T-Angles
This activity belongs to the GeoGebra book GeoGebra Principia.
In the unit T-circle, we can define the T-radian exactly the same way we define an E-radian in the unit E-circle. To T-measure an angle, it's sufficient to measure the T-length of the corresponding (straight) arc on the unit T-circle. A T-circle has 8 T-radians.
Perpendicularity and parallelism are preserved under rotations, but, in general, T-distances are not invariant with respect to E-rotations... nor with respect to T-rotations! In fact, one of the peculiarities of T-distance is that it is sensitive to the orientation of lines: a segment, when T-rotated, no longer measures the same. The same happens with angles.
![[i]Cartoon of Mafalda, by Quino
"It's quite a puzzle, isn't it? How on earth
does time manage to round the corners on square clocks?"[/i]](https://www.geogebra.org/resource/zzvd6yyu/XZQxWjJmg4yMnnsM/material-zzvd6yyu.png)
The sum of the angles in any T-triangle is 4 T-radians. A T-triangle can be equilateral or equiangular, but it can never be regular.
Any E-square is also a T-square. But because the taxicab distance is not uniform in every direction, these two T-squares have the same perimeter (though not the same area):
![[i][i] The square on the left is also a T-circle.
The one on the right is not.[/i][/i]](https://www.geogebra.org/resource/bzkm6nm6/PtUU3jM8glr6kdsA/material-bzkm6nm6.png)
Trigonometric T-functions are much simpler than their Euclidean counterparts. For example, the T-sine function is not only non-transcendent but also piecewise linear. The T-tangent function is composed by piecewise E-hyperbolas.
- Note: One possible expression for the T-sine function is: tsin(x) = 1 – 2 |1/2 - x/4 + 2 floor(1/4 + x/8)|. Thus, the T-cosine function can be defined as tcos(x) = tsin(x+2). The T-tangent function is a piecewise homographic function
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Author of the construction of GeoGebra: Rafael Losada.