Theory
Mirror and slant
Beyond translate, scale, and rotate, two more transformations round out the 2D toolkit: reflection (mirroring a shape) and shearing (slanting it). Both, like the others, transform each point (x, y) by a simple rule, and both are common exam questions.
This final lesson covers them with worked examples. Reflection flips a shape across an axis or line by changing a coordinate's sign or swapping coordinates; shearing pushes points sideways in proportion to their position. Together with the basic three, they give you the full set of standard 2D transformations.
Theory
Reflection: mirroring a shape
Reflection produces a mirror image across an axis or line. The common cases:
- About the x-axis: (x, y) becomes (x, -y), the y-coordinate flips sign.
- About the y-axis: (x, y) becomes (-x, y), the x-coordinate flips sign.
- About the origin: (x, y) becomes (-x, -y), both flip.
- About the line y = x: (x, y) becomes (y, x), the coordinates swap.
Example: reflecting (2, 3) about the x-axis gives (2, -3), the shape flips below the axis. Reflecting (2, 3) about y = x gives (3, 2), the coordinates swap. Reflection is just a sign change or a coordinate swap.
Theory
Shearing: slanting a shape
Shearing slants a shape, turning a rectangle into a parallelogram. Points shift in one direction by an amount proportional to their position in the other.
An x-direction shear pushes points horizontally in proportion to their y (height):
x' = x + shx times y, y' = y
Example: shearing (2, 3) in x with factor 2 gives x' = 2 + 2 times 3 = 8, y' = 3, so (2, 3) becomes (8, 3), the higher the point, the further it slides right. A y-direction shear does the mirror: y' = y + shy times x, x' = x, sliding points up in proportion to their x. Shear is a proportional slide, not a flip.
At a glance
| Transformation | Rule | Effect |
|---|---|---|
| Reflect about x-axis | (x, y) -> (x, -y) | Mirror below/above the x-axis |
| Reflect about y-axis | (x, y) -> (-x, y) | Mirror left/right of the y-axis |
| Reflect about y = x | (x, y) -> (y, x) | Swap coordinates (mirror across the diagonal) |
| Shear in x | x' = x + shx*y, y' = y | Slant horizontally, proportional to y |
| Shear in y | y' = y + shy*x, x' = x | Slant vertically, proportional to x |
Quiz
Applying an x-direction shear with factor 2 to the point (2, 3), where x' = x + 2y and y' = y, gives which point?
- (2, 8), shearing the y instead
- (8, 3), because x' = 2 + 2(3) = 8 and y' = 3
- (2, 3), unchanged
- (2, -3), a reflection
Show the answer
(8, 3), because x' = 2 + 2(3) = 8 and y' = 3
An x-direction shear uses x' = x + shx times y and y' = y. With shx = 2 and the point (2, 3): x' = 2 + 2(3) = 2 + 6 = 8, and y' = 3, giving (8, 3). Option A shears the wrong coordinate (that would be a y-direction shear). Option C is wrong: the shear does change the point (x moves from 2 to 8). Option D, (2, -3), is a reflection about the x-axis, not a shear. In an x-shear the horizontal position shifts by an amount proportional to y (here 2 times 3 = 6), so higher points slide further right.
Think first
How do reflection and shear fit with the matrix approach from the last lesson?
The basic transformations became matrices. Are reflection and shear also matrices, and why does that matter? Then tap.
Show the answer
Yes, reflection and shearing are ALSO linear transformations expressible as matrices, which means the entire 2D toolkit, translate, scale, rotate, reflect, shear, lives in one uniform matrix framework where transformations combine by multiplication. Reflection about the x-axis, for instance, is the matrix that maps (x, y) to (x, -y): its rows are [1 0][0 -1] (or in homogeneous 3x3 form with an extra [0 0 1] row). Reflection about y=x swaps coordinates, matrix rows [0 1][1 0]. An x-shear with factor shx is the matrix [1 shx][0 1], mapping (x,y) to (x + shx*y, y). Because every one of these is a matrix (and translation too, thanks to homogeneous coordinates), you gain the same powerful benefit described in the previous lesson: you can COMBINE any sequence of them, reflect then shear then rotate then translate, by MULTIPLYING their matrices into a single combined matrix, and then apply that one matrix to every point of a shape. This is why the matrix formulation is so central to computer graphics: it turns a whole varied family of geometric operations into one consistent operation (matrix multiply), lets you compose complex transformations from simple ones, and makes applying them to many points fast and uniform. So reflection and shear are not special cases outside the system; they slot right into the same matrix machinery, completing a clean, unified toolkit. Every standard 2D transformation is a matrix, so they all compose by multiplication, which is the elegant heart of transformation geometry in graphics.
Theory
BCA601-03 complete
You have traced computer graphics from the top down: what it is used for, how displays and files represent images (raster vs vector), how a line is computed pixel by pixel (DDA and Bresenham), and how shapes are moved with transformations (translate, scale, rotate, reflect, shear) using matrices and homogeneous coordinates. That is the full arc of the subject. The two algorithm-and-matrix units are the exam heart, practise the worked examples until the pixels and matrices come out right every time. Well done.
Summary
Key takeaways
- Reflection mirrors a shape: about the x-axis (x,y)->(x,-y), about the y-axis (x,y)->(-x,y), about the origin (x,y)->(-x,-y), about y=x (x,y)->(y,x).
- Worked: (2,3) about the x-axis gives (2,-3); (2,3) about y=x gives (3,2).
- Shearing slants a shape: x-shear x' = x + shxy, y' = y (proportional to height); y-shear y' = y + shyx, x' = x.
- Worked: (2,3) with an x-shear of factor 2 gives (2 + 2*3, 3) = (8,3).
- Reflection is a sign change or coordinate swap; shear is a proportional slide (a slant, not a flip).
- Reflection and shear are also matrices, so all standard 2D transformations combine by matrix multiplication.
- Memory hook: reflection flips (sign change/swap), shear slants proportionally; all transformations are matrices.