Parametric equation of a point on the cone A generic point on the cone satisfies the equation: \[ \frac{\left\lVert P - Q \right\rVert}{\left\lVert Q - H \right\rVert} = \frac{r}{\left\lVert C - H \right\rVert} \] The vector \( \mathbf{h} \) represents the axis of the cone. \[ \mathbf{h} = \left( C - H \right) \qquad \mathbf{\hat{h}} = \frac{\left( C - H \right)}{\left\lVert C - H \right\rVert} \] Both sides of the equation can be raised by the power of 2, to simplify the calculations. \[ \frac{\left\lVert P - Q \right\rVert^{2}}{\left\lVert Q - H \right\rVert^{2}} = \frac{r^{2}}{\left\lVert \mathbf{h} \right\rVert^{2}} \] Both the height and the base radius of the cone are caracteristic of the geometry and can be substituted by a constant. \[ m = \frac{r^{2}}{\left\lVert \mathbf{h} \right\rVert^{2}} \] \[ \left\lVert P - Q \right\rVert^{2} = m \left\lVert Q - H \right\rVert^{2} \] The segment \( \overline{HQ} \) can be expressed in terms of \( P \) as: \[ \left\lVert Q - H \right\rVert^{2} = \left[ \left( P - H \right) \cdot \mathbf{\hat{h}} \right]^{2} \] and substituted in the original equation. \[ \left\lVert P - Q \right\rVert^{2} = m \left[ \left( P - H \right) \cdot \mathbf{\hat{h}} \right]^{2} \] According to Pithagoras the squared length of \( \overline{QP} \) can be obtained as the result of the squared length of \( \overline{HP} \) minus the squared length of \( \overline{HQ} \): \[ \left\lVert P - Q \right\rVert^{2} = \left\lVert P - H \right\rVert^{2} - \left\lVert Q - H \right\rVert^{2} \] and substituted in the original equation, along with the squared length of the \( \overline{HQ} \) \[ \left\lVert P - H \right\rVert^{2} - \left[ \left( P - H \right) \cdot \mathbf{\hat{h}} \right]^{2} = m \left[ \left( P - H \right) \cdot \mathbf{\hat{h}} \right]^{2} \] 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 cone and line, substitute the generic point \( P \) on the cone with a generic point on the line \( L_{(t)} \). \[ \left\lVert L_{0} + t \mathbf{v} - H \right\rVert^{2} = \left( m + 1 \right) \left[ \left( L_{0} + t \mathbf{v} - H \right) \cdot \mathbf{\hat{h}} \right]^{2} \] Define a vector \( \mathbf{w} \) equal to the difference of \( L_{0} \) and \( H \): \[ \mathbf{w} = L_{0} - H \] and substitute back. \[ \left\lVert t \mathbf{v} + \mathbf{w} \right\rVert^{2} = \left( m + 1 \right) \left[ \left( t \mathbf{v} + \mathbf{w} \right) \cdot \mathbf{\hat{h}} \right]^{2} \] The length squared of a vector is equal to the dot product of the vector by itself. \[ \left( t \mathbf{v} + \mathbf{w} \right) \cdot \left( t \mathbf{v} + \mathbf{w} \right) = \left( m + 1 \right) \left[ \left( \mathbf{v} \cdot \mathbf{\hat{h}} \right) t + \left( \mathbf{w} \cdot \mathbf{\hat{h}} \right) \right]^{2} \] Expand the dot product and the power. \[ \left( \mathbf{v} \cdot \mathbf{v} \right) t^{2} + 2 \left( \mathbf{v} \cdot \mathbf{w} \right) t + \left( \mathbf{w} \cdot \mathbf{w} \right) = \left( m + 1 \right) \left[ \left( \mathbf{v} \cdot \mathbf{\hat{h}} \right)^{2} t^{2} + 2 \left( \mathbf{v} \cdot \mathbf{\hat{h}} \right) \left( \mathbf{w} \cdot \mathbf{\hat{h}} \right) t + \left( \mathbf{w} \cdot \mathbf{\hat{h}} \right)^{2} \right] \] Group by the coefficients of \( t \). \[ \begin{split} \left[ \left( \mathbf{v} \cdot \mathbf{v} \right) - \left( m + 1 \right) \left( \mathbf{v} \cdot \mathbf{\hat{h}} \right)^{2} \right] t^{2} + 2 \left\{ \left( \mathbf{v} \cdot \mathbf{w} \right) - \left( m + 1 \right) \left[ \left( \mathbf{v} \cdot \mathbf{\hat{h}} \right) \left( \mathbf{w} \cdot \mathbf{\hat{h}} \right) \right] \right\} t + \\ + \left( \mathbf{w} \cdot \mathbf{w} \right) - \left( m + 1 \right) \left( \mathbf{w} \cdot \mathbf{\hat{h}} \right)^{2} = 0 \end{split} \] Substitute the coefficients of \( t \) to obtain a simpler quadratic equation. \[ \begin{cases} a = \left( \mathbf{v} \cdot \mathbf{v} \right) - m \left( \mathbf{v} \cdot \mathbf{\hat{h}} \right)^{2} - \left( \mathbf{v} \cdot \mathbf{\hat{h}} \right)^{2} \\ b = 2 \left[ \left( \mathbf{v} \cdot \mathbf{w} \right) - m \left( \mathbf{v} \cdot \mathbf{\hat{h}} \right) \left( \mathbf{w} \cdot \mathbf{\hat{h}} \right) - \left( \mathbf{v} \cdot \mathbf{\hat{h}} \right) \left( \mathbf{w} \cdot \mathbf{\hat{h}} \right) \right] \\ c = \left( \mathbf{w} \cdot \mathbf{w} \right) - m \left( \mathbf{w} \cdot \mathbf{\hat{h}} \right)^{2} - \left( \mathbf{w} \cdot \mathbf{\hat{h}} \right)^{2} \end{cases} \] Solve for \( t \). \[ a t^{2} + b t + c = 0 \] Intersection points

