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10 arc length parameter and curvature

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The document discusses arc length parameterization and curvature of curves. It gives an example of a curve C(t) = <3t + 1, 4t - 2> describing a particle moving at a constant speed of 5. Reparameterizing this curve by arc length gives C*(s) = <3/5s + 1, 4/5s - 2>, which describes the particle moving at a speed of 1. The relationship between s and t is determined using the arc length formula. The curvature of a curve measures its rate of turning, with straight lines having curvature of 0 and circles having curvature of 1/r, where r is the radius.

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Arc length Parameter and Curvature
Given a curve C(t), there are infinitely many other parametrizations that give the same graph. Arc length Parameter and Curvature
Given a curve C(t), there are infinitely many other parametrizations that give the same graph. Arc length Parameter and Curvature Example: The curve C(t) = <3t +1, 4t – 2> and C*(t) = < > are lines having directional vectors <3, 4 > and < , >. 3 5 t + 1, 4 5 t – 2 3 5 4 5
Given a curve C(t), there are infinitely many other parametrizations that give the same graph. Arc length Parameter and Curvature Example: The curve C(t) = <3t +1, 4t – 2> and C*(t) = < > are lines having directional vectors <3, 4 > and < , >. These are the multiple of each other, so the lines are parallel. 3 5 t + 1, 4 5 t – 2 3 5 4 5
Given a curve C(t), there are infinitely many other parametrizations that give the same graph. Arc length Parameter and Curvature Example: The curve C(t) = <3t +1, 4t – 2> and C*(t) = < > are lines having directional vectors <3, 4 > and < , >. These are the multiple of each other, so the lines are parallel. They both pass through the point (1, -2), hence they have the same graph. 3 5 t + 1, 4 5 t – 2 3 5 4 5
Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula
Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so
Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2         
Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2          For C*(t) from 0 to a, (x'(t))2 +(y'(t))2 = 1,
Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2          For C*(t) from 0 to a, (x'(t))2 +(y'(t))2 = 1, so a dt dt t y t x a t t a t t          0 0 2 2 1 )) ( ' ( )) ( ' (
Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2          For C*(t) from 0 to a, (x'(t))2 +(y'(t))2 = 1, so a dt dt t y t x a t t a t t          0 0 2 2 1 )) ( ' ( )) ( ' ( C(t) describes a moving particle traveling at the constant speed of 5,
Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2          For C*(t) from 0 to a, (x'(t))2 +(y'(t))2 = 1, so a dt dt t y t x a t t a t t          0 0 2 2 1 )) ( ' ( )) ( ' ( C(t) describes a moving particle traveling at the constant speed of 5, C*(t) describe a particle traveling at a constant speed of 1.
Parametrized by Arc length
Parametrized by Arc length Given a curve and a point P on the curve, we may parametrize the curve in the following manner:
Parametrized by Arc length Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P
Parametrized by Arc length Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. 2. Let C(s) = <x(s), y(s)> = Q where the signed distance from P to Q is s. s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P=<x(0),y(0)> arclength=s in s seconds Q = C(s) =<x(s),y(s)>
Parametrized by Arc length s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P=<x(0),y(0)> arclength=s in s seconds C*(s)=< > 3 5 s + 1, 4 5 s – 2 is parametrized by arc-length Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. 2. Let C(s) = <x(s), y(s)> = Q where the signed distance from P to Q is s. Q = C(s) =<x(s),y(s)>
Parametrized by Arc length s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P=<x(0),y(0)> arclength=s in s seconds C*(s)=< > 3 5 s + 1, 4 5 s – 2 is parametrized by arc-length with P = (1, -2) = C*(0) Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. 2. Let C(s) = <x(s), y(s)> = Q where the signed distance from P to Q is s. Q = C(s) =<x(s),y(s)>
Parametrized by Arc length s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P=<x(0),y(0)> arclength=s in s seconds C*(s)=< > 3 5 s + 1, 4 5 s – 2 is parametrized by arc-length with P = (1, -2) = C*(0) since | | = 1. dC* ds Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. 2. Let C(s) = <x(s), y(s)> = Q where the signed distance from P to Q is s. Q = C(s) =<x(s),y(s)>
Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Parametrized by Arc length
Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Re-parametrize C(t) by arc length we get C*(s) = < >. 3 5 s + 1, 4 5 s – 2 Parametrized by Arc length
Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Re-parametrize C(t) by arc length we get C*(s) = < >. x(t) = 3t + 1, y(t) = 4t – 2  3 5 s + 1, 4 5 s – 2 (x'(u))2+(y'(u))2 = 5 Parametrized by Arc length
Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Re-parametrize C(t) by arc length we get C*(s) = < >. x(t) = 3t + 1, y(t) = 4t – 2  The relation between the variable s and t is 3 5 s + 1, 4 5 s – 2 (x'(u))2+(y'(u))2 = 5 t du du u y u x s t u u t u u 5 5 )) ( ' ( )) ( ' ( 0 0 2 2           Parametrized by Arc length
Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Re-parametrize C(t) by arc length we get C*(s) = < >. x(t) = 3t + 1, y(t) = 4t – 2  The relation between the variable s and t is 3 5 s + 1, 4 5 s – 2 (x'(u))2+(y'(u))2 = 5 t du du u y u x s t u u t u u 5 5 )) ( ' ( )) ( ' ( 0 0 2 2           ds dt In particular, = (x'(t))2+(y'(t))2 = 5 Parametrized by Arc length
The curvature (kappa) is a measurment of the rate the curve is turning. Curvature 
The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. Curvature 
The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Curvature  
The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature  
The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature   For a general vector valued curve C(t), the curvature of at any point P is independent of the speed (i.e |C'(t)|) at P.
