Parametric equation of a point on the sphere A generic point on the sphere satisfies the equation: \[ \left\lVert P - C \right\rVert = r \] Both sides of the equation can be raised by the power of 2, to simplify the calculations. \[ \left\lVert P - C \right\rVert^{2} = r^{2} \] Parametric equation of the line A generic line defined by a point \( L_{0} \) and a direction vector \( \mathbf{v} \). It describes all the points along the line as a function of the parameter \( t \). \[ L_{\left( t \right)} = L_{0} + t \mathbf{v} \] Intersections calculation To calculate the intersection between sphere and line, substitute the generic point \( P \) on the sphere with a generic point on the line \( L_{ \left( t \right) } \). \[ \left\lVert L_{0} + t \mathbf{v} - C \right\rVert^{2} = r^{2} \] Define a vector \( \mathbf{w} \) equal to the difference of \( L_{0} \) and \( C \): \[ \mathbf{w} = L_{0} - C \] and substitute back. \[ \left\lVert t \mathbf{v} + \mathbf{w} \right\rVert^{2} = r^{2} \] The length squared of a vector is equal to the dot product of the vector by itself. \[ \left( t \mathbf{v} + \mathbf{w} \right) \cdot \left( t \mathbf{v} + \mathbf{w} \right) = r^{2} \] Expand the dot product. \[ \left( \mathbf{v} \cdot \mathbf{v} \right) t^{2} + 2 \left( \mathbf{v} \cdot \mathbf{w} \right) t + \left( \mathbf{w} \cdot \mathbf{w} \right) = r^{2} \] Substitute the coefficients of \( t \) to obtain a simpler quadratic equation. \[ \begin{cases} a = \left( \mathbf{v} \cdot \mathbf{v} \right) \\ b = 2 \left( \mathbf{v} \cdot \mathbf{w} \right) \\ c = \left( \mathbf{w} \cdot \mathbf{w} \right) - r^{2} \end{cases} \] Solve for \( t \). \[ a t^{2} + b t + c = 0 \] Intersection points Depending on the values of the discriminant of the quadratic equation in \( t \) there can be two, one or no intersections:

• if \( \left( b^{2} - 4 a c \right) > 0 \)
  The distance between the sphere center \( C \) and the line \( L_{ \left( t \right) } \) is smaller than the sphere radius \( r \). There are two intersections at: \[ L_{int_{1,2}} = L_{0} + t_{1,2} \mathbf{v} \qquad t_{1,2} = \frac{ - b \pm \sqrt{ b^{2} - 4 a c } }{ 2 a } \]
• if \( \left( b^{2} - 4 a c \right) = 0 \)
  The distance between the sphere center \( C \) and the line \( L_{ \left( t \right) } \) is equal to the sphere radius \( r \). The line \( L_{ \left( t \right) } \) is tangent to the sphere. There is one intersection at: \[ L_{int} = L_{0} + t \mathbf{v} \qquad t = \frac{ -b }{ 2 a } \]
• if \( \left( b^{2} - 4 a c \right) < 0 \)
  The distance between the sphere center \( C \) and the line \( L_{ \left( t \right) } \) is greater than the sphere radius \( r \). There are no intersections.