What is the fundamental geometric principle that underpins the Pen Tool's ability to create smooth, curved lines, and how does manipulating its control points achieve this?

The fundamental geometric principle that underpins the Pen Tool's ability to create smooth, curved lines is the mathematical concept of Bézier curves. A Bézier curve is a type of parametric curve defined by a set of anchor points and control points. Anchor points determine the exact positions the curve passes through, acting as the vertices of the curve. Control points, on the other hand, influence the shape and curvature of the line segment *between* anchor points without the curve necessarily passing through them. Each control point is attached to an anchor point by an invisible line. Th....