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?

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.