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The two families of lines on a smooth (split) quadric surface. In 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 examples in algebraic geometry. The theory is simplified by working in projective space rather than affine ...
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections.
Attempting to rotate a QGA quadric surface may result in a different type of quadric surface, or a quadric surface that is rotated and distorted in an unexpected way. Attempting to rotate a QGA point may produce a value that projects as the expected rotated vector, but the produced value is generally not a correct embedding of the rotated vector.
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In geometry, a circular section is a circle on a quadric surface (such as an ellipsoid or hyperboloid). It is a special plane section of the quadric, as this circle is the intersection with the quadric of the plane containing the circle. Any plane section of a sphere is a circular section, if it contains at least 2 points.
In the second case (−1 in the right-hand side of the equation): a two-sheet hyperboloid, also called an elliptic hyperboloid. The surface has two connected components and a positive Gaussian curvature at every point. The surface is convex in the sense that the tangent plane at every point intersects the surface only in this point.
For example, a hyperboloid of one sheet is a quadric surface in ruled by two different families of lines, one line of each passing through each point of the surface; each family corresponds under the Plücker map to a conic section within the Klein quadric in .
Often, a surface is defined by equations that are satisfied by the coordinates of its points. This is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, which is called an implicit surface. [1]