Translations and rotations have many applications—and not only in geometry!
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Roulettes do not exhaust the possibilities offered by plane-on-plane motion—or the geometric wonders hidden within it. Armed once again with drawing paper, tracing paper and pens, we continue our exploration with some somewhat overlooked curves: glissettes.
Some geometric problems, however complicated they may look, quickly become clear once geometric transformations are brought in, and are often solved by composing them. To do this, it helps to recast them in vector terms.
The notion of a group—abstract and purely algebraic? Not at all! In geometry, it keeps us from being overwhelmed by the apparent profusion and diversity of transformations, and helps us understand the different kinds of symmetry we may encounter.
Do you know Kempe's universality theorem? This 19th-century result states that any algebraic curve can be drawn by a linkage. Today, computers and robotics have replaced the ingenious mechanisms devised by scientists of the past.
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