Shear mapping

Under a shear or transvection is understood initially in plane geometry certain affine mappings of the plane onto itself, in which the area is maintained. At a shear a straight line of the plane ( the fixed point line or axis of the shear) remains fixed, that is, every point of this line is mapped to itself. All other points in the plane are displaced parallel to the axis, while the length of the displacement vector of a point is proportional to the distance of that point from the axis. All lines that are parallel to the axis are mapped to themselves, so are Fixgeraden. Routes on these straight lines are mapped isometrically.

At a shear ie, the distance of each point to the axis remains unchanged. Thus rectangles and triangles, where one side is parallel to the axis are mapped to parallelograms or triangles ( perpendicular to the page is parallel to the axis) to have a height equal length (cf. the Figure ).

The notion of ( flat ) shear can be generalized to different affinities in space and in higher dimensions. Two ways in which the generalized shear does not change the volume of the figures depicted are shown here.

The successive application of a shear and a central extension (in any order ) gives a shear stretching, do not remain the same for the generally area and space content.

Shears in the plane

An affinity of a two-dimensional affine space ( "plane" ) is accurate then a shear when

From the condition ( 1) follows with the properties of affinity that the line joining the two fixed points is a fixed point straight line ( axis). Condition (2) allows (with the possibility of a third fixed point outside of ) the identity map as shear or forces all points outside are moved parallel to a point outside the fixed point line.

Generally have an affinity in the plane is determined uniquely when three points that do not lie on a straight line, respectively, the pixels not lying on a straight line to be specified.

Properties

It is noteworthy that for the characterization of shear no distance or area concepts must be used. Shears can thus be defined in any affine plane. The properties mentioned in the introduction are then to clarify this:

Is in the plane, a Euclidean distance and a distance compatible with this area defined, then stay at a shear with axis

Receive.

In the real plane can be obtained by choosing a " unit ellipse " or equivalent by choosing an affine coordinate system as orthonormal basis different Euclidean structures, ie introduce different angle and distance concepts. Shear now said invariance with respect to each of these structures, for example, during a rotation with respect to a Euclidean structure needs to be no rotation with respect to another. This independence from the Euclidean structure share the shears with the parallel shifts.

Image construction

A shear in the plane is determined when its axis and for a given point outside the axis is given its image point. Then, the image of another point which is not on the axis, and not on the connecting line, the following can be constructed, the figure on the right shows the steps ( red) of the structure:

When the point is to be imaged on the axis, it is self- fixed point. If it lies on the connecting line, then the pixel must be produced by the above construction text first for an auxiliary point outside and the axis are determined, or one used for image construction, the fact that the shear to operate as a shift.

Matrix representation

Is chosen in the plane of a Cartesian coordinate system in which the axis coincides with the axis of the shear coincides, then the shear is through the linear transformation

Shown.

If an affine transformation of the plane given by its projection matrix and its displacement, then iff is a shear when

For algebraic investigations it is convenient (enhanced imaging matrices ) represent the considered affine transformations as 3x3 matrices with respect to a fixed base. This corresponds to a representation of the affine transformation in homogeneous coordinates:

Algebraic structure

Concatenating two shearing is generally no more shear. The set of all the shearing plane thus forms no particular group. Your advanced imaging matrices are a subset of the group of displacement maps in special linear group. This is a group. ( It consists of the matrices of the form whose determinant is 1. Exactly the advanced imaging matrices of equal-area and orientation- preserving affinities make up this group.)

The amount of shearing with a common axis forms an abelian group. It is isomorphic to the group of displacements in a fixed direction, because you can choose an affine or Cartesian coordinate system ( with the common axis as axis), in which they all have a representation of the shape. Note to the fact that it does not depend on the position of the origin on the axis of the representation of the shear.

For shears, whose axes are parallel, one can choose a common affine or Cartesian coordinate system in which their extended imaging matrices have the form. ( The offset component must vanish, otherwise the fixed point equation has no solution. ) Multiplying two of these extended matrices, so is:

From this it is obvious:

Shears in higher dimensional spaces

In a one -dimensional space is a shear affinity having a fixed point hyperplane and moved by all the points is not on the hyperplane in a fixed direction parallel to the fixed point hyperplane. The length of the displacement vector is again proportional to the distance from the fixed point hyperplane.

In the figure to the right shows how the image of a cube is deformed in such a figure in a skewed prism if the base side (red) is the fixed point level and the shear acting parallel to the front (yellow). In the picture remain the red and light blue side and the respective opposite sides of rectangles.

The generalization to higher-dimensional space is not uniform in the literature. A shear -dimensional space each of the affinity is even more generally referred to a matrix representation of the form

Permits ( when a suitable basis). Here are identity matrices and is an arbitrary matrix. At a shear such, the fixed space is a linear space of dimension (see Rank ( mathematics) ).

The figure at right shows the effect of the general form of a shear on a cuboid in three-dimensional space. The parallelepiped shown emerged through such a shear of a cuboid. The ( oriented ) volume of the parallelepiped corresponds to that of the original parallelepiped.

Both transmissions of the concept of shear in the plane geometry to higher dimensions describe area-preserving affinities that have a least one-dimensional affine subspace pointwise fixed. Each shear one -dimensional space in the more general sense ( with a fixed point line) can be represented ( with one fixed point hyperplane ) and sequential execution of (at most) shears in the specific sense. In this case, all fixed point hyperplanes of the more specific shears contain the fixed point is precisely the general ones. There are in the -dimensional space, a dot product, and has the generalized shear the above matrix representation with respect to an orthonormal basis of the space, the fixed point hyperplanes can be chosen orthogonal to each other for the special shears, of which the common is assembled.

Shear stretching

A shear stretching arises when a central dilation and shearing are performed sequentially ( in any order). In the plane, it can be brought to the normal form, through choice of suitable coordinates ( → compare the main article Jordan normal form). She has no axis ( fixed point straight line), but a clear center ( fixed point ). If in addition we run a shift, the result is again a shear extension, with the center may change.

Connection with the term shear in the mechanics

In mechanics, especially in continuum mechanics is called the shear deformations of a certain three-dimensional body. In this case, the mass members of the body are displaced in parallel in a common direction at a fixed level in the body and the length of the displacement vector is proportional to the distance of the mass element of the fixed ( Fixed Point ) plane. The term thus coincides (as picture) with the more specific generalization to three dimensions, which is described above. Selects the coordinate system so that the unshifted level is the xy-plane of the Cartesian coordinate system, and carried all the displacements parallel to the X axis, allows the three-dimensional image by the linear shearing

Describe. Here, the displacement of the mass element at a distance from the fixing point plane. This is done in the main article shear ( mechanics).