The theory of plate tectonics postulates that the relative motion between two neighboring plates occurs along three types of boundaries: divergent (spreading center, rift), convergent (subduction, collision), and horizontal (transform). Because the theory assumes rigid behavior of plates, transform plate boundaries must lie along small circles around the pole of rotation of relative motion between two neighboring plates. However, global models of current plate motion (e.g., NUVEL-1A) show that several boundaries with significant horizontal motion (i.e., the Dead Sea Fault and the Eastern Andean Frontal Fault Zone) do not lie along small circles but rather intersect the circles at 45°. The orientation of these faults can be explained by a new theory of intraplate tectonics, which predicts the first-order intraplate stress field in terms of small circles, great circles, and spiral lines that intersect both sets of circles at 45°. According to the theory, these transform faults are situated along the 45°spiral lines and follow the direction of maximum horizontal shear stress. The theory also predicts that the direction of interseismic relative plate motion between the two plates should be oriented at 45°to the transform plate boundary; this prediction can be tested within a few years using space geodesy. The alignment of these faults along spiral lines is explained by the theory's predicted stress field and a plasticity (von Mises) yield stress criterion for earthquake rupture. It is suggested that these faults represent a new class of transform plate boundary between large deformable plates and not between rigid sub-plates, as formerly postulated.