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

line - hyperbolical_paraboloid intersection

Equation for line (cartesian)

x: $t + x_{0}$
y: $k t + y_{0}$
z: $l t + z_{0}$

Equation for hyperbolical_paraboloid (cartesian)

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

0= $- \frac{z}{c} - \frac{y^{2}}{b^{2}} + \frac{x^{2}}{a^{2}}$

Intersection solutions

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

$\frac{1}{c \left(a^{2} k^{2} - b^{2}\right)} \left(- \frac{a^{2} l}{2} b^{2} - a^{2} c k y_{0} + b^{2} c x_{0} - \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} l^{2} - 4 a^{2} c k^{2} z_{0} + 4 a^{2} c k l y_{0} - 4 b^{2} c l x_{0} + 4 b^{2} c z_{0} + 4 c^{2} k^{2} x_{0}^{2} - 8 c^{2} k x_{0} y_{0} + 4 c^{2} y_{0}^{2}\right)}\right)$
$\frac{1}{c \left(a^{2} k^{2} - b^{2}\right)} \left(- \frac{a^{2} l}{2} b^{2} - a^{2} c k y_{0} + b^{2} c x_{0} + \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} l^{2} - 4 a^{2} c k^{2} z_{0} + 4 a^{2} c k l y_{0} - 4 b^{2} c l x_{0} + 4 b^{2} c z_{0} + 4 c^{2} k^{2} x_{0}^{2} - 8 c^{2} k x_{0} y_{0} + 4 c^{2} y_{0}^{2}\right)}\right)$

Points in cartesian coordinates (x, y, z)

- x: $x_{0} + \frac{1}{c \left(a^{2} k^{2} - b^{2}\right)} \left(- \frac{a^{2} l}{2} b^{2} - a^{2} c k y_{0} + b^{2} c x_{0} - \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} l^{2} - 4 a^{2} c k^{2} z_{0} + 4 a^{2} c k l y_{0} - 4 b^{2} c l x_{0} + 4 b^{2} c z_{0} + 4 c^{2} k^{2} x_{0}^{2} - 8 c^{2} k x_{0} y_{0} + 4 c^{2} y_{0}^{2}\right)}\right)$
- y: $y_{0} + \frac{k}{c \left(a^{2} k^{2} - b^{2}\right)} \left(- \frac{a^{2} l}{2} b^{2} - a^{2} c k y_{0} + b^{2} c x_{0} - \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} l^{2} - 4 a^{2} c k^{2} z_{0} + 4 a^{2} c k l y_{0} - 4 b^{2} c l x_{0} + 4 b^{2} c z_{0} + 4 c^{2} k^{2} x_{0}^{2} - 8 c^{2} k x_{0} y_{0} + 4 c^{2} y_{0}^{2}\right)}\right)$
- z: $z_{0} + \frac{l}{c \left(a^{2} k^{2} - b^{2}\right)} \left(- \frac{a^{2} l}{2} b^{2} - a^{2} c k y_{0} + b^{2} c x_{0} - \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} l^{2} - 4 a^{2} c k^{2} z_{0} + 4 a^{2} c k l y_{0} - 4 b^{2} c l x_{0} + 4 b^{2} c z_{0} + 4 c^{2} k^{2} x_{0}^{2} - 8 c^{2} k x_{0} y_{0} + 4 c^{2} y_{0}^{2}\right)}\right)$
- x: $x_{0} + \frac{1}{c \left(a^{2} k^{2} - b^{2}\right)} \left(- \frac{a^{2} l}{2} b^{2} - a^{2} c k y_{0} + b^{2} c x_{0} + \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} l^{2} - 4 a^{2} c k^{2} z_{0} + 4 a^{2} c k l y_{0} - 4 b^{2} c l x_{0} + 4 b^{2} c z_{0} + 4 c^{2} k^{2} x_{0}^{2} - 8 c^{2} k x_{0} y_{0} + 4 c^{2} y_{0}^{2}\right)}\right)$
- y: $y_{0} + \frac{k}{c \left(a^{2} k^{2} - b^{2}\right)} \left(- \frac{a^{2} l}{2} b^{2} - a^{2} c k y_{0} + b^{2} c x_{0} + \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} l^{2} - 4 a^{2} c k^{2} z_{0} + 4 a^{2} c k l y_{0} - 4 b^{2} c l x_{0} + 4 b^{2} c z_{0} + 4 c^{2} k^{2} x_{0}^{2} - 8 c^{2} k x_{0} y_{0} + 4 c^{2} y_{0}^{2}\right)}\right)$
- z: $z_{0} + \frac{l}{c \left(a^{2} k^{2} - b^{2}\right)} \left(- \frac{a^{2} l}{2} b^{2} - a^{2} c k y_{0} + b^{2} c x_{0} + \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} l^{2} - 4 a^{2} c k^{2} z_{0} + 4 a^{2} c k l y_{0} - 4 b^{2} c l x_{0} + 4 b^{2} c z_{0} + 4 c^{2} k^{2} x_{0}^{2} - 8 c^{2} k x_{0} y_{0} + 4 c^{2} y_{0}^{2}\right)}\right)$

