Geometric transformations Notes

1. Geometric transformations describe how a figure (a shape made of points) changes position, orientation, or size.
2. In coordinate geometry, a transformation can be treated like a function: it takes an input point (or whole shape) and produces exactly one output point (or image).

Transformations Can Be Treated As Functions

1. A function matches each input with one and only one output.
2. A geometric transformation does the same thing, but the "inputs" and "outputs" are usually points (or sets of points) in the plane.
3. If we write $T$ for a transformation, then
   1. Input $x$ can be a point like $(x,y)$, or a whole shape (a set of points).
   2. The mapping is $T$, the rule (for example "translate by $(4,2)$").
   3. Output $y$ is the image point (or image shape).
4. So we can think of transformation notation like $$\text{image} = T(\text{pre-image})$$

Isometric Transformations Preserve Distance And Angle

1. Isometric transformations change a figure's position (and sometimes orientation), but do not change size or shape. This means:
   1. Lengths stay the same.
   2. Angles stay the same.
   3. The image is congruent to the original.
2. The three isometric transformations you meet most often are translations, rotations, and reflections.

Translation Moves Every Point By The Same Vector

1. A translation slides a figure without turning or flipping it.
2. It changes position but not shape, size, direction, or orientation.
3. A translation is described by a displacement vector $$\begin{pmatrix}a\\b\end{pmatrix},$$ meaning "$a$ units horizontally and $b$ units vertically."
4. In coordinate form: $$T(x,y)=(x+a,\,y+b)$$

Rotation Turns A Figure About A Fixed Center

1. A rotation turns a shape around a center of rotation by a given angle and direction (clockwise or anticlockwise).
2. It changes position and direction, but not size or shape.
3. Common special cases about the origin $(0,0)$ are:
   1. $90^\circ$ anticlockwise: $(x,y)\mapsto(-y,x)$
   2. $90^\circ$ clockwise: $(x,y)\mapsto(y,-x)$
   3. $180^\circ$: $(x,y)\mapsto(-x,-y)$
4. For rotations about a point $C(h,k)$ (not the origin), a reliable method is:
   1. Translate so $C$ becomes the origin.
   2. Apply the origin-rotation rule.
   3. Translate back.

Reflection Flips A Figure Over A Mirror Line

1. A reflection creates a mirror image across a mirror line.
2. It changes position and orientation (it flips the figure), but does not change size or shape.
3. Key properties of reflections:
   1. The mirror line is the perpendicular bisector of the segment joining a point and its image.
   2. Points on the mirror line stay fixed.
4. Common reflection rules:
   1. In the $x$-axis: $(x,y)\mapsto(x,-y)$
   2. In the $y$-axis: $(x,y)\mapsto(-x,y)$
   3. In the line $y=x$: $(x,y)\mapsto(y,x)$
5. For a slanted mirror line like $y=x-3$, you usually use geometric reasoning (equal perpendicular distances) or coordinate methods.

Dilations Change Size But Preserve Shape

1. A dilation changes the size of a figure but keeps it the same shape.
2. Because distances change, a dilation is not an isometric transformation.
3. A dilation is described by:
   1. a center of dilation (often a point $O$), and
   2. a scale factor $k$.

Enlargement And Reduction Depend On The Scale Factor

1. If $k>1$, the image is an enlargement.
2. If $0<k<1$, the image is a reduction.
3. If the dilation is centered at the origin, the coordinate rule is: $$T(x,y)=(kx,\,ky).$$
4. For a center $C(h,k)$, conceptually each point moves along the ray from $C$ through the point.

Negative Scale Factors Reverse Direction Through The Center

If the scale factor is negative, the image appears on the opposite side of the center of dilation.

1. $k=-1$ gives a half-turn effect (similar to a $180^\circ$ rotation about the center), with no change in size.
2. $|k|>1$ makes a larger image on the opposite side.
3. $|k|<1$ makes a smaller image on the opposite side.

Similarity And Congruence Describe What Is Preserved

Transformations help you classify relationships between figures.

1. After an isometric transformation, the image is congruent to the pre-image.
2. After a dilation, the image is similar to the pre-image (same angles, lengths scaled by $k$), but not usually congruent.

Describing A Transformation Precisely

1. In problems you are often asked to "describe the transformation."
2. Each type has specific information that must be given.

What You Must State

1. Translation: the displacement vector $\begin{pmatrix}a\\b\end{pmatrix}$.
2. Rotation: the center, the angle, and the direction.
3. Reflection: the mirror line.
4. Dilation: the center of dilation and the scale factor.