Evolute and Involute

Evolute and Involute and some Examples

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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 $C$

0,0

–o+←↓↑→

t = 2.00

C

C₁

$$\overrightarrow{t}$$

P

P₁

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Some Examples

1. Involute of circle[evolute] is spiral curve
   0,0

   –o+←↓↑→

   t = 2.00

   C

   C₁

   P

   P₁
2. Involute of cetenary[evolute] is a semicubic parabola
   0,0

   –o+←↓↑→

   t = -0.85

   C₁

   C

   P₁

   P
3. Involute of parabola[evolute] is a semicubic parabola
   0,0

   –o+←↓↑→

   t = -1.73

   C

   C₁

   P

   P₁
4. Involute of parabola[evolute] is a semicubic parabola
   0,0

   –o+←↓↑→

   t = 2.00

   C₁

   $$\overrightarrow{t1}$$

   C

   $$\overrightarrow{t}$$
5. Nephroid
   A nephroid is
   1. an algebraic curve of degree 6.
   2. an epicycloid with two cusps
   3. a plane simple closed curve = a Jordan curve
   with parametric equation
   $x=6\cos (t)-4\cos (t{)}^3,y=4\sin (t{)}^3$
   The nephroid (involute: blue) and its evolute (red) are shown below.
   0,0

   –o+←↓↑→

   t = 2.00

   C₁

   $$\overrightarrow{t1}$$

   C

   $$\overrightarrow{t}$$
6. Involute of cycloid[evolute] is congruent cycloid
   0,0

   –o+←↓↑→

   t = -2.72

   C₁

   C

   P

   P₁

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Involute

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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
$\overrightarrow{t}.\overrightarrow{t_1}=0$...(i)
Let $P$ be a point on $C$ and $P_1$ be its corresponding on $C_1$ with
$\overrightarrow{O{P}_1}=\overrightarrow{r_1}$
$\overrightarrow{OP}=\overrightarrow{r}$
Then,
$\overrightarrow{P{P}_1}$ is tangent to C
or $\overrightarrow{P{P}_1}=\lambda \overrightarrow{t}$
where $\lambda$ is constant to be determined
or $\overrightarrow{O{P}_1}-\overrightarrow{OP}=\lambda \overrightarrow{t}$
or $\overrightarrow{r_1}-\overrightarrow{r}=\lambda \overrightarrow{t}$
or $\overrightarrow{r_1}=\overrightarrow{r}+\lambda \overrightarrow{t}$ ...(ii)
Differentiating (ii) w. r. to. s, we get
$\overrightarrow{t_1}\frac{d{s}_1}{ds}=\overrightarrow{t}+{\lambda }^{\prime }\overrightarrow{t}+\lambda \kappa \overrightarrow{n}$
or $\overrightarrow{t_1}\frac{d{s}_1}{ds}=(1+{\lambda }^{\prime })\overrightarrow{t}+\lambda \kappa \overrightarrow{n}$
Since $\overrightarrow{t}.\overrightarrow{t_1}=0$, we take dot product on both sides by $\overrightarrow{t}$, then we get
$(\overrightarrow{t_1}\frac{d{s}_1}{ds}).\overrightarrow{t}=[(1+{\lambda }^{\prime })\overrightarrow{t}+\lambda \kappa \overrightarrow{n}].\overrightarrow{t}$
or $0=[1+{\lambda }^{\prime }+0]$
or ${\lambda }^{\prime }=-1$
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
$\overrightarrow{r_1}=\overrightarrow{r}+\lambda \overrightarrow{t}$
or $\overrightarrow{r_1}=\overrightarrow{r}+(c-s)\overrightarrow{t}$
This is required equation of involute

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Curvature and torsion of involute

