# Question about the definition of involute of a plane curve

I'm studying about involute of a plane curve from here and there's a small point that is really bothering me and I can not understand it.

Assuming curve is parameterized by arc length, the involute is defined as $$\gamma(t) = \beta(t) - t \beta'(t)$$

Why is there a negative sign? Like it says in the link I added, the bob's position is at distance $$t$$ in direction $$-\beta'(t)$$. I can't see how this negative sign has to be there for every curve in general. If the tangent line of $$\beta$$ at point $$t$$ is $$\beta(t) + \lambda \beta'(t)$$, then starting from $$\beta(t)$$ I can move along the line is either direction depending on $$\lambda$$. Why must it be negative in the case of the involute?

• $s = t$ in your linked page. Why not here as well? Cheers! – Robert Lewis May 9 at 0:47
• Edited it, thanks! – 9Sp May 9 at 0:56

The model for the involute is this: Take a circle, with scotch tape wrapped around it. Start to peel the scotch tape off and follow the point $$P$$ at the end of the tape. As you go counterclockwise around the circle, the tape is tangent, but the line segment from the point of contact to $$P$$ goes in the direction opposite to the (counterclockwise) tangent vector to the circle.
• Thank you! so if I understood correctly, the tangent vector is pointing in the direction of increasing $t$, but in order to actually draw the involute, we start from from point of contact going to $P$ in the opposide direction – 9Sp May 9 at 1:10