The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature   For a general vector valued curve C(t), the curvature of at any point P is independent of the speed (i.e |C'(t)|) at P. Hence, we re-parametrize C(t) by the arc-length variable s, obtain C(t(s)) that has constant speed 1.
The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature   For a general vector valued curve C(t), the curvature of at any point P is independent of the speed (i.e |C'(t)|) at P. Hence, we re-parametrize C(t) by the arc-length variable s, obtain C(t(s)) that has constant speed 1. Since the accerleration governs the turning of the curve, we define the curvature as the absolute value of the accerleration vector of C(t(s)) ,
The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature   For a general vector valued curve C(t), the curvature of at any point P is independent of the speed (i.e |C'(t)|) at P. Hence, we re-parametrize C(t) by the arc-length variable s, obtain C(t(s)) that has constant speed 1. Since the accerleration governs the turning of the curve, we define the curvature as the absolute value of the accerleration vector of C(t(s)) , that is, d2C(t(s)) ds2 = | |
Arc length Parameter and Curvature The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | = | | dT ds The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | = | | dT ds One may show that this is the same as: The Curvature Formula: The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | = | | dT ds One may show that this is the same as: The Curvature Formula: Given C(t) a 2D or a 3D vector valued function with T and C''(t) exist, then the curvature  |C'(t) x C''(t)| |C'(t)|3 = The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | = | | dT ds One may show that this is the same as: The Curvature Formula: Given C(t) a 2D or a 3D vector valued function with T and C''(t) exist, then the curvature  |C'(t) x C''(t)| |C'(t)|3 = (For 2D curve <x(t),y(t)>, treat it as a 3D curve <x(t), y(t), 0 > so the cross produc is well defined.) The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)>
Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0>
Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0>
Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>
Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 =
Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< 0, 0, r2 >  |C'(t) x C"(t)| = r2
Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< 0, 0, r2 >  |C'(t) x C"(t)| = r2 |C'(t)| = r
Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< 0, 0, r2 >  |C'(t) x C"(t)| = r2 |C'(t)| = r Therefore the curvature is r2/r3 = 1/r. 
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t>
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1>
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 =
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4 = r1 + r2
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4 = r1 + r2 |C'(t)| = r2+1
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4 = r1 + r2 |C'(t)| = r2+1 Therefore the curvature is  r1 + r2 (r2+1)3
Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4 = r1 + r2 |C'(t)| = r2+1 Therefore the curvature =  r1 + r2 (r2+1)3 = r r2+1

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10 arc length parameter and curvature

  • 1. Arc length Parameter and Curvature
  • 2. Given a curve C(t), there are infinitely many other parametrizations that give the same graph. Arc length Parameter and Curvature
  • 3. Given a curve C(t), there are infinitely many other parametrizations that give the same graph. Arc length Parameter and Curvature Example: The curve C(t) = <3t +1, 4t – 2> and C*(t) = < > are lines having directional vectors <3, 4 > and < , >. 3 5 t + 1, 4 5 t – 2 3 5 4 5
  • 4. Given a curve C(t), there are infinitely many other parametrizations that give the same graph. Arc length Parameter and Curvature Example: The curve C(t) = <3t +1, 4t – 2> and C*(t) = < > are lines having directional vectors <3, 4 > and < , >. These are the multiple of each other, so the lines are parallel. 3 5 t + 1, 4 5 t – 2 3 5 4 5
  • 5. Given a curve C(t), there are infinitely many other parametrizations that give the same graph. Arc length Parameter and Curvature Example: The curve C(t) = <3t +1, 4t – 2> and C*(t) = < > are lines having directional vectors <3, 4 > and < , >. These are the multiple of each other, so the lines are parallel. They both pass through the point (1, -2), hence they have the same graph. 3 5 t + 1, 4 5 t – 2 3 5 4 5
  • 6. Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula
