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Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as e. The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section.
A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (a cone with two nappes).It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
Hence, only the curvature lines of the cylinder are of interest: A horizontal plane intersects a cylinder at a circle and a vertical plane has lines with the cylinder in common. The idea of threefold orthogonal systems can be seen as a generalization of orthogonal trajectories. Special examples are systems of confocal conic sections.
It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relationship between points and lines of the plane determined by the conic. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not ...
When the directrix has the property that the angle it subtends from the apex is exactly , then each nappe of the conical surface, including the apex, is a developable surface. [ 8 ] A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction.
A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity e. If 0 < e < 1 the conic is an ellipse, if e = 1 the conic is a parabola, and if e > 1 the conic is a hyperbola.
The equation for a conic section with apex at the origin and tangent to the y axis is + (+) = alternately = + (+) where R is the radius of curvature at x = 0. This formulation is used in geometric optics to specify oblate elliptical ( K > 0 ), spherical ( K = 0 ), prolate elliptical ( 0 > K > −1 ), parabolic ( K = −1 ), and hyperbolic ( K ...
The directrix of a conic section can be found using Dandelin's construction. Each Dandelin sphere intersects the cone at a circle; let both of these circles define their own planes. The intersections of these two parallel planes with the conic section's plane will be two parallel lines; these lines are the directrices of the conic section.