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| Given a vector a⃗ and a starting point A,,0,, (e.g., the origin), a parametric equation can be expressed as ''A,,t,, = A,,0,, + a⃗t''. The same can be said when given two points in ''R^3^'' as ''A,,0,,'' and ''A,,1,,''. These trivially yield a vector a⃗ as ''[A,,1x,,-A,,0x,, A,,1y,,-A,,0y,, A,,1z,,-A,,0z,,]''. |
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.
Contents
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).
Given a vector a⃗ and a starting point A0 (e.g., the origin), a parametric equation can be expressed as At = A0 + a⃗t. The same can be said when given two points in R3 as A0 and A1. These trivially yield a vector a⃗ as [A1x-A0x A1y-A0y A1z-A0z].