projective plane
Lines always meet
A projective plane is a geometric surface where every pair of lines intersects at exactly one point—there are no parallel lines.
This is achieved by adding "points at infinity" where parallel lines in ordinary geometry would meet.
The simplest example is the real projective plane, formed by identifying opposite points on a sphere or by extending the Euclidean plane.
Projective planes are fundamental objects in topology and algebraic geometry, studied for their symmetry properties and their role in understanding perspective and projection.
Rate of occurrence
- collinearityPoints sharing a line
- dualityTwofold nature
- axiomsAccepted starting principles
- configurationArrangement of parts
Curves from plane-cone intersections
Attitude or spatial rendering
- vanishing pointWhere parallels appear to meet
- horizon lineApparent earth-sky junction
- architectural drawingScaled plan of structures
- one-point perspectiveDepth from one vanishing point
Many and various
Inward-turning process
- fixed pointsElements mapping to themselves
- invariantRemaining always constant
- correlationStatistical relationship measure
- elationIntense triumphant joy
incidence
Rate of occurrence
- collinearityPoints sharing a line
- dualityTwofold nature
- axiomsAccepted starting principles
configuration · order · synthetic · oval · arc
conic sections
Curves from plane-cone intersections
perspective
Attitude or spatial rendering
- vanishing pointWhere parallels appear to meet
- horizon lineApparent earth-sky junction
- architectural drawingScaled plan of structures
manifold
Many and various
involution
Inward-turning process
- fixed pointsElements mapping to themselves
- invariantRemaining always constant
- correlationStatistical relationship measure
- surfacessurface, sphere, cone, torus, cylinder
- algebraic geometryvariety, varieties, scheme, schemes, divisor
- manifoldssphere, torus, Mobius strip, surface
- projective geometryduality, polarity, conic, configuration, correlation
- Topological SurfacesMobius strip
- algebraic topologyhomology, exactness, linking number
plane, planes, projection, projected, projector, projectile, planar, biplane, projectors