Library/Geometry/Projective geometry/Impossibility of constructions using only a ruler

Impossibility of constructions using only a ruler

Overview
Important

Some geometric constructions that are possible with both a ruler and a compass become impossible if you are only allowed to use a ruler (straightedge) and not a compass. For example, you cannot find the center of a given circle using only a ruler. This is because a ruler can only draw straight lines through known points, but cannot measure distances or draw circles.

Important properties

  • A ruler (straightedge) can only draw straight lines between known points.

  • Without a compass, you cannot transfer distances or draw circles.

  • Some constructions, like finding the center of a circle, require more than just drawing lines.

  • Projective transformations can map circles to other shapes, making the center ambiguous with only a ruler.