  • 7. Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so
  • 8. Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2         
  • 9. Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2          For C*(t) from 0 to a, (x'(t))2 +(y'(t))2 = 1,
  • 10. Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2          For C*(t) from 0 to a, (x'(t))2 +(y'(t))2 = 1, so a dt dt t y t x a t t a t t          0 0 2 2 1 )) ( ' ( )) ( ' (
  • 11. Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2          For C*(t) from 0 to a, (x'(t))2 +(y'(t))2 = 1, so a dt dt t y t x a t t a t t          0 0 2 2 1 )) ( ' ( )) ( ' ( C(t) describes a moving particle traveling at the constant speed of 5,
  • 12. Arc length Parameter and Curvature dt t y t x b t a t     2 2 )) ( ' ( )) ( ' ( Applying the arc length formula to C(t) from t=0 to t=a: |C'(t)| = (x'(t))2 +(y'(t))2 = 32 + 42 = 5, so a dt dt t y t x a t t a t t 5 5 )) ( ' ( )) ( ' ( 0 0 2 2          For C*(t) from 0 to a, (x'(t))2 +(y'(t))2 = 1, so a dt dt t y t x a t t a t t          0 0 2 2 1 )) ( ' ( )) ( ' ( C(t) describes a moving particle traveling at the constant speed of 5, C*(t) describe a particle traveling at a constant speed of 1.
  • 14. Parametrized by Arc length Given a curve and a point P on the curve, we may parametrize the curve in the following manner:
  • 15. Parametrized by Arc length Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P
  • 16. Parametrized by Arc length Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. 2. Let C(s) = <x(s), y(s)> = Q where the signed distance from P to Q is s. s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P=<x(0),y(0)> arclength=s in s seconds Q = C(s) =<x(s),y(s)>
  • 17. Parametrized by Arc length s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P=<x(0),y(0)> arclength=s in s seconds C*(s)=< > 3 5 s + 1, 4 5 s – 2 is parametrized by arc-length Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. 2. Let C(s) = <x(s), y(s)> = Q where the signed distance from P to Q is s. Q = C(s) =<x(s),y(s)>
  • 18. Parametrized by Arc length s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P=<x(0),y(0)> arclength=s in s seconds C*(s)=< > 3 5 s + 1, 4 5 s – 2 is parametrized by arc-length with P = (1, -2) = C*(0) Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. 2. Let C(s) = <x(s), y(s)> = Q where the signed distance from P to Q is s. Q = C(s) =<x(s),y(s)>
  • 19. Parametrized by Arc length s=0 s=1 s=2 s=3 s=-1 s=-2 s=-3 P=<x(0),y(0)> arclength=s in s seconds C*(s)=< > 3 5 s + 1, 4 5 s – 2 is parametrized by arc-length with P = (1, -2) = C*(0) since | | = 1. dC* ds Given a curve and a point P on the curve, we may parametrize the curve in the following manner: 1. Selecting a point P, fix the positive and negative directions. 2. Let C(s) = <x(s), y(s)> = Q where the signed distance from P to Q is s. Q = C(s) =<x(s),y(s)>
  • 20. Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Parametrized by Arc length
  • 21. Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Re-parametrize C(t) by arc length we get C*(s) = < >. 3 5 s + 1, 4 5 s – 2 Parametrized by Arc length
  • 22. Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Re-parametrize C(t) by arc length we get C*(s) = < >. x(t) = 3t + 1, y(t) = 4t – 2  3 5 s + 1, 4 5 s – 2 (x'(u))2+(y'(u))2 = 5 Parametrized by Arc length
  • 23. Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Re-parametrize C(t) by arc length we get C*(s) = < >. x(t) = 3t + 1, y(t) = 4t – 2  The relation between the variable s and t is 3 5 s + 1, 4 5 s – 2 (x'(u))2+(y'(u))2 = 5 t du du u y u x s t u u t u u 5 5 )) ( ' ( )) ( ' ( 0 0 2 2           Parametrized by Arc length
  • 24. Example: The curve C(t) = <3t +1, 4t – 2> describes a moving point at the point C(0) = (1, -2) with a constant speed of 5. Re-parametrize C(t) by arc length we get C*(s) = < >. x(t) = 3t + 1, y(t) = 4t – 2  The relation between the variable s and t is 3 5 s + 1, 4 5 s – 2 (x'(u))2+(y'(u))2 = 5 t du du u y u x s t u u t u u 5 5 )) ( ' ( )) ( ' ( 0 0 2 2           ds dt In particular, = (x'(t))2+(y'(t))2 = 5 Parametrized by Arc length
  • 25. The curvature (kappa) is a measurment of the rate the curve is turning. Curvature 
  • 26. The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. Curvature 
  • 27. The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Curvature  
  • 28. The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature  
  • 29. The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature   For a general vector valued curve C(t), the curvature of at any point P is independent of the speed (i.e |C'(t)|) at P.
  • 30. The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature   For a general vector valued curve C(t), the curvature of at any point P is independent of the speed (i.e |C'(t)|) at P. Hence, we re-parametrize C(t) by the arc-length variable s, obtain C(t(s)) that has constant speed 1.
  • 31. The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature   For a general vector valued curve C(t), the curvature of at any point P is independent of the speed (i.e |C'(t)|) at P. Hence, we re-parametrize C(t) by the arc-length variable s, obtain C(t(s)) that has constant speed 1. Since the accerleration governs the turning of the curve, we define the curvature as the absolute value of the accerleration vector of C(t(s)) ,
  • 32. The curvature (kappa) is a measurment of the rate the curve is turning. For straight lines, the curvature is 0 at every point on the line. For circle of radius r, the turn is uniform with curvature = 1/r. Hence large circles have small curvatures (less curved) and small circles have large curvatures (more curved). Curvature   For a general vector valued curve C(t), the curvature of at any point P is independent of the speed (i.e |C'(t)|) at P. Hence, we re-parametrize C(t) by the arc-length variable s, obtain C(t(s)) that has constant speed 1. Since the accerleration governs the turning of the curve, we define the curvature as the absolute value of the accerleration vector of C(t(s)) , that is, d2C(t(s)) ds2 = | |
  • 33. Arc length Parameter and Curvature The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
  • 34. Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
  • 35. Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
  • 36. Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | = | | dT ds The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
  • 37. Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | = | | dT ds One may show that this is the same as: The Curvature Formula: The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
  • 38. Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | = | | dT ds One may show that this is the same as: The Curvature Formula: Given C(t) a 2D or a 3D vector valued function with T and C''(t) exist, then the curvature  |C'(t) x C''(t)| |C'(t)|3 = The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
  • 39. Arc length Parameter and Curvature  d2C(t(s)) ds2 = | | = | d (dC(t(s))/ds) ds | = | | dT ds One may show that this is the same as: The Curvature Formula: Given C(t) a 2D or a 3D vector valued function with T and C''(t) exist, then the curvature  |C'(t) x C''(t)| |C'(t)|3 = (For 2D curve <x(t),y(t)>, treat it as a 3D curve <x(t), y(t), 0 > so the cross produc is well defined.) The tangents of C(t(s)) = dC(t(s))/ds, have length 1 so it is equal to the unit tangent T of C(t).
  • 40. Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)>
  • 41. Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0>
  • 42. Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0>
  • 43. Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>
  • 44. Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 =
  • 45. Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< 0, 0, r2 >  |C'(t) x C"(t)| = r2
  • 46. Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< 0, 0, r2 >  |C'(t) x C"(t)| = r2 |C'(t)| = r
  • 47. Arc length Parameter and Curvature Example: Find the curvature of the circle of radius r C(t) = <rCos(t), rSin(t)> Embed the curve in 3D as <rCos(t),rSin(t), 0> C'(t) = <-rSin(t), rCos(t), 0> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< 0, 0, r2 >  |C'(t) x C"(t)| = r2 |C'(t)| = r Therefore the curvature is r2/r3 = 1/r. 
  • 48. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t>
  • 49. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1>
  • 50. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>
  • 51. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 =
  • 52. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >
  • 53. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4
  • 54. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4 = r1 + r2
  • 55. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4 = r1 + r2 |C'(t)| = r2+1
  • 56. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4 = r1 + r2 |C'(t)| = r2+1 Therefore the curvature is  r1 + r2 (r2+1)3
  • 57. Arc length Parameter and Curvature Example: Find the curvature of the helix of radius r C(t) = <rCos(t), rSin(t), t> C'(t) = <-rSin(t), rCos(t), 1> C"(t) = <-rCos(t), -rSin(t), 0>  |C'(t) x C''(t)| |C'(t)|3 = C'(t) x C''(t) =< rS(t),-rC(t), r2 >  |C'(t) x C"(t)| = r2 + r4 = r1 + r2 |C'(t)| = r2+1 Therefore the curvature =  r1 + r2 (r2+1)3 = r r2+1