i) Angle between radius vector and tangent vector ii) Angle of intersection between two polar curves

i) Angle between radius vector and tangent vector

Algorithm:

·       For a given curve $$r=f(\theta)$$, find $$\phi=cot^-1(\frac{r'}{r})$$, where $$r'=dr/d\theta$$

clear

clc

close

syms theta

%Get the curve

r1=input("Enter the first curve r1(theta): r1=");

%Find phi1

phi1=simplify(acot(diff(r1)/r1));

%display phi1

fprintf("\n Angle between radius vector and tangent vector to the curve %s:\n phi= %s",r1,phi1)

 

Output

Enter the first curve r1(theta): r1=2*(1+cos(theta))

Angle between radius vector and tangent vector to the curve 2*cos(theta) + 2:

 phi1= -acot(sin(theta)/(cos(theta) + 1))

 

ii) Angle of intersection between two polar curves

Algorithm:

·      For a given curve $$r_1=f_1(\theta)$$ and $$r_2=f_2(\theta)$$ find, $$\phi=cot^{-1}(\frac{r'}{r})$$

 where $$r'=dr/d\theta$$

·       Solve given curves for theta value

·       Get the required appropriate theta value from the solution

·       Find $$|\phi_2-\phi_1|$$ by substituting obtained  theta value

clear

clc

close

syms theta

 

%Get the two curves

r1=input("Enter the first curve r1(theta): r1=");

r2=input("Enter the second curve r2(theta): r2=");

%Find phi1 and phi2

 

phi1=simplify(acot(diff(r1)/r1));

phi2=simplify(acot(diff(r2)/r2));

 

%display phi1 and phi2

fprintf("\n Angle between radius vector and tangent vector to the curve %s:\n phi1= %s",r1,phi1)

fprintf("\n Angle between radius vector and tangent vector to the curve %s:\n phi2= %s",r2,phi2)

 

%Find t(point of intersection)

fprintf('\n The values of theta at the point of intersection are: ')

S=solve(r1==r2,theta,'Real',true)

 

%Get the theta

tt=input("\n Choose the value of theta: ");

 

%Calculate the angle between the given curves

ang1= abs(vpa(subs(phi1-phi2, {theta},{tt})));

ang2=vpa(pi-ang1);

fprintf('\n Angle between given polar curves = %f or %f \n', ang1, ang2);

Output

Enter the first curve r1(theta): r1=2*(1+cos(theta))

Enter the second curve r2(theta): r2=2*(1-cos(theta))

Angle between radius vector and tangent vector to the curve 2*cos(theta) + 2:

phi1= -acot(sin(theta)/(cos(theta) + 1))

Angle between radius vector and tangent vector to the curve 2 - 2*cos(theta):

phi2= -acot(sin(theta)/(cos(theta) - 1))

The values of theta at the point of intersection are: S =

pi/2

Choose the value of theta: pi/2

Angle between given polar curves = 1.570796 or 1.570796