Self-determination theory was proposed by Deci and Ryan, which is a popular theory of motivation. The fundamental concepts are extrinsic and intrinsic motivation, amotivation, their type of regulation, locus of causality, and especially, self-determination. First, we give a geometric description of its concepts for the only regulated case (no amotivation), as the unit 1-simplex, the convex hull of the two unit vectors in the plane. We derive a symmetric definition of self-determination and also include full amotivation, as the origin of the plane. Second, we extend the geometric description to the regulated and unregulated case, based on a more general ternary model, in internal motivation, external motivation, and amotivation. We define gradations of amotivation (and motivation), as 1-simplexes parallel to the unit 1-simplex. The ternary representation implies the three types of strong, weak, and general self-determination, as partial orders on the motivation space. Third, we study the order- and lattice-theoretic and algebraic properties of these self-determinations. In a version of polar coordinates, strong self-determination turns out to be a complete lattice on angular line segments, weak self-determination is a complete lattice on radial line segments, and general self-determination entails a complete lattice on the entire motivation space. In addition, the modified polar coordinates are employed to obtain necessary and sufficient conditions for and characterizations of strong, weak, and general self-determination. We propose measures for the strength of an ordinal dependency in self-determinations, which are partial metrics on the motivation space. The aim of this paper is to lay the mathematical foundations of self-determination motivation theory.