C Code

x = x0 + (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*l - pow(a, 2)*c*k*y0 + pow(b, 2)*c*x0 - 1.0L/2.0L*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(l, 2) - 4*pow(a, 2)*c*pow(k, 2)*z0 + 4*pow(a, 2)*c*k*l*y0 - 4*pow(b, 2)*c*l*x0 + 4*pow(b, 2)*c*z0 + 4*pow(c, 2)*pow(k, 2)*pow(x0, 2) - 8*pow(c, 2)*k*x0*y0 + 4*pow(c, 2)*pow(y0, 2))))/(c*(pow(a, 2)*pow(k, 2) - pow(b, 2)));

y = y0 + k*(-1.0L/2.0L*pow(a, 2)*pow(b, 2)*l - pow(a, 2)*c*k*y0 + pow(b, 2)*c*x0 - 1.0L/2.0L*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(l, 2) - 4*pow(a, 2)*c*pow(k, 2)*z0 + 4*pow(a, 2)*c*k*l*y0 - 4*pow(b, 2)*c*l*x0 + 4*pow(b, 2)*c*z0 + 4*pow(c, 2)*pow(k, 2)*pow(x0, 2) - 8*pow(c, 2)*k*x0*y0 + 4*pow(c, 2)*pow(y0, 2))))/(c*(pow(a, 2)*pow(k, 2) - pow(b, 2)));

z = z0 + l*(-1.0L/2.0L*pow(a, 2)*pow(b, 2)*l - pow(a, 2)*c*k*y0 + pow(b, 2)*c*x0 - 1.0L/2.0L*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(l, 2) - 4*pow(a, 2)*c*pow(k, 2)*z0 + 4*pow(a, 2)*c*k*l*y0 - 4*pow(b, 2)*c*l*x0 + 4*pow(b, 2)*c*z0 + 4*pow(c, 2)*pow(k, 2)*pow(x0, 2) - 8*pow(c, 2)*k*x0*y0 + 4*pow(c, 2)*pow(y0, 2))))/(c*(pow(a, 2)*pow(k, 2) - pow(b, 2)));

x = x0 + (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*l - pow(a, 2)*c*k*y0 + pow(b, 2)*c*x0 + (1.0L/2.0L)*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(l, 2) - 4*pow(a, 2)*c*pow(k, 2)*z0 + 4*pow(a, 2)*c*k*l*y0 - 4*pow(b, 2)*c*l*x0 + 4*pow(b, 2)*c*z0 + 4*pow(c, 2)*pow(k, 2)*pow(x0, 2) - 8*pow(c, 2)*k*x0*y0 + 4*pow(c, 2)*pow(y0, 2))))/(c*(pow(a, 2)*pow(k, 2) - pow(b, 2)));

y = y0 + k*(-1.0L/2.0L*pow(a, 2)*pow(b, 2)*l - pow(a, 2)*c*k*y0 + pow(b, 2)*c*x0 + (1.0L/2.0L)*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(l, 2) - 4*pow(a, 2)*c*pow(k, 2)*z0 + 4*pow(a, 2)*c*k*l*y0 - 4*pow(b, 2)*c*l*x0 + 4*pow(b, 2)*c*z0 + 4*pow(c, 2)*pow(k, 2)*pow(x0, 2) - 8*pow(c, 2)*k*x0*y0 + 4*pow(c, 2)*pow(y0, 2))))/(c*(pow(a, 2)*pow(k, 2) - pow(b, 2)));

