Motion described by a parameter

A polar curve r = f(θ) is a parametric curve x = f(θ)cos θ, y = f(θ)sin θ with θ as the parameter, so polar points are positions given by a parameter.

The identity cos squared + sin squared = 1 turns x = cos t, y = sin t into the equation of a circle.

A rider on a wheel has a height that is a sinusoid of time. Pairing it with a second sinusoid for horizontal position describes the whole circular path as a function of one parameter.

The point a fraction t of the way from A to B is A + t(B - A), which is a parametric line.

The path of an object with constant horizontal speed and constant vertical acceleration is a parabola written with a parameter.

Position after equal steps is a first term plus a number of common differences, as a parametric equation moves at constant velocity.

Position at constant velocity is a linear model in time, which parametric equations write for each coordinate.

A parametric circle is rotated or stretched by applying a matrix to the coordinate pair at each parameter value.

A position (x(t), y(t)) is a vector that changes at a constant velocity vector (a, b), whose length is the speed.

Each coordinate of a circular parametric motion is a sine or cosine of the parameter, with amplitude and period.

A circle or ellipse can be traced by x = h + a cos t, y = k + b sin t, which satisfies the conic's standard equation because cos² t + sin² t = 1.