Ellipse As plane sections of quadrics
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the definition of the focal curves of confocal quadrics. See § Confocal quadrics below. Considering the pencils of confocal ellipses and hyperbolas (see...
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal...
hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek...- mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and...
Ellipsoid (redirect from Ellipsoid plane section)Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded...
Intersection (geometry) (redirect from Plane–sphere intersection)Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic...
cutting planes are perpendicular to a symmetry axis. In more generality, the plane sections of a quadric are conic sections. A cross-section of a solid...
Cylinder (redirect from Volume of a cylinder)element. The right sections are circles and all other planes intersect the cylindrical surface in an ellipse. If a plane intersects a base of the cylinder in...
Paraboloid (redirect from Paraboloid of revolution)elliptic if every other nonempty plane section is either an ellipse, or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic...
Eccentricity (mathematics) (category Conic sections)eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane). But: conic sections may occur...
Parabola (redirect from Derivations of Conic Sections)following quadrics contain parabolas as plane sections: elliptical cone, parabolic cylinder, elliptical paraboloid, hyperbolic paraboloid, hyperboloid of one...
mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental...
Algebraic curve (redirect from Sextic plane curve)algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous...
Elliptic curve (redirect from Discriminant of an elliptic curve)there is a natural representation of real elliptic curves with shape invariant j ≥ 1 as ellipses in the hyperbolic plane H 2 {\displaystyle \mathbb {H} ^{2}}...
Hyperboloid (redirect from Hyperboloid of one sheet)in an ellipse, A plane with a slope equal to 1 containing the origin intersects H 1 {\displaystyle H_{1}} in a pair of parallel lines, A plane with a...- Analytic geometry (redirect from History of analytic geometry)ISBN 0-471-75715-2, Section 3.2, page 45 Silvio Levy Quadrics in "Geometry Formulas and Facts", excerpted from 30th Edition of CRC Standard Mathematical...
other quadrics, such as tri-axial ellipsoids, elliptic cylinders, etc. Nevertheless, it is true that: Any quadric surface which contains ellipses contains...
Cone (redirect from Volume of a cone)image of a conic section is a conic section of the same type (ellipse, parabola,...), one gets: Any plane section of an elliptic cone is a conic section. Obviously...
directrix C {\displaystyle C} is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of C {\displaystyle C} , one obtains...- Stodola's cone law (redirect from Law of the Ellipse)parameters as represented in the Cartesian coordinate system has the shape of a degenerate quadric surface, the cone directrix being an ellipse. For a constant...
- Encyclopædia Britannica, Volume 9 — Ellipsoid ELLIPSOID, a quadric surface whose sections are ellipses. Analytically, it has for its equation x2/a2 + y2/b2
- rectangular normal sections at P, and the spherical surface be bent slightly about x so as to alter the curvature of the section in the normal plane at right angles
- like an ellipse on the x y {\displaystyle xy} -plane but infinitely stretched in the z {\displaystyle z} direction. General equation for quadric surfaces