Curves and paths in space Example : Define ( t ) := (cos t, sin t ) - - PowerPoint PPT Presentation

Curves and paths in space

Example: Define γ(t) := (cos πt, sin πt), t ∈ [0, 1]. This defines a function or mapping γ : [0, 1] → R2. This function is called a path in the plane R2. If we plot γ(t) versus 0 ≤ t ≤ 1 then we get the image or range of γ(t), which is called the curve or trace of the path.

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Parametric representation of curves

Definition A two-dimensional path (or parametric function for a two-dimensional curve) is a given function or mapping γ : [0, 1] → R2, from a specified interval [a, b] into R2, which maps each a ≤ t ≤ b into the vector γ(t) ∈ R2. In terms of scalar components, we usually write: γ(t) = (x(t), y(t)) = x(t)i + y(t)j, ∀t ∈ [a, b]. The curve of the path is the set of points Γ ⊂ R2 traced by γ(t) as t traverses the interval [a, b]. Formally: Γ :=

• γ(t) ∈ R2 | t ∈ [a, b]
• .

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Parametric representation of curves (cont.)

The interval a ≤ t ≤ b is called the parametric interval, t is called the parametric variable of the path, and the whole function γ(t) is called a parametric representation of the curve Γ. The starting point of the path is the vector γ(a), while the ending point of the path is the vector γ(b). The curve Γ has a direction from the starting to the ending point corresponding to t increasing from t = a until t = b.

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Parametric curves in R3

Definition A three-dimensional path (or parametric function for a three-dimen- sional curve) is a given function or mapping γ : [0, 1] → R3, from a specified interval [a, b] into R3, which maps each a ≤ t ≤ b into the vector γ(t) ∈ R3. Usually, γ(t) is written in the scalar component form γ(t) = (x(t), y(t), z(t)) = x(t)i + y(t)j + z(t)k, ∀t ∈ [a, b]. The curve of the path is the set of points Γ ⊂ R3 traced by γ(t) as t traverses the interval a ≤ t ≤ b. Formally: Γ :=

• γ(t) ∈ R3 | t ∈ [a, b]
• .

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Parametric curves in R3 (cont.)

The remaining definitions are word-for-word identical to the R2 case, except that everywhere we just replace R2 with R3.

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More definitions

Definition If the path γ : [a, b] → R3 is such that the first derivatives dx(t) dt , dy(t) dt , dz(t) dt exist and are continuous for each t ∈ [a, b], then γ is called a C1-path and the curve corresponding to γ is called a C1-curve. Definition If γ : [a, b] → R3 is a C1 path and additionally the derivatives d2x(t) dt2 , d2y(t) dt2 , d2z(t) dt2 exist and are continuous for each t ∈ [a, b], then γ is called a C2-path and the curve corresponding to γ is called a C2-curve.

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Paths and curves

Different paths can have identical curves! For example, define γ(t) := (cos(2πt), sin(2πt)), 0 ≤ t ≤ 1/2, which results in a mapping γ : [0, 1/2] → R2. The curve corresponding to this path is identical to that of (cos πt, sin πt) with t ∈ [0, 1].

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Paths and curves (cont.)

To further illustrate the idea, let us define γ1(t) = (cos πt, sin πt) with t ∈ [0, 1]. Now define the function ψ : [1, √ 2] → R, ψ(s) := s2 − 1. We see that ψ(1) = 0, ψ( √ 2) = 1, ψ′(s) = 2s > 0, ∀s ∈ [1, √ 2]. The function ψ(s) increases strictly monotonically through the interval [0, 1] as s increases through the interval 1 ≤ s ≤ √ 2.

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Paths and curves (cont.)

Now define the path γ2 : [1, √ 2] → R2 as γ2(s) := γ1(ψ(s)) =

• cos(π(s2 − 1)), sin(π(s2 − 1))
• , 1 ≤ s ≤
• (2).

The paths γ1 and γ2 are clearly different, but they nevertheless have the same curve. In fact, γ1 and γ2 are distinct parametric representations of the same curve.

