On-Line Computer Graphics Notes
SHEARING

Overview

Shearing transformations in three-dimensions alter two of the three coordinate values proportionally to the value of the third coordinate.

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The X-Shear Transformation

Given a frame ${\cal F} =( {\vec u} , {\vec v} , {\vec w} , {\bf O} )$. We ``x-shear'' a frame by modifying the first vector of the frame by adding to it a linear combination of the other two vectors. The frame transformation takes the following form

$$\left[ \begin{array}{c} {\vec u} \\ {\vec v} \\ {\vec w} \\ ... ...c v} + b {\vec w} \\ {\vec v} \\ {\vec w} \\ {\bf O} \end{array}\right]$$

An illustration of this process is given in the following figure.

This transform can be implemented by the following $4 \times 4$ matrix:

$$\left[ \begin{array}{cccc} 1 & a & b & 0 \\ 0 & 1 & 0 & 0 \\ ... ...c v} + b {\vec w} \\ {\vec v} \\ {\vec w} \\ {\bf O} \end{array}\right]$$

and so we define the $x$-shear transformation by

$${H} _ {x; a,b } \; = \; \left[ \begin{array}{cccc} 1 & a & b & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right]$$

If this transformation is applied to the point $(u,v,w)$, we obtain

$$\left[ \begin{array}{cccc} u & v & w & 1 \end{array}\right] \left... ... \: = \: \left[ \begin{array}{cccc} u & au + v & bu + w & 1 \end{array}\right]$$

and thus objects can be sheared by applying this matrix to all points of the object.

The Y-Shear Transformation

Given a frame ${\cal F} =( {\vec u} , {\vec v} , {\vec w} , {\bf O} )$. We ``y-shear'' a frame by transforming the second vector by adding a linear combination of the other two vectors. The frame transformation takes the following form

$$\left[ \begin{array}{c} {\vec u} \\ {\vec v} \\ {\vec w} \\ ... ...\vec u} + {\vec v} + b {\vec w} \\ {\vec w} \\ {\bf O} \end{array}\right]$$

An illustration of this process is given in the following figure.

This transform can be implemented by the following $4 \times 4$ matrix:

$$\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ a & 1 & b & 0 \\ ... ...\vec u} + {\vec v} + b {\vec w} \\ {\vec w} \\ {\bf O} \end{array}\right]$$

and so we define the $y$-shear transformation by

$${H} _ {y; a,b } \; = \; \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ a & 1 & b & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right]$$

If this transformation is applied to the point $(u,v,w)$, we obtain

$$\left[ \begin{array}{cccc} u & v & w & 1 \end{array}\right] \left... ... \: = \: \left[ \begin{array}{cccc} u + av & v & w + bv & 1 \end{array}\right]$$

and thus objects can be sheared by applying this matrix to all points of the object.

The Z-Shear Transformation

Given a frame ${\cal F} =( {\vec u} , {\vec v} , {\vec w} , {\bf O} )$. We ``z-shear'' a frame by transforming the third vector by adding a linear combination of the other two vectors. The frame transformation takes the following form

$$\left[ \begin{array}{c} {\vec u} \\ {\vec v} \\ {\vec w} \\ ... ...ec v} \\ a {\vec u} + b {\vec v} + {\vec w} \\ {\bf O} \end{array}\right]$$

which is illustrated by

This transform can be implemented by the following $4 \times 4$ matrix:

$$\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ ... ...ec v} \\ a {\vec u} + b {\vec v} + {\vec w} \\ {\bf O} \end{array}\right]$$

and so we define the $z$-shear transformation by

$${H} _ {z; a,b } \; = \; \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ a & b & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right]$$

If this transformation is applied to the point $(u,v,w)$, we obtain

$$\left[ \begin{array}{cccc} u & v & w & 1 \end{array}\right] \left... ... \: = \: \left[ \begin{array}{cccc} u + aw & v + bw & w & 1 \end{array}\right]$$

and thus objects can be sheared by applying this matrix to all points of the object.

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