Кривые, заданные параметрическими уравнениями

When a baseball is hit, ignoring air resistance, its path through the air can be shown by a downward-facing parabola.

Here, x is the horizontal position along this path, and y is the height above the ground.

To describe this movement accurately, the ball’s position is expressed using a third variable: time t.

Instead of a single equation like y = f(x), x and y are each written as separate functions of time, x = f(t) and y = g(t). These are known as parametric equations.

This approach is essential because a spatial function like y = f(x) only shows the curve, not the ball's progress along it. Parametric equations allow an observer to track the ball's exact location at any specific time.

This becomes particularly useful when the ball reaches the same height at two different times—once while rising at t₁ and again while falling at t₂. By defining each point as the ordered pair (f(t), g(t)), the parameter t traces the entire trajectory and simultaneously models the motion.