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| 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''. | 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''. |
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| 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 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). |
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| 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. |
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| * ...and is also equivalent to this vector form... | * ...and is also equivalent to this '''vector form'''... |
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| * Generically this form is expressed as ''r = [f(t), g(t), h(t)]'' or ''r = [x(t), y(t), z(t)]''. | |
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| * ''r(t) = f(t)i + g(t)j + h(t)k = (1+2t)i + (3-t)j + (5t)k'' | * ''(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). === 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 [[Calculus/CoordinateSystem|spherical coordinates]] as ''r(φ,θ) = [ρcos(θ)sin(φ), sin(θ)sin(φ), ρcos(φ)]'' for ''0 ≤ θ ≤ 2π'' and ''0 ≤ φ ≤ π''. 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 [[Calculus/VectorOperations#Cross_Product|cross product]], is never zero. |
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).
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), sin(θ)sin(φ), ρcos(φ)] for 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π.
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.