2d_line - circle intersection -- Intersection formulas for 3d objects

2d_line - circle intersection

Equation for 2d_line (cartesian)

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

Equation for circle (cartesian)

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

0= $-1 + \frac{x^{2}}{R^{2}} + \frac{y^{2}}{R^{2}}$

Intersection solutions

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

$\frac{1}{k^{2} + 1} \left(- k y_{0} - x_{0} + \sqrt{R^{2} k^{2} + R^{2} - k^{2} x_{0}^{2} + 2 k x_{0} y_{0} - y_{0}^{2}}\right)$
$- \frac{1}{k^{2} + 1} \left(k y_{0} + x_{0} + \sqrt{R^{2} k^{2} + R^{2} - k^{2} x_{0}^{2} + 2 k x_{0} y_{0} - y_{0}^{2}}\right)$

Points in cartesian coordinates (x, y, z)

- x: $x_{0} + \frac{1}{k^{2} + 1} \left(- k y_{0} - x_{0} + \sqrt{R^{2} k^{2} + R^{2} - k^{2} x_{0}^{2} + 2 k x_{0} y_{0} - y_{0}^{2}}\right)$
- y: $\frac{k}{k^{2} + 1} \left(- k y_{0} - x_{0} + \sqrt{R^{2} k^{2} + R^{2} - k^{2} x_{0}^{2} + 2 k x_{0} y_{0} - y_{0}^{2}}\right) + y_{0}$
- x: $x_{0} - \frac{1}{k^{2} + 1} \left(k y_{0} + x_{0} + \sqrt{R^{2} k^{2} + R^{2} - k^{2} x_{0}^{2} + 2 k x_{0} y_{0} - y_{0}^{2}}\right)$
- y: $- \frac{k}{k^{2} + 1} \left(k y_{0} + x_{0} + \sqrt{R^{2} k^{2} + R^{2} - k^{2} x_{0}^{2} + 2 k x_{0} y_{0} - y_{0}^{2}}\right) + y_{0}$

C Code

x = x0 + (-k*y0 - x0 + sqrt(pow(R, 2)*pow(k, 2) + pow(R, 2) - pow(k, 2)*pow(x0, 2) + 2*k*x0*y0 - pow(y0, 2)))/(pow(k, 2) + 1);

y = k*(-k*y0 - x0 + sqrt(pow(R, 2)*pow(k, 2) + pow(R, 2) - pow(k, 2)*pow(x0, 2) + 2*k*x0*y0 - pow(y0, 2)))/(pow(k, 2) + 1) + y0;

x = x0 - (k*y0 + x0 + sqrt(pow(R, 2)*pow(k, 2) + pow(R, 2) - pow(k, 2)*pow(x0, 2) + 2*k*x0*y0 - pow(y0, 2)))/(pow(k, 2) + 1);

y = -k*(k*y0 + x0 + sqrt(pow(R, 2)*pow(k, 2) + pow(R, 2) - pow(k, 2)*pow(x0, 2) + 2*k*x0*y0 - pow(y0, 2)))/(pow(k, 2) + 1) + y0;

Distance inside

Distance between crossing points.

$2 \left\lvert{\frac{1}{k^{2} + 1} \sqrt{R^{2} k^{2} + R^{2} - k^{2} x_{0}^{2} + 2 k x_{0} y_{0} - y_{0}^{2}}}\right\rvert$

C Code

sol1 = (-k*y0 - x0 + sqrt(pow(R, 2)*pow(k, 2) + pow(R, 2) - pow(k, 2)*pow(x0, 2) + 2*k*x0*y0 - pow(y0, 2)))/(pow(k, 2) + 1);

sol2 = -(k*y0 + x0 + sqrt(pow(R, 2)*pow(k, 2) + pow(R, 2) - pow(k, 2)*pow(x0, 2) + 2*k*x0*y0 - pow(y0, 2)))/(pow(k, 2) + 1);

distance = 2*fabs(sqrt(pow(R, 2)*pow(k, 2) + pow(R, 2) - pow(k, 2)*pow(x0, 2) + 2*k*x0*y0 - pow(y0, 2))/(pow(k, 2) + 1));

Equation for 2d_line (spherical)

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

Equation for circle (cartesian)

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

0= $-1 + \frac{x^{2}}{R^{2}} + \frac{y^{2}}{R^{2}}$

Intersection solutions

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

$- x_{0} \cos{\left (\phi \right )} - y_{0} \sin{\left (\phi \right )} - \sqrt{R^{2} + \frac{x_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{x_{0}^{2}}{2} + x_{0} y_{0} \sin{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2}}$
$- x_{0} \cos{\left (\phi \right )} - y_{0} \sin{\left (\phi \right )} + \sqrt{R^{2} + \frac{x_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{x_{0}^{2}}{2} + x_{0} y_{0} \sin{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2}}$