z = z0 + l*(-1.0L/2.0L*pow(a, 2)*pow(b, 2)*l - pow(a, 2)*c*k*y0 + pow(b, 2)*c*x0 + (1.0L/2.0L)*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(l, 2) - 4*pow(a, 2)*c*pow(k, 2)*z0 + 4*pow(a, 2)*c*k*l*y0 - 4*pow(b, 2)*c*l*x0 + 4*pow(b, 2)*c*z0 + 4*pow(c, 2)*pow(k, 2)*pow(x0, 2) - 8*pow(c, 2)*k*x0*y0 + 4*pow(c, 2)*pow(y0, 2))))/(c*(pow(a, 2)*pow(k, 2) - pow(b, 2)));

Distance inside

Distance between crossing points.

$\left\lvert{\frac{1}{c \left(a^{2} k^{2} - b^{2}\right)} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} l^{2} - 4 a^{2} c k^{2} z_{0} + 4 a^{2} c k l y_{0} - 4 b^{2} c l x_{0} + 4 b^{2} c z_{0} + 4 c^{2} k^{2} x_{0}^{2} - 8 c^{2} k x_{0} y_{0} + 4 c^{2} y_{0}^{2}\right)}}\right\rvert$

C Code

sol1 = (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*l - pow(a, 2)*c*k*y0 + pow(b, 2)*c*x0 - 1.0L/2.0L*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(l, 2) - 4*pow(a, 2)*c*pow(k, 2)*z0 + 4*pow(a, 2)*c*k*l*y0 - 4*pow(b, 2)*c*l*x0 + 4*pow(b, 2)*c*z0 + 4*pow(c, 2)*pow(k, 2)*pow(x0, 2) - 8*pow(c, 2)*k*x0*y0 + 4*pow(c, 2)*pow(y0, 2))))/(c*(pow(a, 2)*pow(k, 2) - pow(b, 2)));

sol2 = (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*l - pow(a, 2)*c*k*y0 + pow(b, 2)*c*x0 + (1.0L/2.0L)*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(l, 2) - 4*pow(a, 2)*c*pow(k, 2)*z0 + 4*pow(a, 2)*c*k*l*y0 - 4*pow(b, 2)*c*l*x0 + 4*pow(b, 2)*c*z0 + 4*pow(c, 2)*pow(k, 2)*pow(x0, 2) - 8*pow(c, 2)*k*x0*y0 + 4*pow(c, 2)*pow(y0, 2))))/(c*(pow(a, 2)*pow(k, 2) - pow(b, 2)));

distance = fabs(sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(l, 2) - 4*pow(a, 2)*c*pow(k, 2)*z0 + 4*pow(a, 2)*c*k*l*y0 - 4*pow(b, 2)*c*l*x0 + 4*pow(b, 2)*c*z0 + 4*pow(c, 2)*pow(k, 2)*pow(x0, 2) - 8*pow(c, 2)*k*x0*y0 + 4*pow(c, 2)*pow(y0, 2)))/(c*(pow(a, 2)*pow(k, 2) - pow(b, 2))));

Equation for line (spherical)

x: $t \sin{\left (\theta \right )} \cos{\left (\phi \right )} + x_{0}$
y: $t \sin{\left (\phi \right )} \sin{\left (\theta \right )} + y_{0}$
z: $t \cos{\left (\theta \right )} + z_{0}$

Equation for hyperbolical_paraboloid (cartesian)

