Final answer:
The string, pins, and pencil method works for drawing an ellipse because as the pencil moves around the pins, the length of the string, which determines the sum of the distances from the foci to any point on the curve, stays constant. This constant distance results in an elliptical shape, accurately fulfilling the geometric definition of an ellipse.
Step-by-step explanation:
The method of drawing an ellipse using string, pins, and pencil is based on the geometric definition of an ellipse. An ellipse is a closed curve where the sum of the distances from any point on the curve to two fixed points, known as foci, is a constant.
This property is what allows us to draw an accurate ellipse using this method. By fixing two pins in place to represent the foci and attaching a string to them, we create a loop. Placing a pencil inside the loop and pulling the string taut creates a constant total distance from the pencil to both pins as we trace the curve. This constant sum of the distances ensures that the resulting shape is an ellipse.
It is important to note that this method does not involve principles such as constant angular momentum, Kepler's laws, gravitational forces, or centripetal acceleration, which are related to physics, not the geometry of drawing an ellipse. Instead, the elliptical shape emerges because the length of the string, representing the sum of the distances from the foci to any point on the ellipse, remains unchanged as the pencil moves, maintaining the geometric definition of an ellipse.