Points in cartesian coordinates (x, y, z)

- x: $x_{0} + \left(- x_{0} \cos{\left (\phi \right )} - y_{0} \sin{\left (\phi \right )} - \sqrt{R^{2} + \frac{x_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{x_{0}^{2}}{2} + x_{0} y_{0} \sin{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2}}\right) \cos{\left (\phi \right )}$
- y: $y_{0} + \left(- x_{0} \cos{\left (\phi \right )} - y_{0} \sin{\left (\phi \right )} - \sqrt{R^{2} + \frac{x_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{x_{0}^{2}}{2} + x_{0} y_{0} \sin{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2}}\right) \sin{\left (\phi \right )}$
- x: $x_{0} + \left(- x_{0} \cos{\left (\phi \right )} - y_{0} \sin{\left (\phi \right )} + \sqrt{R^{2} + \frac{x_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{x_{0}^{2}}{2} + x_{0} y_{0} \sin{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2}}\right) \cos{\left (\phi \right )}$
- y: $y_{0} + \left(- x_{0} \cos{\left (\phi \right )} - y_{0} \sin{\left (\phi \right )} + \sqrt{R^{2} + \frac{x_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{x_{0}^{2}}{2} + x_{0} y_{0} \sin{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2}}\right) \sin{\left (\phi \right )}$

C Code

x = x0 + (-x0*cos(phi) - y0*sin(phi) - sqrt(pow(R, 2) + (1.0L/2.0L)*pow(x0, 2)*cos(2*phi) - 1.0L/2.0L*pow(x0, 2) + x0*y0*sin(2*phi) - 1.0L/2.0L*pow(y0, 2)*cos(2*phi) - 1.0L/2.0L*pow(y0, 2)))*cos(phi);

y = y0 + (-x0*cos(phi) - y0*sin(phi) - sqrt(pow(R, 2) + (1.0L/2.0L)*pow(x0, 2)*cos(2*phi) - 1.0L/2.0L*pow(x0, 2) + x0*y0*sin(2*phi) - 1.0L/2.0L*pow(y0, 2)*cos(2*phi) - 1.0L/2.0L*pow(y0, 2)))*sin(phi);

x = x0 + (-x0*cos(phi) - y0*sin(phi) + sqrt(pow(R, 2) + (1.0L/2.0L)*pow(x0, 2)*cos(2*phi) - 1.0L/2.0L*pow(x0, 2) + x0*y0*sin(2*phi) - 1.0L/2.0L*pow(y0, 2)*cos(2*phi) - 1.0L/2.0L*pow(y0, 2)))*cos(phi);

y = y0 + (-x0*cos(phi) - y0*sin(phi) + sqrt(pow(R, 2) + (1.0L/2.0L)*pow(x0, 2)*cos(2*phi) - 1.0L/2.0L*pow(x0, 2) + x0*y0*sin(2*phi) - 1.0L/2.0L*pow(y0, 2)*cos(2*phi) - 1.0L/2.0L*pow(y0, 2)))*sin(phi);

Distance inside

Distance between crossing points.

$2 \left\lvert{\sqrt{R^{2} + \frac{x_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{x_{0}^{2}}{2} + x_{0} y_{0} \sin{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2} \cos{\left (2 \phi \right )} - \frac{y_{0}^{2}}{2}}}\right\rvert$

C Code

sol1 = -x0*cos(phi) - y0*sin(phi) - sqrt(pow(R, 2) + (1.0L/2.0L)*pow(x0, 2)*cos(2*phi) - 1.0L/2.0L*pow(x0, 2) + x0*y0*sin(2*phi) - 1.0L/2.0L*pow(y0, 2)*cos(2*phi) - 1.0L/2.0L*pow(y0, 2));

sol2 = -x0*cos(phi) - y0*sin(phi) + sqrt(pow(R, 2) + (1.0L/2.0L)*pow(x0, 2)*cos(2*phi) - 1.0L/2.0L*pow(x0, 2) + x0*y0*sin(2*phi) - 1.0L/2.0L*pow(y0, 2)*cos(2*phi) - 1.0L/2.0L*pow(y0, 2));

distance = 2*fabs(sqrt(pow(R, 2) + (1.0L/2.0L)*pow(x0, 2)*cos(2*phi) - 1.0L/2.0L*pow(x0, 2) + x0*y0*sin(2*phi) - 1.0L/2.0L*pow(y0, 2)*cos(2*phi) - 1.0L/2.0L*pow(y0, 2)));

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