\( C \) = Point at the center of the base of the cone
\( H \) = Point at the tip of the cone
\( r \) = Cone base radius
\( P \) = Point on the cone surface

\( L_{0} \) = Point on the line
\( \mathbf{v} \) = Vector that defines the line direction
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 length squared of \( \overline{HP} \) minus the length squared 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 wth 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 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 \( C \).