Planes can intersect cones to make plane curves {conic section}|. Plane can be parallel to base and intersect cone at right angle to axis {circle, cone}. Plane can intersect cone at angle less than vertex angle {ellipse, cone}. Plane can intersect cone at angle equal to vertex angle and parallel to element {parabola, cone}. Plane can intersect cone parallel to axis {hyperbola, cone}. Plane can intersect cone so plane includes vertex and bisector {intersecting lines}. Plane can be tangent to cone {tangent line, cone}.
slope
Circles and ellipses are closed curves and have same slope at diameter ends. Parabolas are not closed curves and approach maximum slope as they go farther from axis. Hyperbolas are not closed curves and approach maximum slope as they go farther from axis.
pole
Two tangents to conic can meet at point {pole, cone}.
conic points
Two conics intersect at four points. Two real conics that do not intersect share two imaginary chords.
generation: point and curve
For conic sections, line goes through fixed point {generator, cone} and closed curve {directrix, cone}.
generation: lines
Conics can be line series and pencils, as can ruled quadrics.
Mathematical Sciences>Geometry>Solid>Cone>Conic Section
3-Geometry-Solid-Cone-Conic Section
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Date Modified: 2022.0224