Shear mapping
In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance from a given line parallel to that direction.The transformations can be applied with a shear matrix or transvection, an elementary matrix that represents the addition of a multiple of one row or column to another.An example is the linear map that takes any point with coordinatesIn this case, the displacement is horizontal by a factor of 2 where the fixed line is the x-axis, and the signed distance is the y-coordinate.Applying a shear map to a set of points of the plane will change all angles between them (except straight angles), and the length of any line segment that is not parallel to the direction of displacement.Therefore, it will usually distort the shape of a geometric figure, for example turning squares into parallelograms, and circles into ellipses.However a shearing does preserve the area of geometric figures and the alignment and relative distances of collinear points.A shear mapping is the main difference between the upright and slanted (or italic) styles of letters.The same definition is used in three-dimensional geometry, except that the distance is measured from a fixed plane.A three-dimensional shearing transformation preserves the volume of solid figures, but changes areas of plane figures (except those that are parallel to the displacement).This transformation is used to describe laminar flow of a fluid between plates, one moving in a plane above and parallel to the first. the distance is measured from a fixed hyperplane parallel to the direction of displacement. that preserves the n-dimensional measure (hypervolume) of any set.; where m is a fixed parameter, called the shear factor.The effect of this mapping is to displace every point horizontally by an amount proportionally to its y-coordinate.Any point above the x-axis is displaced to the right (increasing x) if m > 0, and to the left if m < 0.Straight lines parallel to the x-axis remain where they are, while all other lines are turned (by various angles) about the point where they cross the x-axis.If the coordinates of a point are written as a column vector (a 2×1 matrix), the shear mapping can be written as multiplication by a 2×2 matrix: A vertical shear (or shear parallel to the y-axis) of lines is similar, except that the roles of x and y are swapped.It corresponds to multiplying the coordinate vector by the transposed matrix: The vertical shear displaces points to the right of the y-axis up or down, depending on the sign of m. It leaves vertical lines invariant, but tilts all other lines about the point where they meet the y-axis.Horizontal lines, in particular, get tilted by the shear angleto become lines with slope m. Two or more shear transformations can be combined.This matrix shears parallel to the x axis in the direction of the fourth dimension of the underlying vector space.The determinant will always be 1, as no matter where the shear element is placed, it will be a member of a skew-diagonal that also contains zero elements (as all skew-diagonals have length at least two) hence its product will remain zero and will not contribute to the determinant.Hence, raising a shear matrix to a power n multiplies its shear factor by n. If S is an n × n shear matrix, then: For a vector space V and subspace W, a shear fixing W translates all vectors in a direction parallel to W. To be more precise, if V is the direct sum of W and W′, and we write vectors as correspondingly, the typical shear L fixing W is where M is a linear mapping from W′ into W. Therefore in block matrix terms L can be represented asThe following applications of shear mapping were noted by William Kingdon Clifford: The area-preserving property of a shear mapping can be used for results involving area.[4][5][6] An algorithm due to Alan W. Paeth uses a sequence of three shear mappings (horizontal, vertical, then horizontal again) to rotate a digital image by an arbitrary angle.The algorithm is very simple to implement, and very efficient, since each step processes only one column or one row of pixels at a time.[7] In typography, normal text transformed by a shear mapping results in oblique type.In pre-Einsteinian Galilean relativity, transformations between frames of reference are shear mappings called Galilean transformations.These are also sometimes seen when describing moving reference frames relative to a "preferred" frame, sometimes referred to as absolute time and space.
fluid dynamicsplane geometryaffine transformationsigned distanceparallelelementary matrixmatrixidentity matrixlinear mapcoordinatesrotationsanglesstraight anglesline segmentparallelogramscirclesellipsescollinearslanted (or italic)lettersthree-dimensional geometrylaminar flowCartesian spacehyperplanelinear transformationmeasurecotangentcolumn vectormultiplicationtransposed matrixpositive definite matrixdeterminantinverseinvertibleeigenvalueeigenspacedefectiveblock matrixvolumepolytopevector spacesubspacedirect sumWilliam Kingdon CliffordPythagorean theoremcomputer graphicsAlan W. Paethdigital imagepixelstypographyoblique typeGalilean relativityframes of referenceGalilean transformationsabsolute time and spaceTransformation matrixGeoGebraAddison-WesleyVector graphicsDiffusion curve2D graphicsAlpha compositingLayersText-to-imageIsometric graphicsMode 7Parallax scrollingRay castingSkybox3D graphics3D projection3D renderingImage-basedSpectralUnbiasedAliasingAnisotropic filteringCel shadingFluid animationLightingGlobal illuminationHidden-surface determinationPolygon meshTriangle meshShadingDeferredSurface triangulationWire-frame modelBack-face cullingClippingCollision detectionPlanar projectionReflectionRenderingBeam tracingCone tracingCheckerboard renderingRay tracingPath tracingScanline renderingRotationScalingShadow mappingShadow volumeShear matrixShaderTranslationVolume renderingGraphics software3D computer graphics softwareanimationmodelingRaster graphics editorsVector graphics editorsList of computer graphics algorithmsAlternantAnti-diagonalAnti-HermitianAnti-symmetricArrowheadBidiagonalBisymmetricBlock-diagonalBlock tridiagonalBooleanCauchy