Geometry of a cubic form

In this paper, we construct a geometric scheme based on a group approach. According to this approach, in addition to a set of figures, a certain group of transformations (“motions”) is introduced; this group determines the content of the geometry. Namely, within the framework of the corresponding geometric scheme, properties of figures that are invariant under the actions of the group are examined. To specify the transformation group, as a rule, a set of transformation that preserves some “fundamental object” is chosen. For example, the group of motions (i.e., transformations that preserve the distance between points) is used for constructing Euclidean geometry, the group of affine transformations (i.e., transformations that preserve the simple relation of three points) is used for affine geometry, the group of projective transformations (i.e., transformations that preserve the double (or complex) ratio of four points) is used for projective geometry, and so on. In this paper, a certain cubic form serves as the “fundamental object” of the “motion group.”