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

0= $- \frac{z}{c} - \frac{y^{2}}{b^{2}} + \frac{x^{2}}{a^{2}}$

Intersection solutions

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

$\frac{1}{c \left(a^{2} \sin^{2}{\left (\phi \right )} - b^{2} \cos^{2}{\left (\phi \right )}\right) \sin^{2}{\left (\theta \right )}} \left(- \frac{a^{2} b^{2}}{2} \cos{\left (\theta \right )} - a^{2} c y_{0} \sin{\left (\phi \right )} \sin{\left (\theta \right )} + b^{2} c x_{0} \sin{\left (\theta \right )} \cos{\left (\phi \right )} - \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} \cos^{2}{\left (\theta \right )} + a^{2} c y_{0} \left(\cos{\left (\phi - 2 \theta \right )} - \cos{\left (\phi + 2 \theta \right )}\right) - 4 a^{2} c z_{0} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - b^{2} c x_{0} \left(- \sin{\left (\phi - 2 \theta \right )} + \sin{\left (\phi + 2 \theta \right )}\right) + 4 b^{2} c z_{0} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )} + 4 c^{2} x_{0}^{2} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - 8 c^{2} x_{0} y_{0} \sin{\left (\phi \right )} \sin^{2}{\left (\theta \right )} \cos{\left (\phi \right )} + 4 c^{2} y_{0}^{2} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )}\right)}\right)$
$\frac{1}{c \left(a^{2} \sin^{2}{\left (\phi \right )} - b^{2} \cos^{2}{\left (\phi \right )}\right) \sin^{2}{\left (\theta \right )}} \left(- \frac{a^{2} b^{2}}{2} \cos{\left (\theta \right )} - a^{2} c y_{0} \sin{\left (\phi \right )} \sin{\left (\theta \right )} + b^{2} c x_{0} \sin{\left (\theta \right )} \cos{\left (\phi \right )} + \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} \cos^{2}{\left (\theta \right )} + a^{2} c y_{0} \left(\cos{\left (\phi - 2 \theta \right )} - \cos{\left (\phi + 2 \theta \right )}\right) - 4 a^{2} c z_{0} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - b^{2} c x_{0} \left(- \sin{\left (\phi - 2 \theta \right )} + \sin{\left (\phi + 2 \theta \right )}\right) + 4 b^{2} c z_{0} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )} + 4 c^{2} x_{0}^{2} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - 8 c^{2} x_{0} y_{0} \sin{\left (\phi \right )} \sin^{2}{\left (\theta \right )} \cos{\left (\phi \right )} + 4 c^{2} y_{0}^{2} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )}\right)}\right)$

Points in cartesian coordinates (x, y, z)

