Evolute and Involute and some Examples
Let \( C \) and \( C_1 \) are two space curves such that tangent to \( C \) is normal to \( C_1 \) then
\( C \) is called evolute of \( C_1 \)
\( C_1 \) is called involute of \( C \)
Some Examples
- Involute of circle[evolute] is spiral curve
- Involute of cetenary[evolute] is a semicubic parabola
- Involute of parabola[evolute] is a semicubic parabola
- Involute of parabola[evolute] is a semicubic parabola
- Nephroid
A nephroid is- an algebraic curve of degree 6.
- an epicycloid with two cusps
- a plane simple closed curve = a Jordan curve
\( x=6 \cos(t)-4 \cos(t)^3 ,y=4 \sin(t)^3 \)
The nephroid (involute: blue) and its evolute (red) are shown below. - Involute of cycloid[evolute] is congruent cycloid
Involute
Equation of Involute
Let \( C \) is given spaace curve and \( C_1 \) is involute of \( C \), then tangent to \( C \) is normal to \( C_1 \) , so
\( \vec{t}.\vec{t_1}=0 \)...(i)
Let \( P \) be a point on \( C \) and \( P_1 \) be its corresponding on \( C_1 \) with
\( \overrightarrow{OP_1} =\vec{r_1} \)
\( \overrightarrow{OP} =\vec{r} \)
Then,
\( \overrightarrow{PP_1} \) is tangent to C
or \( \overrightarrow{PP_1} = \lambda \vec{t} \)
where \( \lambda \) is constant to be determined
or \( \overrightarrow{OP_1}- \overrightarrow{OP} = \lambda \vec{t} \)
or \( \vec{r_1}- \vec{r} = \lambda \vec{t} \)
or \( \vec{r_1}= \vec{r} + \lambda \vec{t} \) ...(ii)
Differentiating (ii) w. r. to. s, we get
\( \vec{t_1} \frac{ds_1}{ds}= \vec{t} + \lambda' \vec{t}+\lambda \kappa \vec{n} \)
or \( \vec{t_1} \frac{ds_1}{ds}= (1+ \lambda') \vec{t}+\lambda \kappa \vec{n} \)
Since \( \vec{t}.\vec{t_1}=0 \), we take dot product on both sides by \( \vec{t} \), then we get
\( (\vec{t_1} \frac{ds_1}{ds}).\vec{t} = [(1+ \lambda') \vec{t}+\lambda \kappa \vec{n}] .\vec{t}\)
or \( 0 = [1+ \lambda' +0] \)
or \( \lambda' =-1 \)
Integrating w. r. to s, we get
\( \lambda =c-s \)...(iii)
where c is constant of integration
Thus substituting \( \lambda =c-s \) in (ii), we get
\( \vec{r_1}= \vec{r} + \lambda \vec{t} \)
or \( \vec{r_1}= \vec{r} + (c-s) \vec{t} \)
This is required equation of involute
Curvature and torsion of involute
Let \( C \) is given spaace curve and \( C_1 \) is involute of \( C \).
Then equation of involute \( C_1 \) is
\( \vec{r_1}= \vec{r} + (c-s) \vec{t} \) ...(i)
Differentiating (i) w. r. to. s, we get
\( \vec{t_1}\frac{ds_1}{ds}= \vec{t} + (c-s) \kappa \vec{n} + (-1)\vec{t} \)
or\( \vec{t_1}\frac{ds_1}{ds}= (c-s) \kappa \vec{n} \)
Taking magnitude, we get
\(\frac{ds_1}{ds}= (c-s) \kappa \) ...(A)
Substituting value of \(\frac{ds_1}{ds}= (c-s) \kappa \) , we get
\( \vec{t_1}= \vec{n} \) ...(ii)
Differentiating (ii) w. r. to. s, we get
\( \kappa_1 \vec{n_1}\frac{ds_1}{ds}= \tau \vec{b} -\kappa \vec{t} \)
Taking magnitude, we get
\( \kappa_1 \frac{ds_1}{ds}= \sqrt{\tau^2+ \kappa^2} \)
or\( \frac{ds_1}{ds}= \frac{\sqrt{\tau^2+ \kappa^2}}{\kappa_1 } \) ...(B)
Substituting value of \( \frac{ds_1}{ds}= \frac{\sqrt{\tau^2+ \kappa^2}}{\kappa_1 } \) , we get
\( \kappa_1 \vec{n_1}\frac{ds_1}{ds}= \tau \vec{b} -\kappa \vec{t} \)
or\( \kappa_1 \vec{n_1} \frac{\sqrt{\tau^2+ \kappa^2}}{\kappa_1 } = \tau \vec{b} -\kappa \vec{t} \)
or\( \vec{n_1} = \frac{\tau \vec{b} -\kappa \vec{t}}{\sqrt{\tau^2+ \kappa^2}} \) ...(iii)
Taking cross product between \( \vec{t_1} \) and \( \vec{n_1} \), we get
\( \vec{t_1} \times \vec{n_1} = \vec{n} \times \frac{\tau \vec{b} -\kappa \vec{t}}{\sqrt{\tau^2+ \kappa^2}} \)
or\( \vec{b_1} = \frac{\tau \vec{t}+ \kappa \vec{b}}{\sqrt{\tau^2+ \kappa^2}} \) ...(iv)
Differentiating (iv) w. r. to. s, we get
\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= \frac{ [\frac{d}{ds} (\tau \vec{t}+ \kappa \vec{b})] \sqrt{\tau^2+ \kappa^2} - [\frac{d}{ds}\sqrt{\tau^2+ \kappa^2}](\tau \vec{t}+ \kappa \vec{b})}{\tau^2+ \kappa^2} \)
or\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= \frac{ [(\tau' \vec{t}+\tau \kappa \vec{n}+ \kappa' \vec{b}- \kappa \tau \vec{n})] \sqrt{\tau^2+ \kappa^2} - [\frac{1}{2 \sqrt{\tau^2+ \kappa^2}} (2 \tau \tau' +2 \kappa \kappa')](\tau \vec{t}+ \kappa \vec{b})}{\tau^2+ \kappa^2} \)
or\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= \frac{ [(\tau' \vec{t}+ \kappa' \vec{b})] (\tau^2+ \kappa^2) - (\tau \tau' +\kappa \kappa')(\tau \vec{t}+ \kappa \vec{b})}{\sqrt{\tau^2+ \kappa^2} (\tau^2+ \kappa^2)} \)
Collecting the cofficients of \( \vec{t} \) and \( \vec{b} \) in the roght we get
\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= \frac{ [\tau'(\tau^2+ \kappa^2)- (\tau \tau' +\kappa \kappa')\tau] \vec{t} + [\kappa'(\tau^2+ \kappa^2)-(\tau \tau' +\kappa \kappa')\kappa] \vec{b}}{(\tau^2+ \kappa^2) \sqrt{\tau^2+ \kappa^2} } \)
or\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= \frac{ (\tau'\tau^2+ \tau' \kappa^2- \tau^2 \tau' - \kappa \kappa'\tau) \vec{t} + (\kappa' \tau^2+\kappa' \kappa^2- \kappa \tau \tau' -\kappa^2 \kappa' ) \vec{b}}{(\tau^2+ \kappa^2) \sqrt{\tau^2+ \kappa^2} } \)
