point at infinity

geometry

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July 1, 2026

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In projective geometry, a "point at infinity" is an ideal mathematical point added to a plane or space. It is specifically used to represent the assumed intersection of two parallel lines, thereby eliminating special cases in geometric theorems. This concept was first introduced by Girard Desargues in the 17th century.12

In three-dimensional projective geometry, the "plane at infinity" is the collection of all points at infinity. These concepts help preserve the magnitudes of angles in transformed planes. Projective geometry, which originated in the Italian Renaissance with perspective drawing, focuses on the relationships between geometric figures and their projections, often by disregarding measurements like distances and angles.342

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projective geometry
Parallel lines and the projection of infinity
…who first introduced a single point at infinity to represent the projected intersection of parallel lines. Furthermore, he collected all the points along the horizon in one line at infinity.) With the introduction of Ω, the projected figure corresponds to a theorem discovered by Menelaus of Alexandria in the 1st…
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…who first introduced a single point at infinity to represent the projected intersection of parallel lines. Furthermore, he collected all the points along the horizon in one line at infinity.) With the introduction of Ω, the projected figure corresponds to a theorem discovered by Menelaus of Alexandria in the 1st…
Read More