Transform matrix from unit radius and height cylinder based in the origin to non uniformly scaled cylinder A simpler way to calculate the intersection of a line and a cylinder arbitrarily scaled in space is to build the matrix that transforms from the scaled cylinder to a unit radius and height cylinder based in the origin. Apply the transformation to the line, calculate the intersection with a unit radius and height cylinder, then apply the results to the original line and obtain the intersection points if any are found.

The transform matrix is the result of the composition of the matrices that describe translation, rotation and scale of the scaled cylinder. Its inverse maps the non uniform scaled space to the uniform one. \[ \mathbf{T} = \begin{bmatrix} 1 & 0 & 0 & C_{x} \\ 0 & 1 & 0 & C_{y} \\ 0 & 0 & 1 & C_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix} \qquad \mathbf{R} = \begin{bmatrix} \hat{a}_{x} & \hat{b}_{x} & \hat{c}_{x} & 0 \\ \hat{a}_{y} & \hat{b}_{y} & \hat{c}_{y} & 0 \\ \hat{a}_{z} & \hat{b}_{z} & \hat{c}_{z} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \qquad \mathbf{S} = \begin{bmatrix} \left\lVert \mathbf{a} \right\rVert & 0 & 0 & 0 \\ 0 & \left\lVert \mathbf{b} \right\rVert & 0 & 0 \\ 0 & 0 & \left\lVert \mathbf{c} \right\rVert & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \] \[ \mathbf{M} = \mathbf{TRS} \] Point on unit radius and height cylinder based in the origin A generic point on the cylinder satisfies the equation: \[ \left\lVert P - C \right\rVert = \sqrt{\left\lVert Q - C' \right\rVert^{2} + r^{2}} \] Where \( Q \) is the projection of \( P \) on the cylinder axis.