or\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= \frac{ (\tau' \kappa^2- \kappa \kappa'\tau) \vec{t} + (\kappa' \tau^2- \kappa \tau \tau' ) \vec{b}}{(\tau^2+ \kappa^2) \sqrt{\tau^2+ \kappa^2} } \)
or\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= \frac{ (\tau' \kappa- \kappa'\tau) \kappa\vec{t} + (\kappa' \tau- \kappa \tau' ) \tau \vec{b}}{(\tau^2+ \kappa^2) \sqrt{\tau^2+ \kappa^2} } \)
Substituting the value of \( \vec{n_1} = \frac{\tau \vec{b} -\kappa \vec{t}}{\sqrt{\tau^2+ \kappa^2}} \), we get
\( -\tau_1 \frac{\tau \vec{b} -\kappa \vec{t}}{\sqrt{\tau^2+ \kappa^2}} \frac{ds_1}{ds}= \frac{ (\kappa \tau' - \kappa'\tau) \kappa\vec{t} - (\kappa \tau' -\kappa' \tau) \tau \vec{b}}{(\tau^2+ \kappa^2) \sqrt{\tau^2+ \kappa^2} } \)
or\( \tau_1 \frac{\kappa \vec{t}-\tau \vec{b}}{\sqrt{\tau^2+ \kappa^2}} \frac{ds_1}{ds}= \frac{ (\kappa \tau' - \kappa'\tau) (\kappa\vec{t} - \tau \vec{b})}{(\tau^2+ \kappa^2) \sqrt{\tau^2+ \kappa^2} } \)
Removing \( \frac{\kappa \vec{t}-\tau \vec{b}}{\sqrt{\tau^2+ \kappa^2}} \) from both side, we get
\( \tau_1 \frac{ds_1}{ds}= \frac{ \kappa \tau' - \kappa'\tau}{\tau^2+ \kappa^2} \)
or\( \frac{ds_1}{ds}= \frac{ \kappa \tau' - \kappa'\tau}{\tau_1 (\tau^2+ \kappa^2)} \) ...(C)
Equating (A) and (B), we get
\( (c-s) \kappa =\frac{\sqrt{\tau^2+ \kappa^2}}{\kappa_1 } \)
or\( \kappa_1=\frac{\sqrt{\tau^2+ \kappa^2}}{(c-s) \kappa } \)
This is required curvature of involute.
Equating (A) and (C), we get
\( (c-s) \kappa =\frac{ \kappa \tau' - \kappa'\tau}{\tau_1 (\tau^2+ \kappa^2)} \)
or\( \tau_1=\frac{ \kappa \tau' - \kappa'\tau}{\kappa(c-s)(\tau^2+ \kappa^2)} \)
This is required torsion of involute.
Evolute
Let \( C \) and \( C_1 \) are two space curves such that tangent to \( C \) is normal to \( C_1 \) then
\( C \) is called evolute of \( C_1 \)
Also
\( \vec{t}. \vec{t_1}=0 \)
NOTE
Evolute of the curve is the locus of the centre of curvature for that curve.
Equation of Evolute
Let \( C \) is given space curve and \( C_1 \) is evolute of \( C \). Then we have to find equation of \( C_1 \).
Here, tangent to \( C_1 \) is normal to \( C \) then
\( \vec{t_1}=(\lambda \vec{n}+ \mu \vec{b}) A \) where A is constant
or
\( \vec{t_1}=A \lambda \vec{n}+ A \mu \vec{b} \)…(i)
Let \( P \) be a point on \( C \) and \( P_1 \) be its corresponding on \( C_1 \) with
\( \vec{OP_1} =\vec{r_1} \)
\( \vec{OP} =\vec{r} \)
Then,
\( \vec{PP_1} \) is normal to C
or
\( \vec{PP_1} = \lambda \vec{n}+ \mu \vec{b} \)
where \( \lambda \) and \( \mu \) are constants to be determinedor
or
\( \vec{OP_1}- \lambda \vec{n}+ \mu \vec{b} \)
or
\( \vec{r_1}- \vec{r} = \lambda \vec{n}+ \mu \vec{b} \)
or
\( \vec{r_1}= \vec{r} + \lambda \vec{n}+ \mu \vec{b} \) …(ii)
