Meaning of distance between the two lines

Question

I am reading this sentence that I can't understand very well

For a polygon $C$ that is convex hull of a set of points, $width_{\theta}(C), $ where $ 0 \le\theta<\pi$, denotes the width of $C $ in direction $\theta$ that is $width_{\theta}(C)$ is the distance between the two tangent lines of $C$ making an angle $\theta + \pi/2$ with the positive x-axis.

I can't understand what the distance between the two tangent lines of $C$ means. For a given $C$, and set of points in the following image

and $\theta=\pi/4$, two tangent lines making an angle $3\pi/4$ to the positive x-axis is almost this

But I don't know the author means by the distance between the two tangent lines of $C$

Thanks.

Answer 1

Since both lines form the same angle with a fixed line, they're parallel. The distance between the lines is then just the minimum distance between a point on one line and a point on the other, which you can find by measuring the distance along a line perpendicular to the lines. (If they weren't parallel, then of course the distance would just be zero since they would intersect somewhere.)

Answer 2

The distance between two lines is usually the minimum distance between two points on either line. $$ d(s, t) = \min_{x\in s, y\in t}d(x, y) $$ In your example the tangents cut the $x$-axis in the same angle $\theta +\pi/2$, so they are paralell. If the intersections are $\Delta x$ apart, you have $$ \cos(\theta) = \frac{\Delta x}{d} $$