Intersection of a line with a cylinder

The equation describing a right circular cylinder of radius $a$, aligned along the z-axis, is $x^2 + y^2 = a^2$. The intersection of a line with this cylinder will lead to a quadratic in $t$ with solutions of the form:

$$t = \frac{- ((\underline{u}.\underline{R})- wz_0) \pm \sqrt{... ...ne{R})- wz_0)^2 - (1 - w^2)(x_0^2 + y_0^2 - a^2)}} {1 - w^2}$$ (13.2)

This, of course, assumes an infinite length cylinder. If the cylinder is bounded, then a check as to whether the predicted value of the z coordinate lies within the bounds will show whether the intersection found using the above equation is valid. A bounded cylinder also has ends, which also must be checked. In this case the distance $t$ is defined: $t=(z_{plane} - z_0)/w$. Provided $t$ is positive, this can be used to form a proposed solution $(x_0 + tu,y_0+tv, z_0+tw)$ where the condition for intersection is that $(x_0 + tu)^2+(y_0+tv)^2 \le a^2$.