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Let $C$ is given spaace curve and $C_1$ is involute of $C$.
Then equation of involute $C_1$ is
$\overrightarrow{r_1}=\overrightarrow{r}+(c-s)\overrightarrow{t}$ ...(i)
Differentiating (i) w. r. to. s, we get
$\overrightarrow{t_1}\frac{d{s}_1}{ds}=\overrightarrow{t}+(c-s)\kappa \overrightarrow{n}+(-1)\overrightarrow{t}$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}=(c-s)\kappa \overrightarrow{n}$
Taking magnitude, we get
$\frac{d{s}_1}{ds}=(c-s)\kappa$ ...(A)
Substituting value of $\frac{d{s}_1}{ds}=(c-s)\kappa$ , we get
$\overrightarrow{t_1}=\overrightarrow{n}$ ...(ii)
Differentiating (ii) w. r. to. s, we get
${\kappa }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=\tau \overrightarrow{b}-\kappa \overrightarrow{t}$
Taking magnitude, we get
${\kappa }_1\frac{d{s}_1}{ds}=\sqrt{{\tau }^2+{\kappa }^2}$
or$\frac{d{s}_1}{ds}=\frac{\sqrt{{\tau }^2+{\kappa }^2}}{{\kappa }_1}$ ...(B)
Substituting value of $\frac{d{s}_1}{ds}=\frac{\sqrt{{\tau }^2+{\kappa }^2}}{{\kappa }_1}$ , we get
${\kappa }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=\tau \overrightarrow{b}-\kappa \overrightarrow{t}$
or${\kappa }_1\overrightarrow{n_1}\frac{\sqrt{{\tau }^2+{\kappa }^2}}{{\kappa }_1}=\tau \overrightarrow{b}-\kappa \overrightarrow{t}$
or$\overrightarrow{n_1}=\frac{\tau \overrightarrow{b}-\kappa \overrightarrow{t}}{\sqrt{{\tau }^2+{\kappa }^2}}$ ...(iii)
Taking cross product between $\overrightarrow{t_1}$ and $\overrightarrow{n_1}$, we get
$\overrightarrow{t_1}\times \overrightarrow{n_1}=\overrightarrow{n}\times \frac{\tau \overrightarrow{b}-\kappa \overrightarrow{t}}{\sqrt{{\tau }^2+{\kappa }^2}}$
or$\overrightarrow{b_1}=\frac{\tau \overrightarrow{t}+\kappa \overrightarrow{b}}{\sqrt{{\tau }^2+{\kappa }^2}}$ ...(iv)
Differentiating (iv) w. r. to. s, we get
$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=\frac{[\frac{d}{ds}(\tau \overrightarrow{t}+\kappa \overrightarrow{b})]\sqrt{{\tau }^2+{\kappa }^2}-[\frac{d}{ds}\sqrt{{\tau }^2+{\kappa }^2}](\tau \overrightarrow{t}+\kappa \overrightarrow{b})}{{\tau }^2+{\kappa }^2}$
or$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=\frac{[({\tau }^{\prime }\overrightarrow{t}+\tau \kappa \overrightarrow{n}+{\kappa }^{\prime }\overrightarrow{b}-\kappa \tau \overrightarrow{n})]\sqrt{{\tau }^2+{\kappa }^2}-[\frac{1}{2\sqrt{{\tau }^2+{\kappa }^2}}(2\tau {\tau }^{\prime }+2\kappa {\kappa }^{\prime })](\tau \overrightarrow{t}+\kappa \overrightarrow{b})}{{\tau }^2+{\kappa }^2}$
or$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=\frac{[({\tau }^{\prime }\overrightarrow{t}+{\kappa }^{\prime }\overrightarrow{b})]({\tau }^2+{\kappa }^2)-(\tau {\tau }^{\prime }+\kappa {\kappa }^{\prime })(\tau \overrightarrow{t}+\kappa \overrightarrow{b})}{\sqrt{{\tau }^2+{\kappa }^2}({\tau }^2+{\kappa }^2)}$
