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In affine geometry, uniform scaling (or isotropic scaling[1]) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions (isotropically). The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so ...
Conversely, it is easily shown that if a, b, c, and d are constants and a, b, and c are not all zero, then the graph of the equation + + + =, is a plane having the vector n = (a, b, c) as a normal. [5] This familiar equation for a plane is called the general form of the equation of the plane or just the plane equation. [6]
Scale factor (cosmology) The expansion of the universe is parametrized by a dimensionless scale factor . Also known as the cosmic scale factor or sometimes the Robertson–Walker scale factor, [1] this is a key parameter of the Friedmann equations. In the early stages of the Big Bang, most of the energy was in the form of radiation, and that ...
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. [9] Such a drawing is called a plane graph or planar embedding of the graph.
This implies that the vertical scale factor, h, equals the horizontal scale factor, k. Since k = sec φ, so must h. The graph shows the variation of this scale factor with latitude. Some numerical values are listed below. at latitude 30° the scale factor is k = sec 30° = 1.15, at latitude 45° the scale factor is k = sec 45° = 1.41,
Plane section of the unit sphere (see example) Solution: The scaling u = x / a , v = y / b , w = z / c transforms the ellipsoid onto the unit sphere u 2 + v 2 + w 2 = 1 and the given plane onto the plane with equation + + =. Let m u u + m v v + m w w = δ be the Hesse normal form of the new plane and
Scale bar: 5 μm. In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or surface is contained in a larger space, curvature can be defined extrinsically relative to ...