Both sides of the equation can be raised by the power of 2, to simplify the calculations. \[ \left\lVert P - C \right\rVert^{2} = \left\lVert Q - C' \right\rVert^{2} + r^{2} \] The rescaled cylinder base center \( C' \) is in the origin and its top center \( H' \) is one unit farther along Z axis. The radius is equal to 1: \[ C' = \mathbf{M}^{-1} C = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \qquad H' = \mathbf{M}^{-1} H = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix} \qquad r = 1 \] The height vector \( \mathbf{h} \) is defined by the difference of \( H' \) and \( C' \). \[ \mathbf{h} = H' - C' = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \] The segment \( \overline{CQ} \) can be expressed in terms of \( P \) as: \[ \left\lVert Q - C' \right\rVert = \left( P - C' \right) \cdot \mathbf{h} \] and substituted back. \[ \left\lVert P - C' \right\rVert^{2} - \left[ \left( P - C' \right) \cdot \mathbf{h} \right]^{2} = 1 \] Parametric 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 \( t \). \[ L_{\left( t \right)} = L_{0} + t \mathbf{v} \] To perform the intersection with the unit radius and height cylinder based in the origin, the point \( L_{0} \) and the vector \( \mathbf{v} \) are transformed by the matrix \( \mathbf{M}^{-1} \). \[ L'_{\left( t \right)} = \mathbf{M}^{-1} L_{0} + \mathbf{M}^{-1} t \mathbf{v} \] \[ L'_{0} = \mathbf{M}^{-1} L_{0} \qquad \mathbf{v}' = \mathbf{M}^{-1} \mathbf{v} \] \[ L'_{\left( t \right)} = L'_{0} + t \mathbf{v}' \] Intersections calculation To calculate the intersection between cylinder and line, substitute the generic point \( P \) on the cylinder with a generic point on the line \( L'_{(t)} \). \[ \left\lVert L'_{0} + t \mathbf{v}' - C' \right\rVert^{2} - \left[ \left( L'_{0} + t \mathbf{v}' - C' \right) \cdot \mathbf{h} \right]^{2} = 1 \] Define a vector \( \mathbf{w} \) equal to the difference of \( L'_{0} \) and \( C' \): \[ \mathbf{w} = L'_{0} - C' \] and substitute back. \[ \left\lVert t \mathbf{v}' + \mathbf{w} \right\rVert^{2} - \left[ \left( t \mathbf{v}' + \mathbf{w} \right) \cdot \mathbf{h} \right]^{2} = 1 \] The length squared of a vector is equal to the dot product of the vector by itself. \[ \left( t \mathbf{v}' + \mathbf{w} \right) \cdot \left( t \mathbf{v}' + \mathbf{w} \right) - \left[ \left( \mathbf{v}' \cdot \mathbf{h} \right) t + \left( \mathbf{w} \cdot \mathbf{h} \right) \right]^{2} = 1 \] Expand the dot product and the power. \[ \left( \mathbf{v}' \cdot \mathbf{v}' \right) t^{2} + 2 \left( \mathbf{v}' \cdot \mathbf{w} \right) t + \left( \mathbf{w} \cdot \mathbf{w} \right) - \left( \mathbf{v}' \cdot \mathbf{h} \right)^{2} t^{2} - 2 \left( \mathbf{v}' \cdot \mathbf{h} \right) \left( \mathbf{w} \cdot \mathbf{h} \right) t - \left( \mathbf{w} \cdot \mathbf{h} \right)^{2} = 1 \] Group by the coefficients of \( t \). \[ \left[ \left( \mathbf{v}' \cdot \mathbf{v}' \right) - \left( \mathbf{v}' \cdot \mathbf{h} \right)^{2} \right] t^{2} + 2 \left[ \left( \mathbf{v}' \cdot \mathbf{w} \right) - \left( \mathbf{v}' \cdot \mathbf{h} \right) \left( \mathbf{w} \cdot \mathbf{h} \right) \right] t + \left( \mathbf{w} \cdot \mathbf{w} \right) - \left( \mathbf{w} \cdot \mathbf{h} \right)^{2} - 1 = 0 \] Substitute the coefficients of \( t \) to obtain a simpler quadratic equation. \[ \begin{cases} a = \left( \mathbf{v}' \cdot \mathbf{v}' \right) - \left( \mathbf{v}' \cdot \mathbf{h} \right)^{2} \\ b = 2 \left[ \left( \mathbf{v}' \cdot \mathbf{w} \right) - \left( \mathbf{v}' \cdot \mathbf{h} \right) \left( \mathbf{w} \cdot \mathbf{h} \right) \right] \\ c = \left( \mathbf{w} \cdot \mathbf{w} \right) - \left( \mathbf{w} \cdot \mathbf{h} \right)^{2} - 1 \end{cases} \] Solve for \( t \). \[ a t^{2} + b t + c = 0 \] Intersection points

• if \( \left( b^{2} - 4 a c \right) > 0 \)
  The distance between the cylinder axis \( \overline{CH} \) and the line \( L_{ \left( t \right) } \) is smaller than the cylinder 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 cylinder axis \( \overline{CH} \) and the line \( L_{ \left( t \right) } \) is equal to the cylinder radius \( r \). The line \( L_{ \left( t \right) } \) is tangent to the cylinder.


• if \( \left| \mathbf{\hat{v}} \cdot \mathbf{\hat{h}} \right| \neq 1 \)
  There is one intersection at: \[ L_{int} = L_{0} + t \mathbf{v} \qquad t = - \frac{ b }{ 2 a } \]
• if \( \left| \mathbf{\hat{v}} \cdot \mathbf{\hat{h}} \right| = 1 \)
  The line \( L_{ \left( t \right) } \) lies on the cylinder surface. There are infinite intersections.


• if \( \left( b^{2} - 4 a c \right) < 0 \)
  The distance between the cylinder axis \( \overline{CH} \) and the line \( L_{ \left( t \right) } \) is greater than the cylinder radius \( r \). There are no intersections.

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

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


• if \( \left( L_{int} - C \right) \cdot \mathbf{h} < 0 \)
  The intersection is below the base of the cylinder. Test the intersection with the base cap.


• if \( \left( L_{int} - C \right) \cdot \mathbf{h} > \left\lVert \mathbf{h} \right\rVert \)
  The intersection is above the top of the cylinder. Test the intersection with the top cap.