Collecting the cofficients of $\overrightarrow{t}$ and $\overrightarrow{b}$ in the roght we get
$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=\frac{[{\tau }^{\prime }({\tau }^2+{\kappa }^2)-(\tau {\tau }^{\prime }+\kappa {\kappa }^{\prime })\tau ]\overrightarrow{t}+[{\kappa }^{\prime }({\tau }^2+{\kappa }^2)-(\tau {\tau }^{\prime }+\kappa {\kappa }^{\prime })\kappa ]\overrightarrow{b}}{({\tau }^2+{\kappa }^2)\sqrt{{\tau }^2+{\kappa }^2}}$
or$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=\frac{({\tau }^{\prime }{\tau }^2+{\tau }^{\prime }{\kappa }^2-{\tau }^2{\tau }^{\prime }-\kappa {\kappa }^{\prime }\tau )\overrightarrow{t}+({\kappa }^{\prime }{\tau }^2+{\kappa }^{\prime }{\kappa }^2-\kappa \tau {\tau }^{\prime }-{\kappa }^2{\kappa }^{\prime })\overrightarrow{b}}{({\tau }^2+{\kappa }^2)\sqrt{{\tau }^2+{\kappa }^2}}$
or$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=\frac{({\tau }^{\prime }{\kappa }^2-\kappa {\kappa }^{\prime }\tau )\overrightarrow{t}+({\kappa }^{\prime }{\tau }^2-\kappa \tau {\tau }^{\prime })\overrightarrow{b}}{({\tau }^2+{\kappa }^2)\sqrt{{\tau }^2+{\kappa }^2}}$
or$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=\frac{({\tau }^{\prime }\kappa -{\kappa }^{\prime }\tau )\kappa \overrightarrow{t}+({\kappa }^{\prime }\tau -\kappa {\tau }^{\prime })\tau \overrightarrow{b}}{({\tau }^2+{\kappa }^2)\sqrt{{\tau }^2+{\kappa }^2}}$
Substituting the value of $\overrightarrow{n_1}=\frac{\tau \overrightarrow{b}-\kappa \overrightarrow{t}}{\sqrt{{\tau }^2+{\kappa }^2}}$, we get
$-{\tau }_1\frac{\tau \overrightarrow{b}-\kappa \overrightarrow{t}}{\sqrt{{\tau }^2+{\kappa }^2}}\frac{d{s}_1}{ds}=\frac{(\kappa {\tau }^{\prime }-{\kappa }^{\prime }\tau )\kappa \overrightarrow{t}-(\kappa {\tau }^{\prime }-{\kappa }^{\prime }\tau )\tau \overrightarrow{b}}{({\tau }^2+{\kappa }^2)\sqrt{{\tau }^2+{\kappa }^2}}$
or${\tau }_1\frac{\kappa \overrightarrow{t}-\tau \overrightarrow{b}}{\sqrt{{\tau }^2+{\kappa }^2}}\frac{d{s}_1}{ds}=\frac{(\kappa {\tau }^{\prime }-{\kappa }^{\prime }\tau )(\kappa \overrightarrow{t}-\tau \overrightarrow{b})}{({\tau }^2+{\kappa }^2)\sqrt{{\tau }^2+{\kappa }^2}}$
Removing $\frac{\kappa \overrightarrow{t}-\tau \overrightarrow{b}}{\sqrt{{\tau }^2+{\kappa }^2}}$ from both side, we get
${\tau }_1\frac{d{s}_1}{ds}=\frac{\kappa {\tau }^{\prime }-{\kappa }^{\prime }\tau }{{\tau }^2+{\kappa }^2}$
or$\frac{d{s}_1}{ds}=\frac{\kappa {\tau }^{\prime }-{\kappa }^{\prime }\tau }{{\tau }_1({\tau }^2+{\kappa }^2)}$ ...(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
$(c-s)\kappa =\frac{\kappa {\tau }^{\prime }-{\kappa }^{\prime }\tau }{{\tau }_1({\tau }^2+{\kappa }^2)}$
or${\tau }_1=\frac{\kappa {\tau }^{\prime }-{\kappa }^{\prime }\tau }{\kappa (c-s)({\tau }^2+{\kappa }^2)}$
This is required torsion of involute.

