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The base pairs form a parallelogram with half the area of the quadrilateral, A q, as the sum of the areas of the four large triangles, A l is 2 A q (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, A s is a quarter of A l (half linear dimensions yields quarter area), and the area of the parallelogram is A ...
The rectangle is a more convenient unit to consider as fundamental domain (or set of two of them) than a parallelogram consisting of part of a tile and part of another one. In 2D there may be translational symmetry in one direction for vectors of any length. One line, not in the same direction, fully defines the whole object.
the distance between the two lines can be found by locating two points (one on each line) that lie on a common perpendicular to the parallel lines and calculating the distance between them. Since the lines have slope m, a common perpendicular would have slope −1/m and we can take the line with equation y = −x/m as a common perpendicular ...
Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the p {\displaystyle p} -norm if and only if p = 2 , {\displaystyle p=2,} the so-called Euclidean norm or standard norm.
Parallel lines are mapped on parallel lines, or on a pair of points (if they are parallel to ). The ratio of the length of two line segments on a line stays unchanged. As a special case, midpoints are mapped on midpoints. The length of a line segment parallel to the projection plane remains unchanged. The length of any line segment is shortened ...
A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of ...
A property of Euclidean spaces is the parallelogram property of vectors: If two segments are equipollent, then they form two sides of a parallelogram: If a given vector holds between a and b, c and d, then the vector which holds between a and c is the same as that which holds between b and d.
For a given parallelogram consider an arbitrary inner parallelogram having as a diagonal as well. Furthermore there are two uniquely determined parallelograms G F H D {\displaystyle GFHD} and I B J F {\displaystyle IBJF} the sides of which are parallel to the sides of the outer parallelogram and which share the vertex F {\displaystyle F} with ...