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 curve can often be described with a single equation. The following graph however requires two: 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).

Parametric Curve

If a system can be parameterized to one variable, t, then it describes a curve.

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]

  • Generically this form is expressed as r = [f(t), g(t), h(t)] or r = [x(t), y(t), z(t)].

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

  • (1+2t)i + (3-t)j + (5t)k

  • Generically this form is expressed as r(t) = f(t)i + g(t)j + h(t)k.

A parametric curve is said to be smooth if r'(t) is continuous and exists everywhere. It is said to be regular if there is no point where r'(t) is zero (i.e., all of f'(t), g'(t), and h'(t) are zero).

Some curves are not smooth everywhere. This isn't an issue however, because they generally are piecewise smooth, and integrals can be summed without issue.

Parametric Surfaces

If a system can be parameterized to two variables, u and v, then it describes a surface.

To parameterize the surface described by z = f(x,y), try r(u,v) = [u, v, f(u,v)].

Generically parametric surfaced are expressed either as:

• Vector form: r = [f(u,v), g(u,v), h(u,v)] or r = [x(u,v), y(u,v), z(u,v)]

• Vector-valued function: r(u,v) = f(u,v)i + g(u,v)j + h(u,v)k

A sphere with radius ρ is parameterized into spherical coordinates as r(φ,θ) = [ρcos(θ)sin(φ), ρsin(θ)sin(φ), ρcos(φ)] for 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π. Incidentally, the normal vector to such a sphere is given as [ρ²cos(θ)sin²(φ), -ρ²sin(θ)sin²(φ), ρ²sin(φ)cos(φ)]. Note that in some texts, θ and φ are reversed. Here φ is the polar angle and θ is the azimuthal angle.

A parametric surface is said to be smooth if ∂r/∂u and ∂r/rv are both continuous and both exist everywhere. It is said to be regular if ∂r/∂u × ∂r/rv, i.e., the normal vector as given by the cross product, is never zero. Lastly, a surface is said to be orientable if the normal vector varies continuously.

Orientability can be conceptualized as saying that there are discrete 'upper' and 'lower' faces. For some surfaces, there is a clear 'outer' face; consider a closed surface forming a sphere. For other surfaces, the orientation is interchangeable; consider a plane. Consider instead a nonorientable shape like a Möbius strip.

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