Znanstveno-stručni časopis
Hrvatskog društva za geometriju i grafiku

Scientific and Professional Journal
of the Croatian Society for Geometry and Graphics

Temel Ermiş, Özcan Gelişgen, Rustem Kaya(termis@ogu.edu.tr, gelisgen@ogu.tr, rkaya@ogu.edu.tr)

On Taxicab Incircle and Circumcircle of a Triangle

In this work, we study existence of taxicab incircle and cir-
cumcircle of a triangle in the taxicab plane and give the
functional relationship between them in terms of slope of
sides of the triangle. Finally, we show that the point of
intersection of taxicab inside angle bisectors of a triangle
is the center of taxicab incircle of the triangle.

Key words: taxicab distance, taxicab circle, taxicab incircle, taxicab circumcircle, taxicab plane and taxicab geometry

Article in PDF.

Ana Sliepčević, Ivana Božić (anas@grad.hr, ivana.bozic@tvz.hr)

Steiner Curve in a Pencil of Parabolas

Using the facts from the theory of conics, two theorems that are analogous to the theorems in triangle geometry are proved. If the pencil of parabolas is given by three lines a, b, c, it is proved that, the vertex tangents of all the parabolas in the pencil, envelop the Steiner deltoid curve δ, and the axes of all parabolas in the same pencil envelop further deltoid curve α. Furthermore, the deltoid curves are homeothetic. It is proved that all the vertices in the same pencil of parabolas are located at the 4th degree curve. The above mentioned curves are constructed and treated by synthetic methods.

Key words: Steiner deltoid curve, Wallace-Simson line,
pencil of parabolas, vertex tangent

Article in PDF.

Hiroshi Okumura (hiroshiokmr@gmail.com)

Ubiquitous Archimedean Circles of the Collinear Arbelos

We generalize the arbelos and its Archimedean circles, and
show the existence of the generalized Archimedean circles
which cover the plane.

Key words: arbelos, collinear arbelos, ubiquitous
Archimedean circles

Article in PDF.

Nikolina Kovačević, Ana Sliepčević (nkovacevic@rgn.hr, anas@grad.hr)

On the Certain Families of Triangles

In the present paper, we study a set T={T(r,d) : d ∈R}
of the certain one-parameter families of triangles. The
traces of some triangle points within the set are analyzed
and described.

Key words: tangential triangle, hyperosculating circle,
pencil of conics

Article in PDF.


Marijana Babić, Srđan Vukmirović (marijana@matf.bg.ac.rs, vsrdjan@matf.bg.ac.rs)

Central Projection of Hyperbolic Space onto a Horosphere

Horosphere is surface in hyperbolic space that is isometric to the Euclidean plane. In order to correctly visualize hyperbolic space we embed flat computer screen as horosphere and investigate geometry of central projection of hyperbolic space onto horosphere. We also discuss realization of hyperbolic isometries. Corresponding algorithms are implemented in Mathematica package L3toHorospere.
We briefly present the package and obtain some interesting
pictures of hyperbolic polyhedra.

Key words: hyperbolic space, horosphere, central projection

Article in PDF.




János Pallagi, Benedek Schultz, Jenö Szirmai (jpallagi@math.bme.hu, schultz.benedek@gmail.com, szirmai@math.bme.hu)

On Regular Square Prism Tilings in ^SL2R Space (tilda)

In [9] and [10] we have studied the regular prisms and prism
tilings and their geodesic ball packings in SL2R space (tilda) that is one among the eight Thurston geometries. This geometry can be derived from the 3-dimensional Lie group of all 2×2 real matrices with determinant one.
In this paper we consider the regular infinite and bounded
square prism tilings whose existence was proved in [9]. We
determine the data of the above tilings and visualize them
in the hyperboloid model of SL2R space (tilda).
We use for the computations and visualization of the SL2R space (tilda) space its projective model introduced by E. Molnár.

Key words: Thurston geometries, SL2R space geometry, tiling, prism tiling

Article in PDF.


Norman John Wildberger, Ali Alkhaldi (n.wildberger@unsw.edu.au, aalkaldy@hotmail.com)

Universal Hyperbolic Geometry IV: Sydpoints and Twin Circumcircles

We introduce the new notion of sydpoints into projective triangle geometry with respect to a general bilinear form. These are analogs of midpoints, and allow us to extend hyperbolic triangle geometry to non-classical triangles with points inside and outside of the null conic. Surprising analogs of ircumcircles may be defined, involving the appearance of pairs of twin circles, yielding in general eight circles with interesting intersection properties.

Key words: universal hyperbolic geometry, triangle geo- metry, projective geometry, bilinear form, sydpoints, twin circumcircles

Article in PDF.



Nguyen Le, Norman John Wildberger (n.h.le@unsw.edu.au, n.wildberger@unsw.edu.au)

AUniversal Affine Triangle Geometry and Four-fold Incenter Symmetry

We develop a generalized triangle geometry, using an ar-
bitrary bilinear form in an affine plane over a general field.
By introducing standardized coordinates we find canonical
forms for some basic centers and lines. Strong concurren-
cies formed by quadruples of lines from the Incenter hi-
erarchy are investigated, including joins of corresponding
Incenters, Gergonne, Nagel, Spieker points, Mittenpunkts
and the New points we introduce. The diagrams are taken
from relativistic (green) geometry.

Key words: Triangle geometry, affine geometry, Rational trigonometry, bilinear form, incenter hierarchy, Euler line, Gergonne, Nagel, Mittenpunkt, chromogeometry

Article in PDF.

Tatjana Slijepčević-Manger (tmanger@grad.hr)

The Volume of a Solid of Revolution

In this paper we present classical methods (disk and shell
integration) to compute the volume of a solid of revolu-
tion. We also give a method to compute the volume of
a solid of revolution as a double integral. In the end we
show how Guldin-Pappus' theorem follows from the third

Key words: volume, solid of revolution, disk method, shell method, double integral
Article in PDF.