Rigid Motions in Geometry

Definition

• Rigid motion: a transformation that preserves the shape and size of an object, without changing its orientation or scaling.

Types of Rigid Motions

• Translations: movements that shift the object from one location to another without rotating or scaling it.
• Rotations: movements that rotate the object around a fixed point or axis, without changing its size or shape.
• Reflections: movements that mirror the object over a fixed line or plane, without changing its size or shape.
• Combinations: combinations of translations, rotations, and reflections.

Properties of Rigid Motions

• Preserves distances: rigid motions do not change the distances between points on the object.
• Preserves angles: rigid motions do not change the angles between lines or planes on the object.
• Preserves orientation: rigid motions do not change the orientation of the object in space.
• Reversible: rigid motions can be reversed by applying the inverse transformation.

Composition of Rigid Motions

• The result of applying two or more rigid motions in sequence is also a rigid motion.
• The order of composition may affect the result, but the final result is always a rigid motion.

Rigid Motions in Different Dimensions

• In 2D space, rigid motions can be described using matrices or complex numbers.
• In 3D space, rigid motions can be described using 3x3 matrices or quaternions.
• In higher dimensions, rigid motions can be described using larger matrices or other mathematical representations.

Rigid Motions

• A rigid motion is a transformation that preserves the shape and size of an object, without changing its orientation or scaling.

Types of Rigid Motions

• Translations: movements that shift the object from one location to another without rotating or scaling it.
• Rotations: movements that rotate the object around a fixed point or axis, without changing its size or shape.
• Reflections: movements that mirror the object over a fixed line or plane, without changing its size or shape.
• Combinations: combinations of translations, rotations, and reflections.

Properties of Rigid Motions

• Rigid motions preserve distances between points on the object.
• Rigid motions preserve angles between lines or planes on the object.
• Rigid motions preserve the orientation of the object in space.
• Rigid motions are reversible, meaning they can be reversed by applying the inverse transformation.

Composition of Rigid Motions

• The result of applying two or more rigid motions in sequence is always a rigid motion.
• The order of composition may affect the result, but the final result is always a rigid motion.

Rigid Motions in Different Dimensions

• In 2D space, rigid motions can be described using matrices or complex numbers.
• In 3D space, rigid motions can be described using 3x3 matrices or quaternions.
• In higher dimensions, rigid motions can be described using larger matrices or other mathematical representations.

Learn about rigid motions, including translations, rotations, and reflections, and how they preserve the shape and size of an object.