A plane cutting a cylinder always produces an ellipse:
+ +A cylinder along the x-axis has the equation of a circle, independent of this variable:
+ +\[ y^2 + z^2 = 1 \] + +A vertical plane through the origin has no z-dependence:
+ +\[ n_x x + n_y y = x \cos\phi + y \sin\phi = 0 \] + +Taking y as the independent variable, a parametrization of the intersection is
+ +\[ \left[ -y \tan\phi, y, z \right] = \left[ -y \tan\phi, y, \sqrt{1-y^2} \right] \] + +Squaring and summing the first two coordinates on the left-hand side gives
+ +\[ y^2 (\tan^2\phi + 1) = y^2 \sec^2\phi \] + +so that the parametrization represents the ellipse
+ +\[ \frac{ x^2 + y^2 }{ \sec^2\phi } + z^2 = 1 \] + +For numerical efficiency the variable y is replaced by a circular function in the code.
+ +Complete code for this example:
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