Let \(M\) be the midpoint of \(A\) and \(A'\).To \(C_\infty\) if and only if the inverted point \(A'\) lies on \(c\). A circle \(c\) through a point \(A\) is perpendicular.Let \(C_\infty\) be the circle defining the Poincaré disc and let \(O\) be its center. Some of the properties discussed on the previous page is The basic construction used when constructing geodesics is inversion in circle. The constructions should pass the dragging test. In the static case, where the points do not move, it is fairly easy to make hyperbolic constructions.
Origin will be used as \(C_\infty \) in all constructions. To make it easy, the unit circle centered at the We will make basic hyperbolic tools as GeoGebra constructions and then use them to make basic hyperbolic ruler and compass constructions. A geodesic has a finite EuclideanĪrc length, but it is an infinitely long hyperbolic line. The Euclidean length of an arc however, does not correspond to the hyperbolic length. When doing hyperbolic geometry using the Poincaré disc model, angles are measured the same way as in Euclidean geometry, but distances are not.Īn angle between two geodesics that intersect, is measured as the angle between the tangent lines at the intersection point. Lines through the point that are parallel to the given hyperbolic line. Given a hyperbolic line and a point not on it, there are infinitely many hyperbolic Two points are on a diameter of \(C_\infty \), that diameter is the geodesic through the points.Ī geodesic is the hyperbolic version of a line in Euclidean geometry. Let's call it the circle at infinity, \(C_\infty \).Ī geodesic through two points is an arc through the points that is perpendicular to \(C_\infty \). When doing hyperbolic geometry using the Poincaré disc model,Īll points are in the Poincaré disc, i.e. The Poincaré disc is the interior of a circle.
Note that several points on a hyperbolic circle have the same hyperbolic distance to the centre of the circle. Load GeoGebra worksheet Use the hyperbolic tools under the leftmost menu item! The hyperbolic distance is created as a variable and can be seen in the algebra view.
0 kommentar(er)
