Pythagorean theorems on rectangular tetrahedron

Pythagorean Property and Best-Proximity Point Theorems

The aim of this paper is to prove the existence and convergence theorems for cyclic contractions. We introduce a notion called proximally complete pair (A,B) on a metric space, which unify the earlier notions that are used to prove the existence of a best proximity point for a cyclic contraction. By observing geometrical properties on a Hilbert space, we introduce Pythagorean property and use t...

The Pythagorean Theorem and Area: Postulates into Theorems

Weierstrass-type approximation theorems with Pythagorean hodograph curves

We prove the Weierstrass-type approximation theorem that states every C1 curve in the 2-dimensional or 3-dimensional Euclidean space or in the 3-dimensional Minkowski space can be uniformly approximated by Pythagorean hodograph curves in the corresponding space. This abundance of PH curves is another theoretical confirmation of the usefulness and the versatility of the PH curves. We also addres...

Proper forcing and rectangular Ramsey theorems

I prove forcing preservation theorems for products of definable partial orders preserving cofinality of the meager or null ideal. Rectangular Ramsey theorems for the related ideals follow from the proofs.

Fast Tetrahedron-Tetrahedron Overlap Algorithm

We present an algorithm to test two tetrahedra for overlap. The algorithm is based on a dimension reduction technique that allows to apply the Separating Axis Theorem avoiding part of the computation needed to perform the Separating Axis Test. Source code is available online.