Projective geometry studies projective spaces and incidence relations such as which points lie on which lines, with transformations induced by invertible linear maps. Over a field , points of are one-dimensional vector subspaces of , represented by nonzero homogeneous coordinates up to a common nonzero scalar.
The chart gives affine coordinates . Points with form a hyperplane at infinity. In the projective plane, distinct lines meet in one point: parallel affine lines acquire their shared direction point at infinity. This statement concerns the projective plane, not arbitrary pairs of lines in higher dimension.
A projective transformation preserves incidence and the cross-ratio, but generally not Euclidean lengths or angles. This makes the subject natural for perspective and for homogeneous equations in algebraic geometry.
Classical Laguerre geometry studies oriented hyperplanes, oriented spheres and their oriented tangency in Euclidean space. Its transformations preserve that contact relation. In the plane the objects are oriented straight lines and oriented circles; points can be represented as radius-zero circles. The orientation and signed radius are part of the data, so ordinary unoriented tangency alone loses information.[ref]
Laguerre geometry in space forms and incircular nets by Alexander I. Bobenko and others. Abstract, introduction and Appendix B.1
For example, fix a unit normal and an oriented line . Under a consistent signed-radius convention, a circle with centre and signed radius is in oriented contact when . The absolute-distance equation forgets which side and orientation participate. This linear signed-contact relation motivates coordinate models for tangency and the use of Laguerre geometry in studying incircular nets.
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Projective geometry is a branch of mathematics that studies the properties and relationships of geometric objects that are invariant under projection. It is particularly concerned with the properties of figures that remain unchanged when viewed from different perspectives, making it a fundamental area in both pure mathematics and applications such as computer graphics and art.
