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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). Applications of parametric equations then have three parts: * ''does'' a line pass through some point, line, or plane? * ''when'' does it pass through? * ''where'' does it pass through? |
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| Given a system described by ''x=2t-1'', ''y=t+2'', and ''z=-3t+2'' and a plane ''x+2y+4z=7'', the first and second questions are answered by substitution. | |
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| ''(2t-1)+2(t+2)+4(-3t+2)≟7'' | === Parametric Curve === |
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| ''2t-1+2t+4-12t+8≟7'' | If a system can be parameterized to one variable, ''t'', then it describes a curve. |
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| ''-8t+11≟7'' | 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''. |
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| ''-8t≟-4'' | 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). |
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| ''t=.5'' | Some curves are not smooth everywhere. This isn't an issue however, because they generally are [[Calculus/PiecewiseParameterization|piecewise smooth]], and integrals can be summed without issue. |
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| If there was no solution to satisfy equality, then the line must not pass through. (The third question is then answered by substituting this solution into the original system.) | |
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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 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), 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 ''[ρ^2^cos(θ)sin^2^(φ), -ρ^2^sin(θ)sin^2^(φ), ρ^2^sin(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 [[Calculus/VectorOperations#Cross_Product|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. |
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.