Linear transformation grid

Parametric equations

A linear transformation sends a point $P(x,y)$ in the plane to $P^{\prime }(x^{\prime },y^{\prime })$:

Here $b$ tilts the vertical lines sideways, and $d$ stretches or compresses the grid vertically. As time $t$ advances, $a$ oscillates slightly with $\sin (\omega t)$, so the grid keeps opening and closing a little. The lower-left entry stays $0$, so this motion is not a rotation. A line still maps to a line, and the origin stays fixed.

Interaction

• X shear b: drag the slider to tilt the vertical lines, and watch the square grid become a parallelogram
• Y scale d: stretch or compress the vertical direction
• Transform speed ω: the grid breathes slightly in time, and a line still maps to a line

What to notice

• Under this linear transformation the origin stays fixed, and a line still maps to a line.
• Shear changes angles and does not change parallelism.
• Shear and vertical scaling together turn the square grid into a parallelogram, and the origin stays fixed.

Related

• Matrix and linear transformation
• Affine transformation pattern
• Rotation and scaling, composed

Further reading

• Linear map (Wikipedia)
• Transformation matrix (Wikipedia)