line - plane intersection -- Intersection formulas for 3d objects

line - plane intersection

Assumed to be centred at 0, the coordinate system origin.

Parametric solution (t). Solutions were derived automatically using sympy.

Points in cartesian coordinates (x, y, z)

- x: $x_{0} - \frac{a x_{0} + b y_{0} + c z_{0}}{a + b k + c l}$
- y: $- \frac{k \left(a x_{0} + b y_{0} + c z_{0}\right)}{a + b k + c l} + y_{0}$
- z: $- \frac{l \left(a x_{0} + b y_{0} + c z_{0}\right)}{a + b k + c l} + z_{0}$

x = x0 - (a*x0 + b*y0 + c*z0)/(a + b*k + c*l);

y = -k*(a*x0 + b*y0 + c*z0)/(a + b*k + c*l) + y0;

z = -l*(a*x0 + b*y0 + c*z0)/(a + b*k + c*l) + z0;

sol = -(a*x0 + b*y0 + c*z0)/(a + b*k + c*l);

Assumed to be centred at 0, the coordinate system origin.

Parametric solution (t). Solutions were derived automatically using sympy.

$- \frac{a x_{0} + b y_{0} + c z_{0}}{a \sin{\left (\theta \right )} \cos{\left (\phi \right )} + b \sin{\left (\phi \right )} \sin{\left (\theta \right )} + c \cos{\left (\theta \right )}}$

Points in cartesian coordinates (x, y, z)

- x: $x_{0} - \frac{\left(a x_{0} + b y_{0} + c z_{0}\right) \sin{\left (\theta \right )} \cos{\left (\phi \right )}}{a \sin{\left (\theta \right )} \cos{\left (\phi \right )} + b \sin{\left (\phi \right )} \sin{\left (\theta \right )} + c \cos{\left (\theta \right )}}$
- y: $y_{0} - \frac{\left(a x_{0} + b y_{0} + c z_{0}\right) \sin{\left (\phi \right )} \sin{\left (\theta \right )}}{a \sin{\left (\theta \right )} \cos{\left (\phi \right )} + b \sin{\left (\phi \right )} \sin{\left (\theta \right )} + c \cos{\left (\theta \right )}}$
- z: $z_{0} - \frac{\left(a x_{0} + b y_{0} + c z_{0}\right) \cos{\left (\theta \right )}}{a \sin{\left (\theta \right )} \cos{\left (\phi \right )} + b \sin{\left (\phi \right )} \sin{\left (\theta \right )} + c \cos{\left (\theta \right )}}$

x = x0 - (a*x0 + b*y0 + c*z0)*sin(theta)*cos(phi)/(a*sin(theta)*cos(phi) + b*sin(phi)*sin(theta) + c*cos(theta));

y = y0 - (a*x0 + b*y0 + c*z0)*sin(phi)*sin(theta)/(a*sin(theta)*cos(phi) + b*sin(phi)*sin(theta) + c*cos(theta));

z = z0 - (a*x0 + b*y0 + c*z0)*cos(theta)/(a*sin(theta)*cos(phi) + b*sin(phi)*sin(theta) + c*cos(theta));

sol = -(a*x0 + b*y0 + c*z0)/(a*sin(theta)*cos(phi) + b*sin(phi)*sin(theta) + c*cos(theta));

By Johannes Buchner | Source code: https://github.com/JohannesBuchner/intersection | Open a issue or pull request if you would like somthing added