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Transcript
GeoGebra Konferencia
Budapest, január 2014
GEOGEBRA FOR SIMILARITIES AND DIFFERENCES
BETWEEN DIFFERENT GEOMETRIES
GeoGebra Konferencia Budapest, január 2014
• GeoGebra software can be used for constructing figures and studying
and illustrating different properties. Using GeoGebra software we can
- construct hyperbolic figures
- perform measurements of lengths and of angles by using the
respective GeoGebra virtual tools to investigate if the properties
satisfied by some particular figures of Euclidean geometry are satisfied
by the homologues figures in hyperbolic geometry.
For example, consider the case of the hyperbolic triangle constructed
with GeoGebra software as displayed in following figure.
GeoGebra Konferencia Budapest, január 2014
• For the Euclidean triangle the sum of the angles is 180°. As to
hyperbolic triangle (triangle BCE in the following figure with blue
wider arches) the angles are defined by the tangents to the arches
passing through the vertices. The measurements are done using the
angle virtual tool and, for clarity reasons, in figure are shown the
values of the obtuse angles defined by the two respective tangents.
The measurements show that their sum is:
• (180° - 163.87°) + (180° - 176.11°) + (180° - 178.59°) = 31.43°
• The result: the sum of its angular measures is less than 180° .
Sum of the interior angles of the hyperbolic
triangle
GeoGebra Konferencia Budapest, január 2014
• In the next figure is shown the hyperbolic triangle with sides DG, DF
and GF that are arches with centre C, E and F respectively. The
tangents are not shown for reasons of clarity of the figure. In figure
are shown the exterior angles (denoted by 𝛼, 𝛽 𝑎𝑛𝑑 𝛾) of the angles
between two tangents passing through the vertices.
• The sum of the interior angles of the hyperbolic triangle can be
computed using the expression: 𝛿 = 540° − (𝛼 + 𝛽 + 𝛾). The
address for the applet created for purpose of demonstration of the
property with respect to the sum of the angles is:
• http://www.geogebratube.org/material/show/id/61143
Applet of observing the sum of the angles
Using the mouse or the touch pad, displace each
centre of the arch to observe the property.
GeoGebra Konferencia Budapest, január 2014
• Many other properties are not preserved such as those related to the
distance. Also, in Euclidean geometry the sum of the angles of the
triangle is constant. Not so in hyperbolic geometry. Play with the
above applet and observe the sum.
• The definitions and perceptions of vertical angles and supplementary
angles in hyperbolic geometry are similar to those of Euclidean
geometry. There are properties of Euclidean geometry that are
preserved in hyperbolic geometry. Look at the next figure: it is
obvious that the vertical angles are equal and the sum of
supplementary angles is 180°. This derives from the fact that the
angle between two intersecting arches is defined by the angle formed
by the two tangents passing through the intersecting point.
The property of vertical and supplementary
angles
GeoGebra Konferencia Budapest, január 2014
• Is Pythagorean Theorem true in hyperbolic geometry? The hyperbolic
triangle with vertices B, C, E (light brown colour, in the next figure) is
selected in a special way in order to make easy the calculation of the
lengths of its arches. The hyperbolic triangle BCE (the one consisted of and
bounded by the three arches) is got by the construction of three arches of
angular measures 90° each one of them in such way that the angle with
vertex B, of the hyperbolic triangle, be 90°. The radii of the arches are 2, 4
and 6 units, respectively.
• Looking at the notes in figure can be easy checked that the Pythagorean
Theorem does not hold true for the hyperbolic triangle. The lengths of the
arches are , 2𝜋 𝑎𝑛𝑑 3𝜋 , respectively but:
• 𝜋 2 + 2𝜋 2 ≠ 3𝜋 2
Right-angle hyperbolic triangle
GeoGebra Konferencia Budapest, január 2014
• Are there other formulas used in Euclidean geometry that are valid in
hyperbolic geometry? Consider the formula for the area of a triangle
1
which in Euclidean geometry is: 𝑎 ∆ = 𝑏 ∙ ℎ. For simplicity, we
2
consider the hyperbolic triangle BCE of the above figure and, which is
a right-angle triangle. Its hypotenuse is represented by the arch CE,
whereas arch BL serves as its height. The centre of the arch is point K
and it is chosen in such a way that this arch be perpendicular to arch
CE. The difference between the length of arch BL and the respective
chord BL (1.53) is very small or negligible. Assume now that the
length of arch BL is 1.56. Calculate the area of the hyperbolic triangle
BCE in two ways:
GeoGebra Konferencia Budapest, január 2014
• Formally applying the formula : a(BCE) = 1/2·[arch(CE)] [arch(BL)] =
1/2·3π·1.56 = 2.34·π ≈ 7.35
• Doing more precise calculations in Euclidean geometry:
a(BCE) = 1/4·π·36 – 1/4·π·4 - 1/4·π·16 - 2·4 = 4·π – 8 ≈ 4.56 ˂ 7.35
• If we calculate the area of hyperbolic triangle BCE using the two
perpendicular arches that represent the catheti of the right-angle
triangle in Euclidean geometry we have:
• a(BCE) = 1/2·[arch(BE)] [arch(BC)] = 1/2·π·2π = π2 ≈ 9.86 ≠ 7.35
GeoGebra Konferencia Budapest, január 2014
• Conclusion
• Properties of Euclidean geometry that are preserved in hyperbolic geometry are
mainly those linked with the angles. But: in Euclidean geometry the sum of the
angles of the triangle is constant. Not so in hyperbolic geometry.
• Many other properties are not preserved such as those related to the distance.
Formulas used and valid in Euclidean geometry are not valid in hyperbolic
geometry. The Euclidean geometry satisfies the independence property w.r.t. the
calculations of the area of the triangle (the area does not depend on the option:
which side is taken as a base and the height perpendicular to the base
constructed from the vertex in front of it). In hyperbolic geometry the products of
the lengths of the arches taken as bases, with the lengths of the respective
arches, taken as heights, give different results. That is, the hyperbolic geometry
does not satisfy the independence property w.r.t. the calculation of the area of
the triangle in different ways. The same result is true in the case of calculating the
area of a hyperbolic polygon.
See you next GeoGebra Conference!
Many thanks for your attention:
Pellumb Kllogjeri and Qamil Kllogjeri
ALBANIA