qx-px)),y: (py+t*(qy-py)),z: (pz+t*(qz-pz))
};
}
// Example (plane at y = 10 and perpendicular line from the origin)
console.log(JSON.stringify(findLinePlaneIntersectionCoords(0,1,-10)));
// Example (no intersection,plane and line are parallel)
console.log(JSON.stringify(findLinePlaneIntersectionCoords(0,-10)));/**
* 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);
}