• if \( \left( b^{2} - 4 a c \right) > 0 \)
  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 line \( L_{ \left( t \right) } \) is tangent to the cone. \[ \left\lVert \mathbf{h} \right\rVert = cos \left( \alpha \right) \sqrt{\left\lVert \mathbf{h} \right\rVert^{2} + r^{2} } \qquad cos \left( \alpha \right) = \frac{\left\lVert \mathbf{h} \right\rVert}{\sqrt{\left\lVert \mathbf{h} \right\rVert^{2} + r^{2} }} \]
• if \( \left| \mathbf{\hat{v}} \cdot \mathbf{\hat{h}} \right| \neq cos \left( \alpha \right) \)
  There is one intersection at: \[ L_{int} = L_{0} + t \mathbf{v} \qquad t = - \frac{ b }{ 2 a } \]


• if \( \left| \mathbf{\hat{v}} \cdot \mathbf{\hat{h}} \right| = cos \left( \alpha \right) \)
  The line \( L_{ \left( t \right) } \) lies on the cone surface. There are infinite intersections.


• if \( \left( b^{2} - 4 a c \right) < 0 \)
  There are no intersections.

If any intersections are found and the cone is not infinite, the projection of the intersections on the cone axis must be between the two extremities \( H \) and \( C \).

• if \( 0 \leq \left( L_{int} - H \right) \cdot \mathbf{\hat{h}} \leq \left\lVert \mathbf{h} \right\rVert \)
  The intersection is on the cone surface.


• if \( \left( L_{int} - H \right) \cdot \mathbf{\hat{h}} > \left\lVert \mathbf{h} \right\rVert \)
  The intersection is below the base of the cone. Test the intersection with the cone base cap.


• if \( \left( L_{int} - H \right) \cdot \mathbf{\hat{h}} < 0 \)
  The intersection is above the top of the cone.