Self-determination theory (SDT) is a theory of motivation (Deci and Ryan, 1985, Deci and Ryan, 2000), a theory about external and internal factors and their interrelationships that drive people to act. External factors could be reward or punishment, internal factors fun or curiosity. A detailed review of the theory can be found in Ünlü (2023), this reference is subsequently referred to as MSDT1, which is the first of the two papers on the mathematical underpinnings of the theory. This follow-up second paper is referred to as MSDT2.

An informal representation of SDT can be found in Table 1 (for details on the concepts contained therein, see MSDT1).

Briefly, intrinsic motivation or intrinsic regulation is behavior characterized by volitional internal factors, such as genuine interest. Extrinsic motivation can be the regulation types of external regulation, introjected regulation, and identified regulation. External regulation describes behavior that is dictated by fully external factors, such as money. Introjected regulation includes behavior enacted by external societal expectations, such as shame. Identified regulation occurs if a person identifies with the value of an activity, but not enjoys it. Finally, amotivation or non-regulation is the lack of any form of regulated (intrinsic and extrinsic) motivation.

These regulations have their different internalities (locus of causality), which are fully external and fully internal for external regulation and intrinsic regulation, respectively, external and internal for the intermediate regulations, and impersonal for non-regulation. With more internalized regulations, the corresponding behaviors are more self-determined.

In MSDT1, in the real representation, we have redefined and unified these motivations based on their decomposability into internal motivation, external motivation, and amotivation. Thereby, we have embedded the common motivations into a broader continuous spectrum of graded motivations. In this paper, similar results will be derived, but in the more general affine space representation of the theory.

We recapitulate the main results obtained in MSDT1 later in a separate section of this paper, which will discuss the connections between this work and MSDT1. What has been discussed in greater length in MSDT1 were also the limitations of our approach and the motivations for it. This will not be repeated in this paper.

The real representation of the theory in the lower half of the unit square (Fig. 5 in MSDT1) can be generalized. We present a generalization in real affine spaces with underlying vector spaces over the field of real numbers (or reals) R. More abstract fields with successively stronger restricting properties such as ordered, Archimedean ordered, or complete Archimedean ordered fields could be considered. Not all results of the simpler representation can then be retained. For example, in Section 4.2 in MSDT1, to derive the lattice-theoretic properties, we have relied on the assumption that the supremum and infimum of any non-empty and, respectively, bounded from above and below subset of R exist. This may not be satisfied anymore in more general fields.

This paper is rather of theoretical interest. The representation in MSDT1, which is a special case, is a direct and complete conceptualization of SDT, a simplest choice of representation (cf. Section 5). The following concepts about affine spaces are standard in geometry (e.g., Gallier, 2001, Tarrida, 2011).

This paper is structured as follows. In Section 2, we review the necessary basic concepts of affine geometry. In Sections 3 Self-determination theory in affine space, 4 Polar coordinates in affine motivation space, we generalize the model in reals to represent SDT in more abstract affine spaces. This includes, in Section 3, the primitives for the representation and the motivation 2-simplex, regulated and graded motivation, and self-determination, and in Section 4, polar coordinates in (Euclidean) affine motivation spaces, and the study of self-determination on radial and angular line segments. In Section 5, we prove the distributivity of the lattice of general self-determination, then discuss the special case real representation of MSDT1, and show that different choices of primitives for a description of the theory imply equivalent representations. In particular, the representation in reals is seen to be a simplest canonical choice, in the sense that all resulting motivation simplexes are naturally isomorphic to it. In Section 6, we summarize this paper and conclude with a resume.