- x: $x_{0} + \frac{\cos{\left (\phi \right )}}{c \left(a^{2} \sin^{2}{\left (\phi \right )} - b^{2} \cos^{2}{\left (\phi \right )}\right) \sin{\left (\theta \right )}} \left(- \frac{a^{2} b^{2}}{2} \cos{\left (\theta \right )} - a^{2} c y_{0} \sin{\left (\phi \right )} \sin{\left (\theta \right )} + b^{2} c x_{0} \sin{\left (\theta \right )} \cos{\left (\phi \right )} - \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} \cos^{2}{\left (\theta \right )} + a^{2} c y_{0} \left(\cos{\left (\phi - 2 \theta \right )} - \cos{\left (\phi + 2 \theta \right )}\right) - 4 a^{2} c z_{0} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - b^{2} c x_{0} \left(- \sin{\left (\phi - 2 \theta \right )} + \sin{\left (\phi + 2 \theta \right )}\right) + 4 b^{2} c z_{0} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )} + 4 c^{2} x_{0}^{2} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - 8 c^{2} x_{0} y_{0} \sin{\left (\phi \right )} \sin^{2}{\left (\theta \right )} \cos{\left (\phi \right )} + 4 c^{2} y_{0}^{2} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )}\right)}\right)$
- y: $y_{0} + \frac{\sin{\left (\phi \right )}}{c \left(a^{2} \sin^{2}{\left (\phi \right )} - b^{2} \cos^{2}{\left (\phi \right )}\right) \sin{\left (\theta \right )}} \left(- \frac{a^{2} b^{2}}{2} \cos{\left (\theta \right )} - a^{2} c y_{0} \sin{\left (\phi \right )} \sin{\left (\theta \right )} + b^{2} c x_{0} \sin{\left (\theta \right )} \cos{\left (\phi \right )} - \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} \cos^{2}{\left (\theta \right )} + a^{2} c y_{0} \left(\cos{\left (\phi - 2 \theta \right )} - \cos{\left (\phi + 2 \theta \right )}\right) - 4 a^{2} c z_{0} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - b^{2} c x_{0} \left(- \sin{\left (\phi - 2 \theta \right )} + \sin{\left (\phi + 2 \theta \right )}\right) + 4 b^{2} c z_{0} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )} + 4 c^{2} x_{0}^{2} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - 8 c^{2} x_{0} y_{0} \sin{\left (\phi \right )} \sin^{2}{\left (\theta \right )} \cos{\left (\phi \right )} + 4 c^{2} y_{0}^{2} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )}\right)}\right)$
- z: $z_{0} + \frac{\cos{\left (\theta \right )}}{c \left(a^{2} \sin^{2}{\left (\phi \right )} - b^{2} \cos^{2}{\left (\phi \right )}\right) \sin^{2}{\left (\theta \right )}} \left(- \frac{a^{2} b^{2}}{2} \cos{\left (\theta \right )} - a^{2} c y_{0} \sin{\left (\phi \right )} \sin{\left (\theta \right )} + b^{2} c x_{0} \sin{\left (\theta \right )} \cos{\left (\phi \right )} - \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} \cos^{2}{\left (\theta \right )} + a^{2} c y_{0} \left(\cos{\left (\phi - 2 \theta \right )} - \cos{\left (\phi + 2 \theta \right )}\right) - 4 a^{2} c z_{0} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - b^{2} c x_{0} \left(- \sin{\left (\phi - 2 \theta \right )} + \sin{\left (\phi + 2 \theta \right )}\right) + 4 b^{2} c z_{0} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )} + 4 c^{2} x_{0}^{2} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - 8 c^{2} x_{0} y_{0} \sin{\left (\phi \right )} \sin^{2}{\left (\theta \right )} \cos{\left (\phi \right )} + 4 c^{2} y_{0}^{2} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )}\right)}\right)$
- x: $x_{0} + \frac{\cos{\left (\phi \right )}}{c \left(a^{2} \sin^{2}{\left (\phi \right )} - b^{2} \cos^{2}{\left (\phi \right )}\right) \sin{\left (\theta \right )}} \left(- \frac{a^{2} b^{2}}{2} \cos{\left (\theta \right )} - a^{2} c y_{0} \sin{\left (\phi \right )} \sin{\left (\theta \right )} + b^{2} c x_{0} \sin{\left (\theta \right )} \cos{\left (\phi \right )} + \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} \cos^{2}{\left (\theta \right )} + a^{2} c y_{0} \left(\cos{\left (\phi - 2 \theta \right )} - \cos{\left (\phi + 2 \theta \right )}\right) - 4 a^{2} c z_{0} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - b^{2} c x_{0} \left(- \sin{\left (\phi - 2 \theta \right )} + \sin{\left (\phi + 2 \theta \right )}\right) + 4 b^{2} c z_{0} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )} + 4 c^{2} x_{0}^{2} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - 8 c^{2} x_{0} y_{0} \sin{\left (\phi \right )} \sin^{2}{\left (\theta \right )} \cos{\left (\phi \right )} + 4 c^{2} y_{0}^{2} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )}\right)}\right)$
- y: $y_{0} + \frac{\sin{\left (\phi \right )}}{c \left(a^{2} \sin^{2}{\left (\phi \right )} - b^{2} \cos^{2}{\left (\phi \right )}\right) \sin{\left (\theta \right )}} \left(- \frac{a^{2} b^{2}}{2} \cos{\left (\theta \right )} - a^{2} c y_{0} \sin{\left (\phi \right )} \sin{\left (\theta \right )} + b^{2} c x_{0} \sin{\left (\theta \right )} \cos{\left (\phi \right )} + \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} \cos^{2}{\left (\theta \right )} + a^{2} c y_{0} \left(\cos{\left (\phi - 2 \theta \right )} - \cos{\left (\phi + 2 \theta \right )}\right) - 4 a^{2} c z_{0} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - b^{2} c x_{0} \left(- \sin{\left (\phi - 2 \theta \right )} + \sin{\left (\phi + 2 \theta \right )}\right) + 4 b^{2} c z_{0} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )} + 4 c^{2} x_{0}^{2} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - 8 c^{2} x_{0} y_{0} \sin{\left (\phi \right )} \sin^{2}{\left (\theta \right )} \cos{\left (\phi \right )} + 4 c^{2} y_{0}^{2} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )}\right)}\right)$
- z: $z_{0} + \frac{\cos{\left (\theta \right )}}{c \left(a^{2} \sin^{2}{\left (\phi \right )} - b^{2} \cos^{2}{\left (\phi \right )}\right) \sin^{2}{\left (\theta \right )}} \left(- \frac{a^{2} b^{2}}{2} \cos{\left (\theta \right )} - a^{2} c y_{0} \sin{\left (\phi \right )} \sin{\left (\theta \right )} + b^{2} c x_{0} \sin{\left (\theta \right )} \cos{\left (\phi \right )} + \frac{1}{2} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} \cos^{2}{\left (\theta \right )} + a^{2} c y_{0} \left(\cos{\left (\phi - 2 \theta \right )} - \cos{\left (\phi + 2 \theta \right )}\right) - 4 a^{2} c z_{0} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - b^{2} c x_{0} \left(- \sin{\left (\phi - 2 \theta \right )} + \sin{\left (\phi + 2 \theta \right )}\right) + 4 b^{2} c z_{0} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )} + 4 c^{2} x_{0}^{2} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - 8 c^{2} x_{0} y_{0} \sin{\left (\phi \right )} \sin^{2}{\left (\theta \right )} \cos{\left (\phi \right )} + 4 c^{2} y_{0}^{2} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )}\right)}\right)$

