- #1

Cauchy1789

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**The behavior of the tracetrix(need help to verify properties) :(**

## Homework Statement

Howdy

Given the parametric function [tex]\beta(t) = (sin(t), cos(t) + ln(tan(t/2)) [/tex]

where t is the angle between the tangent vector and the y-axis and where

[tex]\beta: (o,\pi) \rightarrow \mathbb{R}^2 [/tex] then show the following two properties of the tracetrix are true.

1) that's its a regular parameterized curve differentiable curve except at

t = [tex]\frac{\pi}{2} [/tex]

2) That the length of the line segment of tangent in a tangent point of the curve, then intersecting with y-axis, this length always will be 1.

## The Attempt at a Solution

1) According to the definition from my textbook a curve is parameterized curve is said to be regular

*if [tex] \beta'(t) \neq 0[/tex] for all t in I*

thusly since my tracetrix has the [tex]\beta'(t) = (cos(t), -sin(t) + \frac{1}{2 \cdot sin(t/2) \cdot cos(t/2)})[/tex] and thusly [tex]\beta'(\pi/2) = 0,0[/tex]. Therefore the derivative at t = [tex]\frac{\pi}{2} [/tex] is zero and according to the definition the curve is not regular at that point.

But what about the end points? From what I can see they don't have a corresponding (x,y) and thusly they tend to [tex]\pm \infty[/tex]. So what about the definition applying in those end points?

2) From I can understand here the point is that do to the geometrical behavior of the tracetrix then tangent is asymptotical to [tex]\beta(t)[/tex](do I need to show this) and thusly the distance of every tangent point on the interval I (except t = pi/2) will always be the same.

(A hint/idear on howto show this would be very much appricated :D)

Sincerely

Cauchy1789

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