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The dilation is commutative, also given by = =. If B has a center on the origin, as before, then the dilation of A by B can be understood as the locus of the points covered by B when the center of B moves inside A. In the above example, the dilation of the square of side 10 by the disk of radius 2 is a square of side 14, with rounded corners ...
Length contraction can also be derived from time dilation, [34] according to which the rate of a single "moving" clock (indicating its proper time) is lower with respect to two synchronized "resting" clocks (indicating ). Time dilation was experimentally confirmed multiple times, and is represented by the relation:
Dilation is commutative, also given by = =. If B has a center on the origin, then the dilation of A by B can be understood as the locus of the points covered by B when the center of B moves inside A. The dilation of a square of size 10, centered at the origin, by a disk of radius 2, also centered at the origin, is a square of side 14, with ...
Opening can be used to find things into which a specific structuring element can fit (edges, corners, ...). One can think of B sweeping around the inside of the boundary of A , so that it does not extend beyond the boundary, and shaping the A boundary around the boundary of the element.
The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is . A cut is made, splitting the region into two rectangular pieces, as shown in the second diagram. The larger piece, at the top, has width a and height a-b.
Spacetime diagrams can show the geometry underlying phenomena like time dilation and length contraction without mathematical equations. The history of an object's location through time traces out a line or curve on a spacetime diagram, referred to as the object's world line .
By choosing a particular structuring element, one sets a way of differentiating some objects (or parts of objects) from others, according to their shape or spatial orientation. Size . For example, one structuring element can be a 3 × 3 {\displaystyle 3\times 3} square or a 21 × 21 {\displaystyle 21\times 21} square.
For a contraction T (i.e., (‖ ‖), its defect operator D T is defined to be the (unique) positive square root D T = (I - T*T) ½. In the special case that S is an isometry, D S* is a projector and D S =0, hence the following is an Sz. Nagy unitary dilation of S with the required polynomial functional calculus property: