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Key concepts

  • The arc-length function for a vector-valued function is calculated using the integral formula s ( t ) = a t r ( u ) d u . This formula is valid in both two and three dimensions.
  • The curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that point. The arc-length parameterization is used in the definition of curvature.
  • There are several different formulas for curvature. The curvature of a circle is equal to the reciprocal of its radius.
  • The principal unit normal vector at t is defined to be
    N ( t ) = T ( t ) T ( t ) .
  • The binormal vector at t is defined as B ( t ) = T ( t ) × N ( t ) , where T ( t ) is the unit tangent vector.
  • The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector.
  • The osculating circle is tangent to a curve at a point and has the same curvature as the tangent curve at that point.

Key equations

  • Arc length of space curve
    s = a b [ f ( t ) ] 2 + [ g ( t ) ] 2 + [ h ( t ) ] 2 d t = a b r ( t ) d t
  • Arc-length function
    s ( t ) = a t ( f ( u ) ) 2 + ( g ( u ) ) 2 + ( h ( u ) ) 2 d u or s ( t ) = a t r ( u ) d u
  • Curvature
    κ = T ( t ) r ( t ) or κ = r ( t ) × r″ ( t ) r ( t ) 3 or κ = | y | [ 1 + ( y ) 2 ] 3 / 2
  • Principal unit normal vector
    N ( t ) = T ( t ) T ( t )
  • Binormal vector
    B ( t ) = T ( t ) × N ( t )

Find the arc length of the curve on the given interval.

r ( t ) = t 2 i + 14 t j , 0 t 7. This portion of the graph is shown here:

This figure is the graph of a curve beginning at the origin and increasing.
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r ( t ) = t 2 i + ( 2 t 2 + 1 ) j , 1 t 3

8 5

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r ( t ) = 2 sin t , 5 t , 2 cos t , 0 t π . This portion of the graph is shown here:

This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the upper right corner of the box and bends through the box to the other side.
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r ( t ) = t 2 + 1 , 4 t 3 + 3 , 1 t 0

1 54 ( 37 3 / 2 1 )

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r ( t ) = e t cos t , e t sin t over the interval [ 0 , π 2 ] . Here is the portion of the graph on the indicated interval:

This figure is the graph of a curve in the first quadrant. It begins approximately at 0.20 on the y axis and increases to approximately where x = 0.3. Then the curve decreases, meeting the x-axis at 1.0.
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Find the length of one turn of the helix given by r ( t ) = 1 2 cos t i + 1 2 sin t j + 3 4 t k .

Length = 2 π

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Find the arc length of the vector-valued function r ( t ) = t i + 4 t j + 3 t k over [ 0 , 1 ] .

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A particle travels in a circle with the equation of motion r ( t ) = 3 cos t i + 3 sin t j + 0 k . Find the distance traveled around the circle by the particle.

6 π

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Set up an integral to find the circumference of the ellipse with the equation r ( t ) = cos t i + 2 sin t j + 0 k .

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Find the length of the curve r ( t ) = 2 t , e t , e t over the interval 0 t 1. The graph is shown here:

This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the upper left corner of the box and bends through the box to the bottom of the other side.

e 1 e

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Find the length of the curve r ( t ) = 2 sin t , 5 t , 2 cos t for t [ −10 , 10 ] .

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The position function for a particle is r ( t ) = a cos ( ω t ) i + b sin ( ω t ) j . Find the unit tangent vector and the unit normal vector at t = 0.

T ( 0 ) = j , N ( 0 ) = i

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Given r ( t ) = a cos ( ω t ) i + b sin ( ω t ) j , find the binormal vector B ( 0 ) .

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Given r ( t ) = 2 e t , e t cos t , e t sin t , determine the unit tangent vector T ( t ) .

T ( t ) = 2 e t , e t cos t e t sin t , e t cos t + e t sin t

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Given r ( t ) = 2 e t , e t cos t , e t sin t , determine the unit tangent vector T ( t ) evaluated at t = 0.

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Given r ( t ) = 2 e t , e t cos t , e t sin t , find the unit normal vector N ( t ) evaluated at t = 0 , N ( 0 ) .

N ( 0 ) = 2 2 , 0 , 2 2

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Given r ( t ) = 2 e t , e t cos t , e t sin t , find the unit normal vector evaluated at t = 0.

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Given r ( t ) = t i + t 2 j + t k , find the unit tangent vector T ( t ) . The graph is shown here:

This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the bottom left corner of the box and bends through the box to the upper left side.

T ( t ) = 1 4 t 2 + 2 < 1 , 2 t , 1 >

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Find the unit tangent vector T ( t ) and unit normal vector N ( t ) at t = 0 for the plane curve r ( t ) = t 3 4 t , 5 t 2 2 . The graph is shown here:

This figure is the graph of a curve above the x-axis. The curve decreases in the second quadrant, passes through the y-axis at y=20. Then it intersects the origin. The curve loops at the origin, increasing back through y=20 into the first quadrant.
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Find the unit tangent vector T ( t ) for r ( t ) = 3 t i + 5 t 2 j + 2 t k

T ( t ) = 1 100 t 2 + 13 ( 3 i + 10 t j + 2 k )

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Find the principal normal vector to the curve r ( t ) = 6 cos t , 6 sin t at the point determined by t = π / 3 .

