System, such as Cartesian space, can have three perpendicular axes {homogeneous coordinate}. Point has coordinates that are perpendicular distances to axes. For homogeneous coordinates u, v, and w {line coordinate}, u*x + v*y + w*z = 0 represents a line.
Using homogeneous coordinates, all curve equations are first-degree homogeneous equations. Equation degree gives curve class. For example, quadratic equations are surfaces.
triangle
In triangle, point has signed perpendicular distances to triangle sides. If triangle side is at infinity, x = x1/x3 and y = x2/x3.
comparison
For point coordinates, equation degree is curve order.
duality
Line coordinates prove the duality principle that points and lines are complementary.
Mathematical Sciences>Geometry>Coordinate System
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Date Modified: 2022.0224