Andromeda
Note

Rotations in Geometry

Definition

A rotation (or “turn”) moves every point of a figure through a specified angle (angle of rotation) around a fixed point called the center of rotation.

Why It Matters

Rotations are fundamental to any system that moves through space; from robotic arms to satellite orientation, mastering the math of turning is essential for ensuring that an object ends up exactly where it needs to be.

Core Concepts

  • 90° clockwise: (x,y)→(y,−x)(x, y) \to (y, -x)
  • How to read: “The point x comma y maps to y comma negative x.”
    • Meaning: Quarter-turn clockwise about the origin.
  • 180° rotation: (x,y)→(−x,−y)(x, y) \to (-x, -y)
  • How to read: “The point x comma y maps to negative x comma negative y.”
    • Meaning: Half-turn; point moves to the diametrically opposite position.
  • 270° clockwise (90° CCW): (x,y)→(−y,x)(x, y) \to (-y, x)
  • How to read: “The point x comma y maps to negative y comma x.”
    • Meaning: Three-quarter turn clockwise equals quarter-turn counterclockwise.
  • General clockwise rotation by θ\theta: P(x,y)→P′(xcos⁡θ+ysin⁡θ,  ycos⁡θ−xsin⁡θ)P(x, y) \to P'(x \cos \theta + y \sin \theta,\; y \cos \theta - x \sin \theta)
  • How to read: “The new x equals x cosine theta plus y sine theta; the new y equals y cosine theta minus x sine theta.”
    • Meaning / when to use: Standard rotation matrix for clockwise turns about the origin. For CCW, flip the sign on the sine terms.
  • Rigid Motion: Rotations are isometries; they preserve congruence.
  • Orientation: The orientation of the figure remains unchanged.
  • Invariance: The center of rotation is the only point mapped to itself.
  • Rotational symmetry: A figure matches itself when rotated by some θ\theta with 0<θ<360∘0 < \theta < 360^\circ.
  • How to read: “The zero is less than theta is less than three-sixty degrees.”
    • Meaning: The figure looks unchanged after a partial turn (e.g., a square has 90∘90^\circ rotational symmetry).

Connected Concepts

Connected notes