Hyperbolic Triangles | |
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are triangles having -3- sides & -3- angles in the hyperbolic plane analogous to Euclidean triangles. |
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the relations among the sides and / or angles are governed by those of spherical trigonometry. |
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Length of the sides is expressed in terms of a unit called R where R= 1/√(-K)^{1/2} , K being Gaussian Curvature of the plane. |
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Each angle of the hyperbolic triangle, say ,A, B, C is slightly less than corresponding angle say A',B',C' in Euclidean triangle. Hence sum of the angles of a hyperbolic triangle is less than the sum of the straight angles in analogous Euclidean triangle & consequently less than π . |
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Mathematically, A+B+C<π and also A+B+C< A'+B'+C'. Difference (A'+B'+C')-(A+B+C)=Defect |
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Area of hyperbolic triangle is given by πR^{2}>area where area=defect*R^{2} |
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In trigonometric relations involving the sides-->hyperbolic functions are applied. angles-->standard trigonometric functions are applied. Further sinA/sinha =sinB/sinhb=sinC/sinhc |
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If C is the right angle in a hyperbolic right-angled
triangle, and a is the side opposite to angle A, c is the side opposite to
angle C and b is the side opposite to angle B, then sinA= sinha/sinhc cosA=tanhb/tanhc tanA=tanha/sinhb |
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The sides of a hyperbolic triangle can become as large as possible but the area of the triangle is always less than π |