MHB How Do You Calculate the Radius of Curvature for Complex Curves?

Click For Summary
SUMMARY

The radius of curvature for complex curves can be calculated using specific formulas. For the cycloid defined by the equations x = a(θ + sinθ) and y = a(1 - cosθ), the radius of curvature can be derived at any point. Additionally, for the curve x³ + y³ = 3axy, the radius of curvature at the point (3a/2, 3a/2) can also be determined. Understanding the concept of curvature and having access to relevant formulas is essential for solving these problems effectively.

PREREQUISITES
  • Understanding of calculus, particularly derivatives and integrals.
  • Familiarity with parametric equations and their applications.
  • Knowledge of curvature and its mathematical definitions.
  • Access to resources such as textbooks or online references on radius of curvature.
NEXT STEPS
  • Study the derivation of the radius of curvature for parametric curves.
  • Learn how to apply the curvature formula to different types of curves.
  • Explore the use of software tools like Mathematica or MATLAB for visualizing curvature.
  • Research additional examples of calculating radius of curvature for various complex curves.
USEFUL FOR

Students in mathematics, engineers dealing with curve design, and anyone interested in advanced calculus applications will benefit from this discussion.

abhik5195
Messages
1
Reaction score
0
1) Find the radius of curvature at any point of the cycloid x = a(\theta + sin\theta)y = a(1- cos\theta).
2) Find the radius of curvature at the point (3a/2 , 3a/2) for the curve x3 + y3 = 3axy
 
Physics news on Phys.org
What do you mean by "help"? Do your homework for you? It is impossible to actually help without knowing what you do understand and what you can do yourself. Do you know what "radius of curvature" means? Do you know how to find the curvature of a one dimensional curve? Have you not at least attempted to do this problem yourself?

Does your textbook give no formulas for "radius of curvature"? Did you "google" "radius of curvature"? When I did I got https://en.wikipedia.org/wiki/Radius_of_curvature
which seems to be exactly what you need.
 

Thread 'How does this derivative work? And why the speed of light remains?'

Relativistic Momentum, Mass, and Energy Momentum and mass (...), the classic equations for conserving momentum and energy are not adequate for the analysis of high-speed collisions. (...) The momentum of a particle moving with velocity ##v## is given by $$p=\cfrac{mv}{\sqrt{1-(v^2/c^2)}}\qquad{R-10}$$ ENERGY In relativistic mechanics, as in classic mechanics, the net force on a particle is equal to the time rate of change of the momentum of the particle. Considering one-dimensional...
View full post »

Similar threads

Undergrad Landscape Fabric Roll
  • · Replies 8 ·
Replies
8
Views
3K
High School Trigonometric substitution, a case I'd like to share
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Having trouble understanding a seemingly simple u-substitution
  • · Replies 5 ·
Replies
5
Views
2K
High School Having trouble deriving the volume of an elliptical ring toroid
  • · Replies 4 ·
Replies
4
Views
4K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad How to evaluate the enclosed area of this implicit curve?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Is My Calculus of Variations Approach Correct?
  • · Replies 12 ·
Replies
12
Views
3K
MHB Using advanced calculus for finding values
  • · Replies 2 ·
Replies
2
Views
1K
High School Why is this definite integral a single number?
  • · Replies 20 ·
Replies
20
Views
4K
Undergrad Troubleshooting a difficult integral
  • · Replies 6 ·
Replies
6
Views
2K