C Code

x = x0 + (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*cos(theta) - pow(a, 2)*c*y0*sin(phi)*sin(theta) + pow(b, 2)*c*x0*sin(theta)*cos(phi) - 1.0L/2.0L*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(cos(theta), 2) + pow(a, 2)*c*y0*(cos(phi - 2*theta) - cos(phi + 2*theta)) - 4*pow(a, 2)*c*z0*pow(sin(phi), 2)*pow(sin(theta), 2) - pow(b, 2)*c*x0*(-sin(phi - 2*theta) + sin(phi + 2*theta)) + 4*pow(b, 2)*c*z0*pow(sin(theta), 2)*pow(cos(phi), 2) + 4*pow(c, 2)*pow(x0, 2)*pow(sin(phi), 2)*pow(sin(theta), 2) - 8*pow(c, 2)*x0*y0*sin(phi)*pow(sin(theta), 2)*cos(phi) + 4*pow(c, 2)*pow(y0, 2)*pow(sin(theta), 2)*pow(cos(phi), 2))))*cos(phi)/(c*(pow(a, 2)*pow(sin(phi), 2) - pow(b, 2)*pow(cos(phi), 2))*sin(theta));

y = y0 + (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*cos(theta) - pow(a, 2)*c*y0*sin(phi)*sin(theta) + pow(b, 2)*c*x0*sin(theta)*cos(phi) - 1.0L/2.0L*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(cos(theta), 2) + pow(a, 2)*c*y0*(cos(phi - 2*theta) - cos(phi + 2*theta)) - 4*pow(a, 2)*c*z0*pow(sin(phi), 2)*pow(sin(theta), 2) - pow(b, 2)*c*x0*(-sin(phi - 2*theta) + sin(phi + 2*theta)) + 4*pow(b, 2)*c*z0*pow(sin(theta), 2)*pow(cos(phi), 2) + 4*pow(c, 2)*pow(x0, 2)*pow(sin(phi), 2)*pow(sin(theta), 2) - 8*pow(c, 2)*x0*y0*sin(phi)*pow(sin(theta), 2)*cos(phi) + 4*pow(c, 2)*pow(y0, 2)*pow(sin(theta), 2)*pow(cos(phi), 2))))*sin(phi)/(c*(pow(a, 2)*pow(sin(phi), 2) - pow(b, 2)*pow(cos(phi), 2))*sin(theta));

z = z0 + (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*cos(theta) - pow(a, 2)*c*y0*sin(phi)*sin(theta) + pow(b, 2)*c*x0*sin(theta)*cos(phi) - 1.0L/2.0L*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(cos(theta), 2) + pow(a, 2)*c*y0*(cos(phi - 2*theta) - cos(phi + 2*theta)) - 4*pow(a, 2)*c*z0*pow(sin(phi), 2)*pow(sin(theta), 2) - pow(b, 2)*c*x0*(-sin(phi - 2*theta) + sin(phi + 2*theta)) + 4*pow(b, 2)*c*z0*pow(sin(theta), 2)*pow(cos(phi), 2) + 4*pow(c, 2)*pow(x0, 2)*pow(sin(phi), 2)*pow(sin(theta), 2) - 8*pow(c, 2)*x0*y0*sin(phi)*pow(sin(theta), 2)*cos(phi) + 4*pow(c, 2)*pow(y0, 2)*pow(sin(theta), 2)*pow(cos(phi), 2))))*cos(theta)/(c*(pow(a, 2)*pow(sin(phi), 2) - pow(b, 2)*pow(cos(phi), 2))*pow(sin(theta), 2));

