判别式是水平曲线的投影 圆柱为什么轴方向投影是圆？

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Introduction to Computing with Geometry page 138

The discriminants of a polynomial are the projections of its horizon curves in some direction. They are found by taking the resultant of the polynomial with its own derivative. The discriminant of a cylinder would be two straight lines (or a circle, if the projection was along the axis); that of a sphere would be a circle in any direction, and so on.

圆柱$x^2+y^2=r^2$的判别式是两条直线，圆柱为什么轴方向投影是圆？
例如$x$方向投影的判别式是两条直线$y=\pm r$，因为Discriminant[x^2+y^2-r^2,x]$=-4 (y^2 - r^2)$

球$x^2+y^2+z^2=r^2$从任意方向的的判别式是圆。
例如$x$方向投影的判别式是$-4(y^2+z^2-r^2)$