Differentiating w. r. to, s, we get
\( \vec{t_1} \frac{ds_1}{ds}= \vec{t} + \lambda' \vec{n}+\lambda (\tau \vec{b}-\kappa \vec{t})+ \mu' \vec{b}+ \mu (-\tau \vec{n}) \)
or
\( \vec{t_1} \frac{ds_1}{ds}= \vec{t} + \lambda' \vec{n}+ \lambda \tau \vec{b}-\lambda \kappa \vec{t}+ \mu' \vec{b}- \mu \tau \vec{n} \)
Collecting cofficients of \( \vec{t}, \vec{n}, \vec{b} \), we get
\( \vec{t_1} \frac{ds_1}{ds}= (1-\lambda \kappa)\vec{t} + (\lambda' - \mu \tau) \vec{n}+ (\lambda \tau + \mu') \vec{b} \)…(iii)
Comparing (i) and (ii), we get
\( \frac{ds_1}{ds}= \frac{1-\lambda \kappa}{0}=\frac{\lambda' - \mu \tau}{A \lambda}=\frac{\lambda \tau + \mu'}{A \mu} \)
Equating first two and last two part, we get
\( \frac{ds_1}{ds}= \frac{1-\lambda \kappa}{0} \) and \( \frac{\lambda' - \mu \tau}{A \lambda}=\frac{\lambda \tau + \mu'}{A \mu} \)
or
\( 1-\lambda \kappa=0 \) and \( \frac{\lambda' - \mu \tau}{\lambda}=\frac{\lambda \tau + \mu'}{\mu} \)
or
\( \lambda=\frac{1}{\kappa} \) and \( \mu(\lambda' - \mu \tau)=\lambda(\lambda \tau + \mu') \)
or
\( \lambda=\rho \) and \( \mu \lambda'- \mu^2 \tau =\lambda^2 \tau + \lambda\mu' \)
or
\( \lambda=\rho \) and \( (\lambda^2+ \mu^2 )\tau= \lambda'\mu - \lambda \mu' \)
or
\( \lambda=\rho \) and \( \tau= \frac{\lambda'\mu - \lambda \mu'}{\lambda^2+ \mu^2} \) \( \frac{d}{ds} tan^{-1} \frac{\lambda}{\mu} =\frac{\lambda'\mu - \lambda \mu'}{\lambda^2+ \mu^2} \) рднрдПрдХोрд▓े
or
\( \lambda=\rho \) and \( \tau= \frac{d}{ds} tan^{-1} \frac{\lambda}{\mu} \)
Integrating right part w. r. to. s,we get
\( \lambda=\rho \) and \( \int{\tau} ds= tan^{-1} \frac{\lambda}{\mu} \)
or
\( \lambda=\rho \) and \( tan(\int{\tau} ds)= \frac{\lambda}{\mu} \)
\( \int{\tau} ds= \phi(s) +c \) рдоाрдиौ, then \(\frac{d}{ds} \phi(s)=\phi'(s)=\phi'=\tau \)
Now we have
\( \lambda=\rho \) and \( tan(\int{\tau} ds)= \frac{\lambda}{\mu} \)
or
\( \lambda=\rho \) and \( tan(\phi+c)= \frac{\lambda}{\mu} \)
or
\( \lambda=\rho \) and \( \mu= \frac{\lambda}{tan(\phi+c)} \)
or
\( \lambda=\rho \) and \( \mu= \lambda cot(\phi+c) \) \( cot(\phi+c) = -tan(\frac{\pi}{2}+(\phi+c)) \) рднрдПрдХोрд▓े
or
\( \lambda=\rho \) and \( \mu= -\lambda tan(\frac{\pi}{2}+(\phi+c)) \)
or
\( \lambda=\rho \) and \( \mu= -\lambda tan(\frac{\pi}{2}+\phi+c) \)
\( \frac{\pi}{2}+\phi+c = \psi \) рдоाрдиौ, then \(\frac{d}{ds} \psi=\psi'=\phi'=\tau \)
or
\( \lambda=\rho \) and \( \mu= -\lambda tan(\psi) \)
or
\( \lambda=\rho \) and \( \mu= -\rho tan(\psi) \)
Now, substituting value of \( \lambda \) and \( \mu \) in (ii) we get,
\( \vec{r_1}= \vec{r} + \lambda \vec{n}+ \mu \vec{b} \)
or
\( \vec{r_1}= \vec{r} + \rho \vec{n}-\rho tan(\psi) \vec{b} \) where \( \psi'=\tau \)
this is the required equation of evolute.
