Important Geometric Transformations

Why This Matters

Geometric transformations are the backbone of discrete geometry—they're how we formalize the intuitive idea that shapes can move, flip, stretch, or warp while maintaining certain properties. You're being tested on your ability to recognize what's preserved under each transformation type: distances, angles, parallelism, or just the basic structure of points and lines. This hierarchy of "what stays the same" is fundamental to understanding everything from congruence proofs to computer graphics algorithms.

The key insight here is that transformations form a nested hierarchy. Isometries preserve everything geometric (distance, angle, area), similarities relax distance but keep shape, affine transformations preserve only parallelism and ratios, and projective transformations preserve only the incidence of points and lines. Don't just memorize definitions—know which transformation class you need when a problem asks about preserving specific properties.

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Rigid Motions (Isometries)

These transformations preserve all metric properties—distances, angles, and area remain unchanged. They're the "congruence transformations" because the output is always congruent to the input.

Translation

• Shifts every point by a constant vector $\mathbf{v}$—mathematically, $P \mapsto P + \mathbf{v}$ for all points
• Preserves orientation along with distance and angle, making it the "gentlest" isometry
• No fixed points exist unless $\mathbf{v} = \mathbf{0}$, distinguishing it from rotation and reflection

Rotation

• Turns all points around a fixed center by angle $\theta$—the center is the unique fixed point
• Preserves orientation (clockwise stays clockwise), unlike reflection
• Composition of two reflections across intersecting lines produces a rotation of twice the angle between them

Reflection

• Flips points across a line of reflection—each point and its image are equidistant from this axis
• Reverses orientation (turns clockwise figures counterclockwise), the key distinction from rotation
• Every point on the reflection line is fixed, giving infinitely many fixed points

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Shape-Preserving Transformations

These transformations maintain the shape of figures (angles stay the same) but allow size to change. The output is always similar to the input.

Scaling

• Multiplies all distances from a center point by factor $k$—uniform scaling uses the same $k$ in all directions
• Preserves angles and shape but changes area by factor $k^2$ (volume by $k^3$ in 3D)
• Non-uniform scaling uses different factors per axis, which does distort angles and breaks similarity

Similarity

• Combines scaling with any isometry—the general form of "same shape, different size"
• Preserves angle measure and ratios of distances but not absolute distances
• Two figures are similar if and only if a similarity transformation maps one to the other

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Linear Structure-Preserving Transformations

These transformations preserve the linear structure of geometry—straight lines stay straight, parallel lines stay parallel—but distances and angles may change.

Shear

• Slides points parallel to a fixed direction by an amount proportional to their distance from a fixed line
• Preserves area despite distorting shape—a rectangle becomes a parallelogram with the same area
• Keeps one family of parallel lines fixed while tilting all others, useful for understanding determinants

Affine Transformation

• Combines any linear transformation with translation—the general form is $P \mapsto AP + \mathbf{b}$ for matrix $A$
• Preserves collinearity, parallelism, and distance ratios along lines, but not angles or absolute distances
• Includes all isometries, similarities, and shears as special cases—it's the broadest "nice" transformation class

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Projective and Complex Transformations

These transformations work on extended geometric structures, allowing even parallel lines to meet and enabling powerful mappings between different geometric spaces.

Projective Transformation

• Maps points between projective planes while preserving only incidence (which points lie on which lines)
• Parallel lines can map to intersecting lines—they meet at a "point at infinity" that becomes finite
• Essential for perspective and computer vision, where camera views create projective distortions

Möbius Transformation

• Maps the extended complex plane to itself via $z \mapsto \frac{az + b}{cz + d}$ where $ad - bc \neq 0$
• Preserves angles (conformal) and maps circles/lines to circles/lines—called circle-preserving
• Forms a group under composition, fundamental in complex analysis and hyperbolic geometry

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Classification Concepts

These aren't individual transformations but rather categories that organize transformations by what they preserve.

Isometry

• Defined as distance-preserving—if $d(P, Q) = d(f(P), f(Q))$ for all points, $f$ is an isometry
• Automatically preserves angles, area, and shape—distance preservation implies everything else
• Classified into four types: translation, rotation, reflection, and glide reflection (reflection + translation along the axis)

Similarity

• Defined as angle-preserving plus uniform distance scaling—ratios of distances are constant
• Generated by isometries plus uniform scaling—any similarity decomposes this way
• The transformation group for "same shape" problems, central to trigonometry and proportional reasoning

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Quick Reference Table

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Self-Check Questions

1. Which two transformations preserve distance but differ in whether they have exactly one fixed point or infinitely many?

2. A rectangle is transformed into a parallelogram with the same area but different angles. Which transformation class does this belong to, and why can't it be a similarity?

3. Compare and contrast affine and projective transformations: what property do affine transformations preserve that projective transformations don't?

4. If you compose two reflections across parallel lines, what type of transformation results? What if the lines intersect?

5. An FRQ asks you to classify a transformation that preserves angles but changes all distances by the same factor. What transformation class is this, and what specific transformations could produce it?