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Evolute

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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
$\overrightarrow{t}.\overrightarrow{t_1}=0$
NOTE
Evolute of the curve is the locus of the centre of curvature for that curve.

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Equation of Evolute

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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
$\overrightarrow{t_1}=(\lambda \overrightarrow{n}+\mu \overrightarrow{b})A$ where A is constant
or $\overrightarrow{t_1}=A\lambda \overrightarrow{n}+A\mu \overrightarrow{b}$…(i)
Let $P$ be a point on $C$ and $P_1$ be its corresponding on $C_1$ with
$\overrightarrow{O{P}_1}=\overrightarrow{r_1}$
$\overrightarrow{OP}=\overrightarrow{r}$
Then,
$\overrightarrow{P{P}_1}$ is normal to C
or $\overrightarrow{P{P}_1}=\lambda \overrightarrow{n}+\mu \overrightarrow{b}$
where $\lambda$ and $\mu$ are constants to be determinedor
or $\overrightarrow{O{P}_1}-\lambda \overrightarrow{n}+\mu \overrightarrow{b}$
or $\overrightarrow{r_1}-\overrightarrow{r}=\lambda \overrightarrow{n}+\mu \overrightarrow{b}$
or $\overrightarrow{r_1}=\overrightarrow{r}+\lambda \overrightarrow{n}+\mu \overrightarrow{b}$ …(ii)
Differentiating w. r. to, s, we get
$\overrightarrow{t_1}\frac{d{s}_1}{ds}=\overrightarrow{t}+{\lambda }^{\prime }\overrightarrow{n}+\lambda (\tau \overrightarrow{b}-\kappa \overrightarrow{t})+{\mu }^{\prime }\overrightarrow{b}+\mu (-\tau \overrightarrow{n})$
or $\overrightarrow{t_1}\frac{d{s}_1}{ds}=\overrightarrow{t}+{\lambda }^{\prime }\overrightarrow{n}+\lambda \tau \overrightarrow{b}-\lambda \kappa \overrightarrow{t}+{\mu }^{\prime }\overrightarrow{b}-\mu \tau \overrightarrow{n}$
Collecting cofficients of $\overrightarrow{t},\overrightarrow{n},\overrightarrow{b}$, we get
$\overrightarrow{t_1}\frac{d{s}_1}{ds}=(1-\lambda \kappa )\overrightarrow{t}+({\lambda }^{\prime }-\mu \tau )\overrightarrow{n}+(\lambda \tau +{\mu }^{\prime })\overrightarrow{b}$…(iii)
Comparing (i) and (ii), we get
$\frac{d{s}_1}{ds}=\frac{1-\lambda \kappa }{0}=\frac{{\lambda }^{\prime }-\mu \tau }{A\lambda }=\frac{\lambda \tau +{\mu }^{\prime }}{A\mu }$
Equating first two and last two part, we get
$\frac{d{s}_1}{ds}=\frac{1-\lambda \kappa }{0}$ and $\frac{{\lambda }^{\prime }-\mu \tau }{A\lambda }=\frac{\lambda \tau +{\mu }^{\prime }}{A\mu }$
or $1-\lambda \kappa =0$ and $\frac{{\lambda }^{\prime }-\mu \tau }{\lambda }=\frac{\lambda \tau +{\mu }^{\prime }}{\mu }$
or $\lambda =\frac{1}{\kappa }$ and $\mu ({\lambda }^{\prime }-\mu \tau )=\lambda (\lambda \tau +{\mu }^{\prime })$
or $\lambda =\rho$ and $\mu {\lambda }^{\prime }-{\mu }^2\tau ={\lambda }^2\tau +\lambda {\mu }^{\prime }$
or $\lambda =\rho$ and $({\lambda }^2+{\mu }^2)\tau ={\lambda }^{\prime }\mu -\lambda {\mu }^{\prime }$
or $\lambda =\rho$ and $\tau =\frac{{\lambda }^{\prime }\mu -\lambda {\mu }^{\prime }}{{\lambda }^2+{\mu }^2}$ $\frac{d}{ds}ta{n}^{-1}\frac{\lambda }{\mu }=\frac{{\lambda }^{\prime }\mu -\lambda {\mu }^{\prime }}{{\lambda }^2+{\mu }^2}$ भएकोले
or $\lambda =\rho$ and $\tau =\frac{d}{ds}ta{n}^{-1}\frac{\lambda }{\mu }$
Integrating right part w. r. to. s,we get
$\lambda =\rho$ and $\int \tau ds=ta{n}^{-1}\frac{\lambda }{\mu }$
or $\lambda =\rho$ and $tan(\int \tau ds)=\frac{\lambda }{\mu }$
$\int \tau ds=\varphi (s)+c$ मानौ, then $\frac{d}{ds}\varphi (s)={\varphi }^{\prime }(s)={\varphi }^{\prime }=\tau$
Now we have
$\lambda =\rho$ and $tan(\int \tau ds)=\frac{\lambda }{\mu }$
or $\lambda =\rho$ and $tan(\varphi +c)=\frac{\lambda }{\mu }$
or $\lambda =\rho$ and $\mu =\frac{\lambda }{tan(\varphi +c)}$
or $\lambda =\rho$ and $\mu =\lambda cot(\varphi +c)$ $cot(\varphi +c)=-tan(\frac{\pi }{2}+(\varphi +c))$ भएकोले
or $\lambda =\rho$ and $\mu =-\lambda tan(\frac{\pi }{2}+(\varphi +c))$
or $\lambda =\rho$ and $\mu =-\lambda tan(\frac{\pi }{2}+\varphi +c)$
$\frac{\pi }{2}+\varphi +c=\psi$ मानौ, then $\frac{d}{ds}\psi ={\psi }^{\prime }={\varphi }^{\prime }=\tau$
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,
$\overrightarrow{r_1}=\overrightarrow{r}+\lambda \overrightarrow{n}+\mu \overrightarrow{b}$
or $\overrightarrow{r_1}=\overrightarrow{r}+\rho \overrightarrow{n}-\rho tan(\psi )\overrightarrow{b}$ where ${\psi }^{\prime }=\tau$
this is the required equation of evolute.

