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Many real-world motions, like a plane’s flight path, involve movement in three-dimensional space. These paths can be modeled using space curves.

A space curve is the set of all points traced by a moving particle as the parameter t varies over an interval. Its position is given by a vector function r(t), where x, y, and z are differentiable functions of t.

A common example is a 3D helix, a smooth curve that spirals upward. It can be described by a vector function with cos(t) in the x-direction, sin(t) in the y-direction, and t in the z-direction.

At t equals zero, the vector simplifies to one, zero, zero, which is the starting point of the curve.

As t increases, the x and y components trace a circle when viewed from above because the cosine of t and the sine of t define circular motion parametrically. At the same time, the z-value increases steadily, causing the curve to rise.

Together, these components form a spiral that wraps around a cylinder.

This shows how a space curve can model a plane’s position as it moves through three-dimensional space.