Problem 71 - Algebra and Trigonometry

Question

What are plane curves and parametric equations?

Step-by-Step Solution

Answer

A plane curve is a curve that lies entirely in one plane and it's visually represented in a 2D space. Parametric equations express quantities as explicit functions of a number of independent variables, called parameters. These equations are used extensively to describe the positional coordinates of a point on a plane curve.

Step 1: Definition of Plane Curves

A plane curve is a curve that lies entirely in one plane. These planes can be the x-y, x-z, y-z for instance. Plane curves are visually represented in a 2-dimensional space.

Step 2: Defintion of Parametric Equations

Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. For instance, in a 2-dimensional space, a parametric equation allows us to represent a curve by a pair of functions \( x=f(t) \) and \( y=g(t) \) where \( x \) and \( y \) are coordinates on the curve and \( t \) is the parameter.

Step 3: Connection between Plane Curves and Parametric Equations

Parametric equations are used to specify the positional coordinates of a point on a plane curve in terms of a single parameter. This makes it possible to 'draw' the curve in terms of this parameter, which typically represents time.