The reciprocal stretch and compression leave the area invariant. For rotation by an angle θ clockwise about the origin the functional form is formula_38 and formula_39. Written in matrix form, this becomes:  Similarly, for a rotation counterclockwise about the origin, the functional form is formula_41 and formula_42 the matrix form is:  These formulae assume that the "x" axis points right and the "y" axis points up. For shear mapping (visually similar to slanting), there are two possibilities. A shear parallel to the "x" axis has formula_44 and formula_45. Written in matrix form, this becomes:  A shear parallel to the "y" axis has formula_47 and formula_48, which has matrix form:  For reflection about a line that goes through the origin, let formula_50 be a vector in the direction of the line. Then use the transformation matrix:  To project a vector orthogonally onto a line that goes through the origin, let formula_52 be a vector in the direction of the line. Then use the transformation matrix:  As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation. Parallel projections are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix, homogeneous coordinates can be used. The matrix to rotate an angle "θ" about any axis defined by unit vector ("l","m","n") is   To reflect a point through a plane formula_55 (which goes through the origin), one can use formula_56, where formula_57 is the 3x3 identity matrix and formula_58 is the three-dimensional unit vector for the vector normal of the plane. If the L2 norm of formula_59 and formula_60 is unity, the transformation matrix can be expressed as:  Note that these are particular cases of a Householder reflection in two and three dimensions. A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4x4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector):  where formula_63 for some point formula_64 on the plane. If the 4th component of the vector is 0 instead of 1, then only the vector's direction is reflected and its length remains unchanged, as if it were mirrored through a parallel plane that passes through the origin.