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. The direction and distance of this invisible line from the anchor point dictate how the curve bends away from or towards that anchor point. When you manipulate a control point, you are essentially changing the tangent of the curve at the connected anchor point. A tangent is a straight line that touches a curve at a single point and indicates the direction of the curve at that exact spot. By adjusting the position of the control points, you precisely control the slope and curvature of the Bézier curve. For instance, if you drag a control point directly away from the previous anchor point, the curve will smoothly extend in that direction. If you pull a control point towards the next anchor point, the curve will become sharper or more pointed in that area. A Bézier curve segment is defined by two anchor points and two control points. The curve starts at the first anchor point and ends at the second anchor point. The first control point, associated with the first anchor point, dictates the curve's direction as it leaves the first anchor point. The second control point, associated with the second anchor point, dictates the curve's direction as it arrives at the second anchor point. The curve itself is pulled towards its control points, creating a smooth transition between anchor points. This mathematical foundation allows for precise, predictable, and infinitely scalable curves.