求曲线在给定点处的曲率x=a(cost+tsint)y=a(sint-tcost)在t=π/2处

Question

求曲线在给定点处的曲率 x=a(cost+tsint) y=a(sint-tcost) 在t=π/2处

Answer 1

求曲线在给定点处的曲率 x=a(cost+tsint)； y=a(sint-tcost) ；在t=π/2处
y′=dy/dx=(dy/dt)/(dx/dt)=a[cost-(cost-tsint)]/a[-sint+(sint+tcost)]=(tsint)/(tcost)=tant
y″=d²y/dx²=(dy′/dt)/(dx/dt)=(1/cos²t)/(atcost)=1/atcos³t
曲率k=y″/(1+y′²)^(3/2)=(1/atcos³t)/(1+tan²t)^(3/2)=(1/atcos³t)/(1/cos²t)^(3/2)=1/(at)
故当t=π/2时,曲率k=2/aπ