Curvature and Torsion of Evolute
Let \( C \) is given space curve and \( C_1 \) is evolute of \( C \).
Then equation of \( C_1 \) is
\( \vec{r_1}= \vec{r} + \rho \vec{n}-\rho tan\psi \vec{b} \)(i)
Differentiating w. r. to, s, we get
\( \vec{t_1}\frac{ds_1}{ds}= \vec{t} + \rho' \vec{n}+\rho (\tau \vec{b}-\kappa \vec{t})-\rho' tan\psi \vec{b}-\rho (sec^2\psi \psi') \vec{b}-\rho tan\psi (-\tau \vec{n}) \)
or\( \vec{t_1}\frac{ds_1}{ds}= \vec{t} + \rho' \vec{n}+\rho \tau \vec{b}-\vec{t}-\rho' tan\psi \vec{b}-\rho sec^2\psi \tau \vec{b}+\rho \tau tan\psi \vec{n} \)
or\( \vec{t_1}\frac{ds_1}{ds}= \rho' \vec{n}+\rho \tau \vec{b}-\rho' tan\psi \vec{b}-\rho sec^2\psi \tau \vec{b}+\rho \tau tan\psi \vec{n} \)
Collecting cofficient of \( \vec{n} \) and \( \vec{b} \), we get
\( \vec{t_1}\frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi )\vec{n}+(\rho \tau-\rho' tan\psi-\rho sec^2\psi \tau)\vec{b} \)
or\( \vec{t_1}\frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi )\vec{n}+(\rho \tau-\rho \tau sec^2\psi-\rho' tan\psi )\vec{b} \)
or\( \vec{t_1}\frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi )\vec{n}+[\rho \tau(1-sec^2\psi)-\rho' tan\psi ]\vec{b} \)
or\( \vec{t_1}\frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi )\vec{n}+[\rho \tau(-tan^2\psi)-\rho' tan\psi ]\vec{b} \)
or\( \vec{t_1}\frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi )\vec{n}+(-\rho \tau tan^2\psi-\rho' tan\psi )\vec{b} \)
or\( \vec{t_1}\frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi )\vec{n}-(\rho \tau tan^2\psi+\rho' tan\psi )\vec{b} \)
or\( \vec{t_1}\frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi )\vec{n}-(\rho \tau tan\psi+\rho' )tan\psi \vec{b} \)
or\( \vec{t_1}\frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi )(\vec{n}-tan\psi \vec{b}) \)
Taking Magnitude, we get
\( \frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi ) \sqrt{1+tan^2\psi} \)
or\( \frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi ) sec\psi \)(A)
Substituting the value of \( \frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi ) sec\psi \), we get
\( \vec{t_1}\frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi )(\vec{n}-tan\psi \vec{b}) \)
or\( \vec{t_1}(\rho'+\rho \tau tan\psi ) sec\psi= (\rho'+\rho \tau tan\psi )(\vec{n}-tan\psi \vec{b}) \)
or\( \vec{t_1}= \frac{\vec{n}-tan\psi \vec{b}}{sec\psi} \)
or\( \vec{t_1}= cos\psi \vec{n}-sin\psi \vec{b} \) (ii)
Differentiating w. r. to, s, we get
\( \kappa_1 \vec{n_1}\frac{ds_1}{ds}= (-sin\psi \tau) \vec{n}+cos\psi (\tau \vec{b}-\kappa \vec{t})- (cos\psi \tau) \vec{b}-sin\psi (-\tau \vec{n}) \)