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Re-parametrization

More generally, suppose we have γ1 : [a1, b1] → R3. In order to change the parametrization to get a different path with exactly the same curve, we fix some interval [a2, b2] along with some strictly increasing C1-function ψ : [a2, b2] → R such that ψ(a2) = a1, ψ(b2) = b1, ψ′(s) > 0, ∀s ∈ [a2, b2]. Then the path γ2 : [a2, b2] → R3 defined by γ2(s) := γ1(ψ(s)), a2 ≤ s ≤ b2 gives a different parametric representation of the same curve.

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Tangent to a curve

Given a path γ : [a, b] → R3 represented as γ(t) = (x(t), y(t), z(t)) = x(t)i + y(t)j + z(t)k, its derivative w.r.t t is the vector in R3 given by γ(1)(t) := dγ(t) dt = dx(t) dt , dy(t) dt , dz(t) dt

• = dx(t)

dt i+dy(t) dt j+dz(t) dt k, for all a ≤ t ≤ b. Similarly, the second derivative of the path w.r.t. t is given by γ(2)(t) := dγ(1)(t) dt = dx2(t) dt2 , dy2(t) dt2 , dz2(t) dt2

• =

= dx2(t) dt2 i + dy2(t) dt2 j + dz2(t) dt2 k, for all a ≤ t ≤ b.

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Tangent to a curve (cont.)

The fact that γ(t) = (x(t), y(t), z(t) along with dx dt = lim

∆t→0

x(t + ∆t) − x(t) ∆t dy dt = lim

∆t→0

y(t + ∆t) − y(t) ∆t dz dt = lim

∆t→0

z(t + ∆t) − z(t) ∆t suggests that dγ dt = lim

∆t→0

γ(t + ∆t) − γ(t) ∆t

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Tangent to a curve (cont.)

The difference vector γ(t + ∆t) − γ(t) is vey close to tangential to the curve of the path. When ∆t is small, and the limit of (γ(t + ∆t) − γ(t)) /∆t is exactly tangent to the curve of the path at γ(t). Thus, the derivative γ(1)(t) is tangent to the curve of the path γ : [a, b] → R3 at γ(t) for each instant a ≤ t ≤ b. Similarly, one can see that the derivative γ(2)(t) is tangent to the curve of the path γ(1) : [a, b] → R3 at γ(1)(t) for each a ≤ t ≤ b.

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Products of curves

Given two paths γ1 : [a, b] → R3 and γ2 : [a, b] → R3, the inner product of the vectors γ1(t) and γ2(t) in R3 for each a ≤ t ≤ b gives a scalar-valued function ϕ(t) := γ1(t) · γ2(t), a ≤ t ≤ b. By the rule for differentiation of products, we have: dϕ(t) dt = γ1(t) · γ(1)

2 (t) + γ1(t)(1) · γ2(t),

∀t ∈ [a, b].

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Velocity and acceleration

The parametric variable for a given path γ : [a, b] → R3 does not have any “physical” interpretation, in general. However, in many applications, t may be interpreted as time (and hence [a, b] as a time interval). Moreover, if γ(t) represents a physical point moving in space, one can introduce the notions of velocity v and acceleration a, namely v(t) := γ(1)(t), a(t) := γ(2)(t), for all a ≤ t ≤ b.

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Velocity and acceleration (cont.)

Now, let F : D → R3 be a vector field (e.g., a force acting on a particle of mass m) defined over the domain D ⊂ R3. Suppose that, in response to this force, the particle follows the path γ : [a, b] → R3. Newton’s second law then says that ma(t) = F(γ(t)), t ∈ [a, b]. Alternatively, one has mγ(2)(t) = F(γ(t)), t ∈ [a, b]. This is a second order vector differential equation which can, in principle, be solved to get the path γ : [a, b] → R3 followed by the particle if one knows the force vector field F.

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Simple curves and closed curves

Definition If the path γ : [a, b] → R3 has the property that, for any t1, t2 ∈ [a, b], γ(t1) = γ(t2) when t1 = t2, the corresponding curve is called simple. Simply put, a simple curve has the property that it does not “cross itself” anywhere (as opposed to non-simple curves).

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Simple curves and closed curves (cont.)

Definition If the path γ : [a, b] → R3 has the property that γ(a) = γ(b), the corresponding curve (simple or not) is called closed. A closed curve which is also simple is called a simple closed curve whereas a closed curve which does cross itself somewhere is a non- simple closed curve.

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