x = x0 + (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*cos(theta) - pow(a, 2)*c*y0*sin(phi)*sin(theta) + pow(b, 2)*c*x0*sin(theta)*cos(phi) + (1.0L/2.0L)*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(cos(theta), 2) + pow(a, 2)*c*y0*(cos(phi - 2*theta) - cos(phi + 2*theta)) - 4*pow(a, 2)*c*z0*pow(sin(phi), 2)*pow(sin(theta), 2) - pow(b, 2)*c*x0*(-sin(phi - 2*theta) + sin(phi + 2*theta)) + 4*pow(b, 2)*c*z0*pow(sin(theta), 2)*pow(cos(phi), 2) + 4*pow(c, 2)*pow(x0, 2)*pow(sin(phi), 2)*pow(sin(theta), 2) - 8*pow(c, 2)*x0*y0*sin(phi)*pow(sin(theta), 2)*cos(phi) + 4*pow(c, 2)*pow(y0, 2)*pow(sin(theta), 2)*pow(cos(phi), 2))))*cos(phi)/(c*(pow(a, 2)*pow(sin(phi), 2) - pow(b, 2)*pow(cos(phi), 2))*sin(theta));

y = y0 + (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*cos(theta) - pow(a, 2)*c*y0*sin(phi)*sin(theta) + pow(b, 2)*c*x0*sin(theta)*cos(phi) + (1.0L/2.0L)*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(cos(theta), 2) + pow(a, 2)*c*y0*(cos(phi - 2*theta) - cos(phi + 2*theta)) - 4*pow(a, 2)*c*z0*pow(sin(phi), 2)*pow(sin(theta), 2) - pow(b, 2)*c*x0*(-sin(phi - 2*theta) + sin(phi + 2*theta)) + 4*pow(b, 2)*c*z0*pow(sin(theta), 2)*pow(cos(phi), 2) + 4*pow(c, 2)*pow(x0, 2)*pow(sin(phi), 2)*pow(sin(theta), 2) - 8*pow(c, 2)*x0*y0*sin(phi)*pow(sin(theta), 2)*cos(phi) + 4*pow(c, 2)*pow(y0, 2)*pow(sin(theta), 2)*pow(cos(phi), 2))))*sin(phi)/(c*(pow(a, 2)*pow(sin(phi), 2) - pow(b, 2)*pow(cos(phi), 2))*sin(theta));

z = z0 + (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*cos(theta) - pow(a, 2)*c*y0*sin(phi)*sin(theta) + pow(b, 2)*c*x0*sin(theta)*cos(phi) + (1.0L/2.0L)*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(cos(theta), 2) + pow(a, 2)*c*y0*(cos(phi - 2*theta) - cos(phi + 2*theta)) - 4*pow(a, 2)*c*z0*pow(sin(phi), 2)*pow(sin(theta), 2) - pow(b, 2)*c*x0*(-sin(phi - 2*theta) + sin(phi + 2*theta)) + 4*pow(b, 2)*c*z0*pow(sin(theta), 2)*pow(cos(phi), 2) + 4*pow(c, 2)*pow(x0, 2)*pow(sin(phi), 2)*pow(sin(theta), 2) - 8*pow(c, 2)*x0*y0*sin(phi)*pow(sin(theta), 2)*cos(phi) + 4*pow(c, 2)*pow(y0, 2)*pow(sin(theta), 2)*pow(cos(phi), 2))))*cos(theta)/(c*(pow(a, 2)*pow(sin(phi), 2) - pow(b, 2)*pow(cos(phi), 2))*pow(sin(theta), 2));

Distance inside

Distance between crossing points.