or\( \kappa_1 \vec{n_1}\frac{ds_1}{ds}= -sin\psi \tau \vec{n}+cos\psi \tau \vec{b}-cos\psi \kappa \vec{t}- cos\psi \tau \vec{b}+ sin\psi \tau \vec{n} \)
or\( \kappa_1 \vec{n_1}\frac{ds_1}{ds}= -cos\psi \kappa \vec{t} \)
Taking Magnitude, we get
\( \kappa_1 \frac{ds_1}{ds}= cos\psi \kappa \)
or\( \frac{ds_1}{ds}= \frac{cos\psi \kappa}{\kappa_1} \) (B)
Substituting the value of \( \frac{ds_1}{ds}= \frac{cos\psi \kappa}{\kappa_1} \), we get
\( \kappa_1 \vec{n_1}\frac{ds_1}{ds}= -cos\psi \kappa \vec{t} \)
or\( \kappa_1 \vec{n_1} \frac{cos\psi \kappa}{\kappa_1}= -cos\psi \kappa \vec{t} \)
or\( \vec{n_1} = -\vec{t} \) (iii)
Taking cross product between \( \vec{t_1} \) and \( \vec{n_1} \), we get
\( \vec{t_1} \times \vec{n_1}=(cos\psi \vec{n}-sin\psi \vec{b}) \times ( -\vec{t}) \)
or\( \vec{b_1} =cos\psi \vec{b}+sin\psi \vec{n} \) (iv)
Differentiating w. r. to, s, we get
\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= (-sin\psi \tau) \vec{b}+cos\psi (-\tau \vec{n}) + (cos\psi \tau) \vec{n}+sin\psi (\tau \vec{b}-\kappa \vec{t}) \)
or\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= -sin\psi \tau \vec{b}- cos\psi \tau \vec{n} + cos\psi \tau \vec{n}+sin\psi \tau \vec{b}-sin\psi \kappa \vec{t} \)
or\( -\tau_1 \vec{n_1} \frac{ds_1}{ds}= -sin\psi \kappa \vec{t} \)
Taking Magnitude, we get
\( \tau_1 \frac{ds_1}{ds}= - sin\psi \kappa \)
or\( \frac{ds_1}{ds}= - \frac{sin\psi \kappa}{\tau_1} \)(C)
Here,
\( \frac{ds_1}{ds}= (\rho'+\rho \tau tan\psi ) sec\psi \)(A)
or\( \frac{ds_1}{ds}= [\frac{-\kappa'}{\kappa^2}+\frac{1}{\kappa} \tau \frac{sin\psi}{cos \psi} ] \frac{1}{cos\psi} \)
or\( \frac{ds_1}{ds}= [\frac{-\kappa'}{\kappa^2}+\frac{\tau sin\psi}{\kappa cos \psi} ] \frac{1}{cos\psi} \)
or\( \frac{ds_1}{ds}= \frac{-\kappa' cos \psi+\kappa \tau sin\psi}{\kappa^2 cos \psi} \times \frac{1}{cos\psi} \)
or\( \frac{ds_1}{ds}= \frac{\kappa \tau sin\psi-\kappa' cos \psi}{\kappa^2 cos^2\psi} \)(A)
Equating (A) and (B), we get
\( \frac{\kappa \tau sin\psi-\kappa' cos \psi}{\kappa^2 cos^2\psi} =\frac{cos\psi \kappa}{\kappa_1} \)
or\( \kappa_1= \frac{\kappa^3 cos^3\psi }{\kappa \tau sin\psi-\kappa' cos \psi} \)
This is required curvature of evolue.
Next,
Equating (A) and (C), we get
\( \frac{\kappa \tau sin\psi-\kappa' cos \psi}{\kappa^2 cos^2\psi} =- \frac{sin\psi \kappa}{\tau_1} \)
or\( \tau_1= - \frac{\kappa^3 sin\psi cos^2\psi }{\kappa \tau sin\psi-\kappa' cos \psi} \)
This is required torsion of evolue.
Find the involute of following curves
- \( (\cos t, \sin t) \)
- \( (\cos t, 2 \sin t) \)
- \( (t, t^2) \)
- \( (6 \cos t -4 \cos t ^3, 4 \sin t ^3) \)
- \( (t - \sin t, 1- \cos t) \)
- \( (t, \cosh t) \)
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