Parametric Equation

A parametric equation is a reformulation of an equations in terms of time, such that there is now a direction associated with movement along the equation.

Description

A line can often be described with a single equation. This below line, however, requires two equation to be expressed: y=√(x+1)+1 and y=-√(x+1)+1.

This can be addressed by reformulating the equation in terms of time t: y=f(t) and x=g(t). In this specific example, the parametric equations are y=t and x=(t-1)²-1.

This has the side effect of associating a direction with movement along the line. At t=0, the solution is (0,0). At t=1, the solution is (-1,1).

Especially with multiple variables, parametric equations are sometimes re-expressed in several ways:

• This triplet...
  • x = 1+2t
  • y = 3-t
  • z = 5t
• ...is equivalent to this symmetric equation...
  • (x-1)/2 = (y-3)/-1 = z/5
• ...and is also equivalent to this vector form...

  • [1, 3, 0] + t[2, -1, 5] = [1, 3, 0] + [2t, -t, 5t] = [1+2t, 3-t, 5t]

• ...and can therefore be expressed as a vector-valued function...

  • r(t) = f(t)i + g(t)j + h(t)k = (1+2t)i + (3-t)j + (5t)k

Usage

A parametric equation is useful for characterizing intersections, e.g. between the parameterized line and a given point or plane.

There is also a straightforward method for calculating arc length of a parametric equation. Given a vector-valued function r(t) = f(t)i + g(t)j + h(t)k:

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