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Other planes may be obtained as weighted sums of the basis planes. for example, + would be the plane midway between the y- and z-plane. In general, combining two geometric objects in plane-based GA will always be as a weighted average of them – combining points will give a point between them, as will combining lines, and indeed rotations.
The archetypical example is the real projective plane, also known as the extended Euclidean plane. [4] This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP 2, or P 2 (R), among other notations.
Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; [49] it can be studied as an affine space, where collinearity and ratios can be studied but not distances; [50] it can be studied as the complex plane using techniques of complex analysis; [51] and so on.
There are two geometric interpretations of the dual-quaternions, both of which can be used to derive the action of the planar quaternions on the plane: As a way to represent rigid body motions in 3D space. The planar quaternions can then be seen to represent a subset of those rigid-body motions.
Interactive geometry software (IGS) or dynamic geometry environments (DGEs) are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. In most IGS, one starts construction by putting a few points and using them to define new objects such as lines , circles or other points.
This term is commonly applied in plane geometry to triangles, parallelograms, trapezoids, and in solid geometry to cylinders, cones, pyramids, parallelepipeds, prisms, and frustums. The side or point opposite the base is often called the apex or summit of the shape.