Projective geometry and metric geometry (4)

Looking at geometry from the perspective of transformation

After Klein summarized various metric geometries as projective geometry, he began to seek the characteristics that distinguish various geometries, not only based on non-metric and metric The distinction between properties and various measures is based on a broader view: the goals that geometry wants to accomplish, to characterize them. He gave this characterization in a speech he gave to the Faculty of Erlangen in 1827, the views of which became known as the Erlangen Programme.

Klein's basic point is that every geometry is characterized by a transformation group, and what every geometry does is to consider its invariants under this transformation group. The subgeometry of a geometry is A family of invariants under a subgroup of the original transformation group. Under this definition, the geometric theorems of a given transformation group are still theorems in the geometry of the subgroup.

Although Klein did not use analytic expressions to state the transformation groups he discussed in his paper, analytic expressions will be used for explanation below. According to his geometric concept, projective geometry (such as two-dimensional) is the study of invariants and transformation forms under the transformation group from a point on a plane to a point on another plane or to a point on the same plane (direct transformation) For example, x1'=a11x1+a12x2+a13x3 (homogeneous coordinates) or x'=(a11x+a12y+a13)/(a31x+a32y+a33) (non-homogeneous coordinates, y' is to change a1i to a2i), coefficient The determinant must not be 0. The invariants under the projective transformation group include: linearity, maximum linearity, intersection ratio, harmonic set, remaining conic section unchanged, etc.

A subset of the photographic group is a family of affine transformations. This subgroup is defined as follows: Suppose any straight line l∞ is fixed on the projective plane, and the point on l∞ is called an ideal point or an infinity point , l∞ is called an infinite straight line, and other points on the projective plane are called ordinary points. The affine group of direct transformation is the subgroup of the photographic group that makes l∞ invariant (but the points on the line do not need to remain unchanged). Affine geometry is the properties and relationships that are invariant under affine transformation. Two-dimensional homogeneous coordinates The affine transformation is expressed algebraically as the above equation, but where a31=a32=0, and has the same determinant conditions. The affine transformation of non-homogeneous coordinates is x'=a11x+a12y+a13, y'=a21x+a22y+a23, and the cofactor of a33 is not 0. Under the affine transformation, the straight line becomes a straight line, and the parallel straight line becomes to parallel lines, however the length and size change. Affine geometry was first noticed by Euler and later pointed out by M?bius in the book "Calculation of Barycentric Coordinates". It is useful in the study of deformation mechanics.

Any metric geometric group is the same as the affine group except that the determinant value above must be ±1. The first metric geometry is Euclidean geometry. To define this geometric group, start from l∞ and assume that there is a fixed involution transformation on l∞. It is required that this involution transformation has no real two points and is virtual at ∞. The dots serve as secondary points. Consider the projective transformation to make l∞ unchanged, and change any point of the involution into any point of the involution, that is, each virtual circle point becomes itself, the algebra of the non-homogeneous two-dimensional coordinates of these transformations of the Euclidean group The expression is x'=ρ(xcosθ-ysinθ+α), y'=ρ(xsinθ-ycosθ+β), ρ=±1. What remains unchanged is the length, the size of the angle, and the size and shape of any figure.

In the terms of this classification, Euclidean geometry is a set of invariants under rotation, translation, and reflection transformations. To obtain invariants about similar shapes, the subgroup of the affine group we introduce is called the parabolic metric group, which is defined as a family of projective transformations that make the involution on l∞ invariant, that is, each pair of corresponding points transforms into the corresponding other A pair of dots. The transformation of the parabolic metric group of non-homogeneous coordinates has the form x'=ax-by+c,y'=bex+aey+d, . These transformations keep the size of the angle constant.