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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.
Sprott [43] found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel [ 44 ] [ 45 ] showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side ...
A 3D cell culture is an artificially created environment in which biological cells are permitted to grow or interact with their surroundings in all three dimensions. Unlike 2D environments (e.g. a Petri dish), a 3D cell culture allows cells in vitro to grow in all directions, similar to how they would in vivo. [1]
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 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.
Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths , bond angles , torsional angles and any other geometrical parameters that determine the position of each atom.
A pseudo-Euclidean space is an affine space with an associated real vector space equipped with a non-degenerate quadratic form (that may be indefinite). A fundamental example of such a space is the Minkowski space, which is the space-time of Einstein's special relativity. It is a four-dimensional space, where the metric is defined by the ...
Given a metric tensor g on an n-dimensional real manifold, the quadratic form q(x) = g(x, x) associated with the metric tensor applied to each vector of any orthogonal basis produces n real values. By Sylvester's law of inertia , the number of each positive, negative and zero values produced in this manner are invariants of the metric tensor ...