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Find T ( t ) for the curve r ( t ) = ( t 3 4 t ) i + ( 5 t 2 2 ) j .

T ( t ) = 1 9 t 4 + 76 t 2 + 16 ( [ 3 t 2 4 ] i + 10 t j )

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Find N ( t ) for the curve r ( t ) = ( t 3 4 t ) i + ( 5 t 2 2 ) j .

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Find the unit normal vector N ( t ) for r ( t ) = 2 sin t , 5 t , 2 cos t .

N ( t ) = sin t , 0 , cos t

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Find the unit tangent vector T ( t ) for r ( t ) = 2 sin t , 5 t , 2 cos t .

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Find the arc-length function s ( t ) for the line segment given by r ( t ) = 3 3 t , 4 t . Write r as a parameter of s.

Arc-length function: s ( t ) = 5 t ; r as a parameter of s : r ( s ) = ( 3 3 s 5 ) i + 4 s 5 j

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Parameterize the helix r ( t ) = cos t i + sin t j + t k using the arc-length parameter s , from t = 0.

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Parameterize the curve using the arc-length parameter s , at the point at which t = 0 for r ( t ) = e t sin t i + e t cos t j .

r ( s ) = ( 1 + s 2 ) sin ( ln ( 1 + s 2 ) ) i + ( 1 + s 2 ) cos [ ln ( 1 + s 2 ) ] j

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Find the curvature of the curve r ( t ) = 5 cos t i + 4 sin t j at t = π / 3 . ( Note: The graph is an ellipse.)

This figure is the graph of an ellipse. The ellipse is oval along the x-axis. It is centered at the origin and intersects the y-axis at -4 and 4.
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Find the x -coordinate at which the curvature of the curve y = 1 / x is a maximum value.

The maximum value of the curvature occurs at x = 5 4 .

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Find the curvature of the curve r ( t ) = 5 cos t i + 5 sin t j . Does the curvature depend upon the parameter t ?

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Find the curvature κ for the curve y = x 1 4 x 2 at the point x = 2.

1 2

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Find the curvature κ for the curve y = 1 3 x 3 at the point x = 1.

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Find the curvature κ of the curve r ( t ) = t i + 6 t 2 j + 4 t k . The graph is shown here:

This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve has a parabolic shape in the middle of the box.

κ 49.477 ( 17 + 144 t 2 ) 3 / 2

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Find the curvature of r ( t ) = 2 sin t , 5 t , 2 cos t .

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Find the curvature of r ( t ) = 2 t i + e t j + e t k at point P ( 0 , 1 , 1 ) .

1 2 2

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At what point does the curve y = e x have maximum curvature?

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What happens to the curvature as x for the curve y = e x ?

The curvature approaches zero.

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Find the point of maximum curvature on the curve y = ln x .

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Find the equations of the normal plane and the osculating plane of the curve r ( t ) = 2 sin ( 3 t ) , t , 2 cos ( 3 t ) at point ( 0 , π , −2 ) .

y = 6 x + π and x + 6 = 6 π

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Find equations of the osculating circles of the ellipse 4 y 2 + 9 x 2 = 36 at the points ( 2 , 0 ) and ( 0 , 3 ) .

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Find the equation for the osculating plane at point t = π / 4 on the curve r ( t ) = cos ( 2 t ) i + sin ( 2 t ) j + t .

x + 2 z = π 2

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Find the radius of curvature of 6 y = x 3 at the point ( 2 , 4 3 ) .

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Find the curvature at each point ( x , y ) on the hyperbola r ( t ) = a cosh ( t ) , b sinh ( t ) .

a 4 b 4 ( b 4 x 2 + a 4 y 2 ) 3 / 2

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Calculate the curvature of the circular helix r ( t ) = r sin ( t ) i + r cos ( t ) j + t k .

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Find the radius of curvature of y = ln ( x + 1 ) at point ( 2 , ln 3 ) .

10 10 3

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Find the radius of curvature of the hyperbola x y = 1 at point ( 1 , 1 ) .

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A particle moves along the plane curve C described by r ( t ) = t i + t 2 j . Solve the following problems.

Find the length of the curve over the interval [ 0 , 2 ] .

38 3

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Find the curvature of the plane curve at t = 0 , 1 , 2.

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Describe the curvature as t increases from t = 0 to t = 2.

The curvature is decreasing over this interval.

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The surface of a large cup is formed by revolving the graph of the function y = 0.25 x 1.6 from x = 0 to x = 5 about the y -axis (measured in centimeters).

[T] Use technology to graph the surface.

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Find the curvature κ of the generating curve as a function of x.

κ = 6 x 2 / 5 ( 25 + 4 x 6 / 5 )

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[T] Use technology to graph the curvature function.

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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