Pohlke's theorem

Pohlke's theorem states that any three coplanar segments sharing an endpoint can represent the parallel projections of three equal, mutually perpendicular segments in space.

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Parallel projection: A projection that maps points along parallel lines onto a plane. The theorem interprets the three planar segments as images under this projection.
Julius Weisbach: A German engineer and mathematician who contributed to mechanics, mining, and descriptive geometry. His work on axonometric representation is associated with the theorem's early formulation.
Technical drawing: The production of precise drawings that communicate an object's form and dimensions. Axonometric drawings rely on planar images of spatial axes of the kind the theorem permits.
Cavalieri's principle: A principle stating that equal cross-sectional areas at corresponding heights imply equal volumes. It shares a name with projection geometry but concerns volume comparison, not projected axis existence.
Axonometric projection: A parallel projection used to depict three-dimensional objects with multiple axes visible. Pohlke's theorem guarantees the geometric basis for constructing axonometric views.
Karl Pohlke: A German mathematician and educator known for work on descriptive geometry. The theorem bears his name and is linked to his treatment of axonometric projection.
Isometric projection: An axonometric projection in which the three principal axes have equal scale and symmetric directions. It is a familiar special case of the projected-axis arrangements covered by the theorem.
Perspective projection: A projection in which rays converge at a finite viewpoint, making distant objects appear smaller. Unlike Pohlke's parallel-projection setting, perspective projection does not preserve parallel lines.
Affine transformation: A map that preserves points, straight lines, and parallelism, but not necessarily lengths or angles. Parallel projection is affine, which explains why spatial frames can yield planar segment triples.
Axonometry: A method of representing three-dimensional objects by parallel projection onto a plane. The theorem is commonly presented as the existence principle underlying axonometric drawing.