Shear (mathematics)
Encyclopedia
In mathematics
, shear mapping or transvection is a particular kind of linear mapping. Linear mapping is a function
between two vector space
s that preserves the operations of vector addition and scalar
multiplication. Shear mapping's effect leaves all points on one axis fixed, while the other points are shifted parallel to the axis by a distance proportional to their perpendicular distance from that axis.
Shear mappings carry area
s into equal areas; see equi-areal mapping for the reason and for other linear mappings that have this property.
This leaves horizontal lines y = c invariant, but for m ≠ 0 maps vertical lines x = a into lines y' = (x' − a)/m having slope
1/m
Substituting 1/m for m in the matrix gives lines y = m(x − a) of slope m, if desired.
A vertical shear (or shear parallel to the y axis) of lines is accomplished by the linear mapping
The vertical shear leaves vertical lines x = a invariant, but maps horizontal lines y = b into lines y' = mx' + b
The matrices above are special cases of shear matrices
, which allow for generalization to higher dimensions. The shear elements here are either m or 1/m, case depending.
The 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. For instance, the Pythagorean theorem
has been illustrated with shear mapping (see external link).
V and subspace W, a shear fixing W translates all vectors 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 fixing W is L where
where M is a linear mapping from W′ into W. Therefore in block matrix
terms L can be represented as
with blocks on the diagonal I (identity matrix
), with M above the diagonal, and 0 below.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, shear mapping or transvection is a particular kind of linear mapping. Linear mapping is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
between two vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s that preserves the operations of vector addition and scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
multiplication. Shear mapping's effect leaves all points on one axis fixed, while the other points are shifted parallel to the axis by a distance proportional to their perpendicular distance from that axis.
Shear mappings carry area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
s into equal areas; see equi-areal mapping for the reason and for other linear mappings that have this property.
Elementary form
In the plane {(x, y): x,y ∈ R }, a horizontal shear (or shear parallel to the x axis) is represented by the linear mapping
This leaves horizontal lines y = c invariant, but for m ≠ 0 maps vertical lines x = a into lines y' = (x' − a)/m having slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
1/m
Substituting 1/m for m in the matrix gives lines y = m(x − a) of slope m, if desired.
A vertical shear (or shear parallel to the y axis) of lines is accomplished by the linear mapping
The vertical shear leaves vertical lines x = a invariant, but maps horizontal lines y = b into lines y' = mx' + b
The matrices above are special cases of shear matrices
Shear matrix
In mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another...
, which allow for generalization to higher dimensions. The shear elements here are either m or 1/m, case depending.
The following applications of shear mapping were noted by William Kingdon Clifford
William Kingdon Clifford
William Kingdon Clifford FRS was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour, with interesting applications in contemporary mathematical physics...
:
- "A succession of shears will enable us to reduce any figure bounded by straight lines to a triangle of equal area."
- "... we may shear any triangle into a right-angled triangle, and this will not alter its area. Thus the area of any triangle is half the area of the rectangle on the same base and with height equal to the perpendicular on the base from the opposite angle."
The area-preserving property of a shear mapping can be used for results involving area. For instance, the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
has been illustrated with shear mapping (see external link).
Advanced form
For a vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V and subspace W, a shear fixing W translates all vectors parallel to W.
To be more precise, if V is the direct sum of W and W′, and we write vectors as
- v = w + w′
correspondingly, the typical shear fixing W is L where
- L(v) = (w + Mw′) + w ′
where M is a linear mapping from W′ into W. Therefore in block matrix
Block matrix
In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...
terms L can be represented as
with blocks on the diagonal I (identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
), with M above the diagonal, and 0 below.