Parametric equation

A parametric equation defines variables as functions of one or more independent parameters. It can describe curves and surfaces, including shapes that are awkward to express with a single Cartesian equation.

Parameterization: A representation of a geometric object or relationship using one or more parameters. It is the general method of expressing a curve or surface through parameter-dependent coordinates.
Cartesian coordinate system: A coordinate system that locates points by signed distances along perpendicular axes. Parametric equations assign Cartesian coordinates separately as functions of a parameter.
Projectile motion: The motion of an object moving under gravity after being launched. Time parameterizes horizontal and vertical position, producing a parabolic trajectory.
Cartesian equation: An equation relating coordinate variables directly, without an explicit parameter. It describes a locus directly, while a parametric equation generates points through parameter values.
Parametric curve: A curve whose coordinates are functions of a parameter. It is the basic geometric object traced by a pair or tuple of parametric equations.
Independent variable: A variable whose values are chosen or specified when describing a relationship. A parameter plays this role while determining the dependent coordinates.
Circular motion: Motion along a circular path. Sine and cosine of time give the changing coordinates of a rotating point.
Polar coordinate system: A coordinate system locating points by distance from an origin and angle from a reference direction. Polar coordinates are a specialized alternative that can simplify rotationally symmetric curves.
Tangent vector: A vector indicating the instantaneous direction of a curve at a point. Differentiating the coordinate functions gives the curve’s tangent vector.
Function: A rule assigning each allowed input exactly one output. Each coordinate in a parametric equation is typically a function of the parameter.

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