$\left\lvert{\frac{1}{c \left(a^{2} \sin^{2}{\left (\phi \right )} - b^{2} \cos^{2}{\left (\phi \right )}\right) \sin^{2}{\left (\theta \right )}} \sqrt{a^{2} b^{2} \left(a^{2} b^{2} \cos^{2}{\left (\theta \right )} + a^{2} c y_{0} \left(\cos{\left (\phi - 2 \theta \right )} - \cos{\left (\phi + 2 \theta \right )}\right) - 4 a^{2} c z_{0} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} + b^{2} c x_{0} \left(\sin{\left (\phi - 2 \theta \right )} - \sin{\left (\phi + 2 \theta \right )}\right) + 4 b^{2} c z_{0} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )} + 4 c^{2} x_{0}^{2} \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} - 8 c^{2} x_{0} y_{0} \sin{\left (\phi \right )} \sin^{2}{\left (\theta \right )} \cos{\left (\phi \right )} + 4 c^{2} y_{0}^{2} \sin^{2}{\left (\theta \right )} \cos^{2}{\left (\phi \right )}\right)}}\right\rvert$

C Code

sol1 = (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*cos(theta) - pow(a, 2)*c*y0*sin(phi)*sin(theta) + pow(b, 2)*c*x0*sin(theta)*cos(phi) - 1.0L/2.0L*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(cos(theta), 2) + pow(a, 2)*c*y0*(cos(phi - 2*theta) - cos(phi + 2*theta)) - 4*pow(a, 2)*c*z0*pow(sin(phi), 2)*pow(sin(theta), 2) - pow(b, 2)*c*x0*(-sin(phi - 2*theta) + sin(phi + 2*theta)) + 4*pow(b, 2)*c*z0*pow(sin(theta), 2)*pow(cos(phi), 2) + 4*pow(c, 2)*pow(x0, 2)*pow(sin(phi), 2)*pow(sin(theta), 2) - 8*pow(c, 2)*x0*y0*sin(phi)*pow(sin(theta), 2)*cos(phi) + 4*pow(c, 2)*pow(y0, 2)*pow(sin(theta), 2)*pow(cos(phi), 2))))/(c*(pow(a, 2)*pow(sin(phi), 2) - pow(b, 2)*pow(cos(phi), 2))*pow(sin(theta), 2));

sol2 = (-1.0L/2.0L*pow(a, 2)*pow(b, 2)*cos(theta) - pow(a, 2)*c*y0*sin(phi)*sin(theta) + pow(b, 2)*c*x0*sin(theta)*cos(phi) + (1.0L/2.0L)*sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(cos(theta), 2) + pow(a, 2)*c*y0*(cos(phi - 2*theta) - cos(phi + 2*theta)) - 4*pow(a, 2)*c*z0*pow(sin(phi), 2)*pow(sin(theta), 2) - pow(b, 2)*c*x0*(-sin(phi - 2*theta) + sin(phi + 2*theta)) + 4*pow(b, 2)*c*z0*pow(sin(theta), 2)*pow(cos(phi), 2) + 4*pow(c, 2)*pow(x0, 2)*pow(sin(phi), 2)*pow(sin(theta), 2) - 8*pow(c, 2)*x0*y0*sin(phi)*pow(sin(theta), 2)*cos(phi) + 4*pow(c, 2)*pow(y0, 2)*pow(sin(theta), 2)*pow(cos(phi), 2))))/(c*(pow(a, 2)*pow(sin(phi), 2) - pow(b, 2)*pow(cos(phi), 2))*pow(sin(theta), 2));

distance = fabs(sqrt(pow(a, 2)*pow(b, 2)*(pow(a, 2)*pow(b, 2)*pow(cos(theta), 2) + pow(a, 2)*c*y0*(cos(phi - 2*theta) - cos(phi + 2*theta)) - 4*pow(a, 2)*c*z0*pow(sin(phi), 2)*pow(sin(theta), 2) + pow(b, 2)*c*x0*(sin(phi - 2*theta) - sin(phi + 2*theta)) + 4*pow(b, 2)*c*z0*pow(sin(theta), 2)*pow(cos(phi), 2) + 4*pow(c, 2)*pow(x0, 2)*pow(sin(phi), 2)*pow(sin(theta), 2) - 8*pow(c, 2)*x0*y0*sin(phi)*pow(sin(theta), 2)*cos(phi) + 4*pow(c, 2)*pow(y0, 2)*pow(sin(theta), 2)*pow(cos(phi), 2)))/(c*(pow(a, 2)*pow(sin(phi), 2) - pow(b, 2)*pow(cos(phi), 2))*pow(sin(theta), 2)));

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