AbstractWe present an axiom system in terms of collinearity and congruence for plane geometry independent of assumptions about parallels. The line reflections and motions of corresponding geometries are studied, and it is shown that the axiom system characterizes the non-elliptic metric planes in the sense of Bachmann with midpoints. If the hyperbolic axiom is added, we show that the familiar algebraic models of hyperbolic planes result.

AbstractWe prove that this formula characterizes the square matrices over commutative rings for which all $$2\times 2$$ 2 × 2 minors equal zero.

AbstractFor a given plane curve, consider a one-parameter family of curves consisting of those points at which two support lines to the initial curve intersect at a constant angle. Such curves are well known in differential and convex geometry and called isoptics. In this paper, we describe parametrizations of orthogonal trajectories to isoptics of ovals. We show that such parametrizations can be obtained using solutions to a specific Cauchy problem constructed from the parametrizations of the oval and its isoptics. Moreover, we provide analytical and numerical examples of orthogonal trajectories to isoptics of some ovals.

AbstractIf we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector $$\tau $$ τ , and finally join corresponding points of the two copies, then we obtain a framework with ‘extrusion’ symmetry in the direction of $$\tau $$ τ . This process may be repeated t times to obtain a framework whose underlying graph has $$\mathbb {Z}_2^t$$ Z 2 t as a subgroup of its automorphism group and which has ‘t-fold extrusion’ symmetry. Extruding a framework is a widely used technique in CAD for generating a 3D model from an initial 2D sketch, and hence it is important to understand the flexibility of extrusion-symmetric frameworks. Using group representation theory, we show that while t-fold extrusion symmetry is not a point-group symmetry, the rigidity matrix of a framework with t-fold extrusion symmetry can still be transformed into a block-decomposed form in the analogous way as for point-group symmetric frameworks. This allows us to establish Fowler-Guest-type character counts to analyse the mobility of such frameworks. We show that this entire theory also extends to the more general point-hyperplane frameworks with t-fold extrusion symmetry. Moreover, we show that under suitable regularity conditions the infinitesimal flexes we detect with our symmetry-adapted counts extend to finite (continuous) motions. Finally, we establish an algorithm that checks for finite motions via linearly displacing framework points along velocity vectors of infinitesimal motions.