Transformations in Geometry

Why This Matters

Transformations are the backbone of algebraic geometry—they reveal what properties of geometric objects are truly fundamental versus what's merely an artifact of how we've positioned things in space. When you study varieties, morphisms, and coordinate systems, you're constantly asking: what stays the same when we move, stretch, or project? Understanding transformation hierarchies—from rigid isometries to flexible projective maps—gives you the conceptual framework to classify geometric objects and recognize when two seemingly different equations describe "the same" shape.

You're being tested on your ability to distinguish which properties each transformation type preserves and how these transformations relate to algebraic structures like groups and rings. Don't just memorize that rotations preserve distance—know why isometries form a group, how affine transformations connect to linear algebra, and what makes projective transformations essential for studying curves at infinity. This hierarchical thinking is exactly what FRQs target.

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Isometries: Distance-Preserving Transformations

Isometries are transformations where every distance measurement stays exactly the same—the transformation acts like picking up a rigid object and repositioning it. These form a group under composition and are fundamental for studying congruence.

Translation

• Vector addition to coordinates—every point $(x, y)$ maps to $(x + a, y + b)$ for fixed constants $a, b$
• No fixed points unless the translation vector is zero, distinguishing it from rotations and reflections
• Generates infinite cyclic subgroups of the isometry group, important for studying periodic structures like lattices

Rotation

• Turning around a fixed center point by angle $\theta$, represented by the matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
• Exactly one fixed point (the center) in the plane, unless $\theta = 0$
• Preserves orientation—a clockwise-labeled triangle stays clockwise, unlike reflections

Reflection

• Mirror image across a line where each point and its image are equidistant from the reflection axis
• Reverses orientation—transforms a clockwise figure to counterclockwise, detectable algebraically by determinant $-1$
• Every point on the reflection line is fixed, giving infinitely many fixed points along a one-dimensional set

Glide Reflection

• Composition of translation and reflection where the translation direction is parallel to the reflection line
• No fixed points despite being an isometry—the translation component prevents any point from staying put
• Only orientation-reversing isometry without fixed points, making it algebraically distinct from simple reflection

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Similarity Transformations: Preserving Shape but Not Size

Similarity transformations maintain angles and ratios of distances—shapes look the same, just zoomed in or out. These are crucial for understanding when geometric objects are "essentially the same" up to scaling.

Dilation (Scaling)

• Uniform scaling from a center point by factor $k$, mapping $(x, y)$ to $(kx, ky)$ when centered at the origin
• Scale factor $k > 1$ enlarges, $0 < k < 1$ shrinks—negative $k$ combines scaling with point reflection
• Preserves angles but changes area by factor $k^2$, distinguishing it from isometries

Similarity Transformation

• Composition of isometry and dilation—any transformation preserving angle measures belongs here
• Ratio of corresponding lengths is constant (the similarity ratio), fundamental for proving triangles similar
• Forms a group containing isometries as the subgroup where scale factor equals 1

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Affine Transformations: Preserving Lines and Parallelism

Affine transformations keep straight lines straight and preserve the property of lines being parallel—but distances and angles can change. These correspond algebraically to maps of the form $\mathbf{x} \mapsto A\mathbf{x} + \mathbf{b}$ where $A$ is an invertible matrix.

Shear

• Slides layers of a figure parallel to a fixed direction—horizontal shear maps $(x, y)$ to $(x + ky, y)$
• Preserves area despite distorting shape, since the transformation matrix has determinant 1
• Parallelograms map to parallelograms but rectangles generally don't stay rectangular—angles aren't preserved

Affine Transformation

• Linear map plus translation—the most general transformation preserving collinearity and parallelism
• Midpoints map to midpoints and ratios along lines are preserved, making barycentric coordinates affine-invariant
• Determinant of the linear part gives the signed area scaling factor—zero determinant means the transformation is degenerate

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Projective Transformations: The Most General Linear Framework

Projective transformations work in projective space where we add "points at infinity"—parallel lines can meet, and the distinction between finite and infinite points disappears. These are essential for studying algebraic curves and their behavior at infinity.

Projective Transformation

• Maps projective space to itself via invertible linear maps on homogeneous coordinates $[x : y : z]$
• Parallel lines can map to intersecting lines—the transformation can send points at infinity to finite points and vice versa
• Cross-ratio is the fundamental invariant—the only numerical quantity preserved, making it central to projective geometry

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The Classification Hierarchy

Understanding how these transformation types nest inside each other is essential—each larger class preserves fewer properties but reveals deeper invariants.

Isometry

• Preserves distances, angles, and area—the most restrictive class of transformations
• Forms a group including translations, rotations, reflections, and glide reflections as generators
• Classifies congruence—two figures are congruent if and only if an isometry maps one to the other

Similarity Transformation

• Preserves angles and distance ratios but allows uniform scaling
• Classifies similarity—two figures are similar iff a similarity transformation relates them
• Contains isometries as the subgroup with scale factor 1

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Quick Reference Table

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Self-Check Questions

1. Which two transformations reverse orientation while still being isometries, and how do their fixed point sets differ?

2. A transformation preserves angles but doubles all distances. What type of transformation is this, and why isn't it an isometry?

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

4. If you apply a shear followed by a dilation, is the result necessarily an affine transformation? Explain using the algebraic definition.

5. An FRQ gives you two congruent triangles and asks what transformations could map one to the other. How would you systematically determine whether you need a rotation, reflection, or glide reflection?