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Using homogeneous coordinates, a non-zero quadratic form in n variables defines an (n − 2)-dimensional quadric in the (n − 1)-dimensional projective space. This is a basic construction in projective geometry. In this way one may visualize 3-dimensional real quadratic forms as conic sections.
(A homogeneous polynomial is also called a form, and so q may be called a quadratic form.) If q is the product of two linear forms, then X is the union of two hyperplanes . It is common to assume that n ≥ 1 {\displaystyle n\geq 1} and q is irreducible , which excludes that special case.
In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hypersurface (of dimension D) embedded in a higher dimensional space (of dimension D + 1) is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D=1 is the case of conic sections ...
and the second fundamental form at the origin in the coordinates (x,y) is the quadratic form L d x 2 + 2 M d x d y + N d y 2 . {\displaystyle L\,dx^{2}+2M\,dx\,dy+N\,dy^{2}\,.} For a smooth point P on S , one can choose the coordinate system so that the plane z = 0 is tangent to S at P , and define the second fundamental form in the same way.
In mathematics, a quadratic equation (from Latin quadratus ' square ') is an equation that can be rearranged in standard form as [1] + + =, where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.)
A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K.The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition [c] = , where the product on the left is that of the algebra, and the 1 on the right is the algebra's ...
In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite .
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