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Curvature and Torsion of Evolute

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Let $C$ is given space curve and $C_1$ is evolute of $C$.
Then equation of $C_1$ is
$\overrightarrow{r_1}=\overrightarrow{r}+\rho \overrightarrow{n}-\rho tan\psi \overrightarrow{b}$(i)
Differentiating w. r. to, s, we get
$\overrightarrow{t_1}\frac{d{s}_1}{ds}=\overrightarrow{t}+{\rho }^{\prime }\overrightarrow{n}+\rho (\tau \overrightarrow{b}-\kappa \overrightarrow{t})-{\rho }^{\prime }tan\psi \overrightarrow{b}-\rho (se{c}^2\psi {\psi }^{\prime })\overrightarrow{b}-\rho tan\psi (-\tau \overrightarrow{n})$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}=\overrightarrow{t}+{\rho }^{\prime }\overrightarrow{n}+\rho \tau \overrightarrow{b}-\overrightarrow{t}-{\rho }^{\prime }tan\psi \overrightarrow{b}-\rho se{c}^2\psi \tau \overrightarrow{b}+\rho \tau tan\psi \overrightarrow{n}$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}={\rho }^{\prime }\overrightarrow{n}+\rho \tau \overrightarrow{b}-{\rho }^{\prime }tan\psi \overrightarrow{b}-\rho se{c}^2\psi \tau \overrightarrow{b}+\rho \tau tan\psi \overrightarrow{n}$
Collecting cofficient of $\overrightarrow{n}$ and $\overrightarrow{b}$, we get
$\overrightarrow{t_1}\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )\overrightarrow{n}+(\rho \tau -{\rho }^{\prime }tan\psi -\rho se{c}^2\psi \tau )\overrightarrow{b}$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )\overrightarrow{n}+(\rho \tau -\rho \tau se{c}^2\psi -{\rho }^{\prime }tan\psi )\overrightarrow{b}$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )\overrightarrow{n}+[\rho \tau (1-se{c}^2\psi )-{\rho }^{\prime }tan\psi ]\overrightarrow{b}$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )\overrightarrow{n}+[\rho \tau (-ta{n}^2\psi )-{\rho }^{\prime }tan\psi ]\overrightarrow{b}$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )\overrightarrow{n}+(-\rho \tau ta{n}^2\psi -{\rho }^{\prime }tan\psi )\overrightarrow{b}$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )\overrightarrow{n}-(\rho \tau ta{n}^2\psi +{\rho }^{\prime }tan\psi )\overrightarrow{b}$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )\overrightarrow{n}-(\rho \tau tan\psi +{\rho }^{\prime })tan\psi \overrightarrow{b}$
or$\overrightarrow{t_1}\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )(\overrightarrow{n}-tan\psi \overrightarrow{b})$
Taking Magnitude, we get
$\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )\sqrt{1+ta{n}^2\psi }$
or$\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )sec\psi$(A)
Substituting the value of $\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )sec\psi$, we get
$\overrightarrow{t_1}\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )(\overrightarrow{n}-tan\psi \overrightarrow{b})$
or$\overrightarrow{t_1}({\rho }^{\prime }+\rho \tau tan\psi )sec\psi =({\rho }^{\prime }+\rho \tau tan\psi )(\overrightarrow{n}-tan\psi \overrightarrow{b})$
or$\overrightarrow{t_1}=\frac{\overrightarrow{n}-tan\psi \overrightarrow{b}}{sec\psi }$
or$\overrightarrow{t_1}=cos\psi \overrightarrow{n}-sin\psi \overrightarrow{b}$ (ii)
Differentiating w. r. to, s, we get
