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A family of conic sections of varying eccentricity share a focus point and directrix line, including an ellipse (red, e = 1/2), a parabola (green, e = 1), and a hyperbola (blue, e = 2). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity ∞ is an infinitesimally separated ...
Define b by the equations c 2 = a 2 − b 2 for an ellipse and c 2 = a 2 + b 2 for a hyperbola. For a circle, c = 0 so a 2 = b 2, with radius r = a = b. For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. In standard form the parabola will always pass through the ...
The case H = 0 can be eliminated as degenerate, so the tangential equation of C can be written as P + fQ = 0 where f is an arbitrary polynomial of degree 2m. [1] For example, let m = 2, P 1 = (1, 0), and P 2 = (−1, 0). The tangential equations are + = = so P = X 2 − 1 = 0. The tangential equations for the circular points at infinity are
If the directrix is a circle , and the apex is located on the circle's axis (the line that contains the center of and is perpendicular to its plane), one obtains the right circular conical surface or double cone. [2] More generally, when the directrix is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of ...
(1) All rulings are parallel to a plane, the directrix plane. (2) All rulings intersect a fixed line, the axis. The conoid is a right conoid if its axis is perpendicular to its directrix plane. Hence all rulings are perpendicular to the axis. Because of (1) any conoid is a Catalan surface and can be represented parametrically by
In algebraic geometry, ruled surfaces were originally defined as projective surfaces in projective space containing a straight line through any given point. This immediately implies that there is a projective line on the surface through any given point, and this condition is now often used as the definition of a ruled surface: ruled surfaces ...
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A cone can be generated by moving a line (the generatrix) fixed at the future apex of the cone along a closed curve (the directrix); if that directrix is a circle perpendicular to the line connecting its center to the apex, the motion is rotation around a fixed axis and the resulting shape is a circular cone. [3]