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Parametric Equations and Polar Coordinates

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Parametric Equations and Polar Coordinates

Key Terms

Limacon - A polar equation of the form r = a + b sin(θ) or r = a + b cos(θ) , where a, b≠ 0 .

Logarithmic Spiral - A polar equation of the form r = ab ^θ .

Orientation - The direction of a plane curve as the parameter increases.

Parameter - A third variable (often time) which determines the values of x and y in parametric equations.

Parametric Equations - Two equations of the form x = f (t) and y = g(t) , which specify the location of a point according to the variable t .

Plane Curve - The set of all points (f (t), g(t)) , where x = f (t) and y = g(t) are parametric equations.

Polar Axis - The ray whose endpoint is the pole and which is the initial side of any angle measure in the polar plane.

Polar Coordinate System - The system in which a point in the plane is specified according to an ordered pair (r, θ) in which r is a length and θ is an angle. The length r refers to the distance from the point to a fixed origin, called the pole. The angle θ is the angle whose initial side is a fixed ray (the polar axis) and whose terminal side contains the point. Under these circumstances, the point (r, θ) is expressed in polar coordinates.

Pole - The fixed point in the polar coordinate system from which every point is r units away.

Rectangular Coordinate System - The coordinate system in which every point is specified by exactly one ordered pair (x, y) . Here x is the distance between the point and a fixed line (the y -axis) and y is the distance between the point and a line fixed perpendicular to the other line (this line is the x -axis). The perpendicular lines are the axes, and the point (x, y) is expressed in rectangular coordinates.