${\kappa }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=(-sin\psi \tau )\overrightarrow{n}+cos\psi (\tau \overrightarrow{b}-\kappa \overrightarrow{t})-(cos\psi \tau )\overrightarrow{b}-sin\psi (-\tau \overrightarrow{n})$
or${\kappa }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=-sin\psi \tau \overrightarrow{n}+cos\psi \tau \overrightarrow{b}-cos\psi \kappa \overrightarrow{t}-cos\psi \tau \overrightarrow{b}+sin\psi \tau \overrightarrow{n}$
or${\kappa }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=-cos\psi \kappa \overrightarrow{t}$
Taking Magnitude, we get
${\kappa }_1\frac{d{s}_1}{ds}=cos\psi \kappa$
or$\frac{d{s}_1}{ds}=\frac{cos\psi \kappa }{{\kappa }_1}$ (B)
Substituting the value of $\frac{d{s}_1}{ds}=\frac{cos\psi \kappa }{{\kappa }_1}$, we get
${\kappa }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=-cos\psi \kappa \overrightarrow{t}$
or${\kappa }_1\overrightarrow{n_1}\frac{cos\psi \kappa }{{\kappa }_1}=-cos\psi \kappa \overrightarrow{t}$
or$\overrightarrow{n_1}=-\overrightarrow{t}$ (iii)
Taking cross product between $\overrightarrow{t_1}$ and $\overrightarrow{n_1}$, we get
$\overrightarrow{t_1}\times \overrightarrow{n_1}=(cos\psi \overrightarrow{n}-sin\psi \overrightarrow{b})\times (-\overrightarrow{t})$
or$\overrightarrow{b_1}=cos\psi \overrightarrow{b}+sin\psi \overrightarrow{n}$ (iv)
Differentiating w. r. to, s, we get
$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=(-sin\psi \tau )\overrightarrow{b}+cos\psi (-\tau \overrightarrow{n})+(cos\psi \tau )\overrightarrow{n}+sin\psi (\tau \overrightarrow{b}-\kappa \overrightarrow{t})$
or$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=-sin\psi \tau \overrightarrow{b}-cos\psi \tau \overrightarrow{n}+cos\psi \tau \overrightarrow{n}+sin\psi \tau \overrightarrow{b}-sin\psi \kappa \overrightarrow{t}$
or$-{\tau }_1\overrightarrow{n_1}\frac{d{s}_1}{ds}=-sin\psi \kappa \overrightarrow{t}$
Taking Magnitude, we get
${\tau }_1\frac{d{s}_1}{ds}=-sin\psi \kappa$
or$\frac{d{s}_1}{ds}=-\frac{sin\psi \kappa }{{\tau }_1}$(C)
Here,
$\frac{d{s}_1}{ds}=({\rho }^{\prime }+\rho \tau tan\psi )sec\psi$(A)
or$\frac{d{s}_1}{ds}=[\frac{-{\kappa }^{\prime }}{{\kappa }^2}+\frac{1}{\kappa }\tau \frac{sin\psi }{cos\psi }]\frac{1}{cos\psi }$
or$\frac{d{s}_1}{ds}=[\frac{-{\kappa }^{\prime }}{{\kappa }^2}+\frac{\tau sin\psi }{\kappa cos\psi }]\frac{1}{cos\psi }$
or$\frac{d{s}_1}{ds}=\frac{-{\kappa }^{\prime }cos\psi +\kappa \tau sin\psi }{{\kappa }^{2}cos\psi }\times \frac{1}{cos\psi }$
or$\frac{d{s}_1}{ds}=\frac{\kappa \tau sin\psi -{\kappa }^{\prime }cos\psi }{{\kappa }^{2}co{s}^2\psi }$(A)
Equating (A) and (B), we get
$\frac{\kappa \tau sin\psi -{\kappa }^{\prime }cos\psi }{{\kappa }^{2}co{s}^2\psi }=\frac{cos\psi \kappa }{{\kappa }_1}$
or${\kappa }_1=\frac{{\kappa }^{3}co{s}^3\psi }{\kappa \tau sin\psi -{\kappa }^{\prime }cos\psi }$
This is required curvature of evolue.
Next,
Equating (A) and (C), we get
$\frac{\kappa \tau sin\psi -{\kappa }^{\prime }cos\psi }{{\kappa }^{2}co{s}^2\psi }=-\frac{sin\psi \kappa }{{\tau }_1}$
or${\tau }_1=-\frac{{\kappa }^{3}sin\psi co{s}^2\psi }{\kappa \tau sin\psi -{\kappa }^{\prime }cos\psi }$
This is required torsion of evolue.

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Find the involute of following curves

1. $(\cos t,\sin t)$
2. $(\cos t,2\sin t)$
3. $(t,t^2)$
4. $(6\cos t-4\cos t^3,4\sin t^3)$
5. $(t-\sin t,1-\cos t)$
6. $(t,\cosh t)$