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In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measurement (such as metres and grams) and tracking these dimensions as calculations or comparisons are performed.
Aside from the Orthographic, six standard principal views (Front; Right Side; Left Side; Top; Bottom; Rear), descriptive geometry strives to yield four basic solution views: the true length of a line (i.e., full size, not foreshortened), the point view (end view) of a line, the true shape of a plane (i.e., full size to scale, or not ...
Scale analysis rules as follows: Rule1-First step in scale analysis is to define the domain of extent in which we apply scale analysis. Any scale analysis of a flow region that is not uniquely defined is not valid. Rule2-One equation constitutes an equivalence between the scales of two dominant terms appearing in the equation. For example,
Planck length; typical scale of hypothetical loop quantum gravity or size of a hypothetical string and of branes; according to string theory, lengths smaller than this do not make any physical sense. [1] Quantum foam is thought to exist at this scale. 10 −24: 1 yoctometer 142 ym Effective cross section radius of 1 MeV neutrinos [2] 10 −21
Measurements are often expressed as a size relative to a theoretically perfect part that has its geometry defined in a print or computer model. A print is a blueprint illustrating the defined geometry of a part and its features. Each feature can have a size, a distance from other features, and an allowed tolerance set for each element.
The atomic length scale is ℓ a ~ 10 −10 m and is given by the size of hydrogen atom (i.e., the Bohr radius, approximately 53 pm).; The length scale for the strong interactions (or the one derived from QCD through dimensional transmutation) is around ℓ s ~ 10 −15 m, and the "radii" of strongly interacting particles (such as the proton) are roughly comparable.
An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points (1, 0) and (-1, 0) can be joined by paths of length arbitrarily close to 2, but not by a path of length 2. An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between ...
For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space of dimension three. Projective geometry has in fact the simplest and most elegant synthetic expression of any geometry. [5]