How can I find the angle of a curve in a shape with a given length and radius?

In summary, the conversation revolved around finding the angle of a curve in a given shape. The person was wondering what information was needed to determine this angle, and the response explained that it would depend on the length of the curve and the radius of the circle. The formula for finding the angle was also provided. Overall, it was a more complex process than expected, but the person appreciated the help.
  • #1
uperkurk
167
0
Hello everyone, I've been googling how to find the angle of a curve but the results are not the kind I'm looking for.

Let's say I have a shape that has a curve in it at some point. Something like this.

w97xc0V.png


I'm curious what I need to be reading in order to find the angle of the curve. what information do I need to know about the top part in order to find the angle at which the line curves?
 
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  • #2
First you will have to tell us what you mean by "angle of a curve". In this picture, does the circle meet the triangle "smoothly"?
 
  • #3
I've labled the shape a bit better.

L8yFNfP.png


I want to measure the angle of [itex]a[/itex] and [itex]b[/itex] and then I want to find out the degree at which the line curves.

Is is a smooth 70 degree curve? Maybe a smooth 60 degree curve?

This is basically what I'm asking myself. Sorry I'm not being that clear, I'm not even sure if what I'm asking can be solved. I'm just playing with what I'm learned so far.
 
  • #4
No, this is not going to have any simple answer like '60' or '70'. The angles will depend both on the length of ab and the radius of the circle.

I am going to assume ab is less than the diameter of the circle. In particular, drawing lines from a and b to the center of the circle, call it O, gives a triangle with two sides of length Oa= Ob= r, the radius of the circle and one side of length ab. If we call the angle Oa and Ob make, then, by the cosine law, [itex]ab^2= 2r^2- 2r^2cos(\theta)= 2r^2(1- cos(\theta))[/itex]. From that, [itex]cos(\theta)= (2r^2- ab^2)/2r^2[/itex] so that the angle between Oa and ab is [itex]\theta= arccos(2r^2- ab^2)/2r^2[/itex]. Since the angle between a tangent to a circle and a radius is 90 degrees, to find the angle between ab and the tangent, add 90 degrees to that:
[itex]arccos(2r^2- ab^2)/2r^2)+ 90[/itex] degrees.
 
  • #5
HallsofIvy said:
No, this is not going to have any simple answer like '60' or '70'. The angles will depend both on the length of ab and the radius of the circle.

I am going to assume ab is less than the diameter of the circle. In particular, drawing lines from a and b to the center of the circle, call it O, gives a triangle with two sides of length Oa= Ob= r, the radius of the circle and one side of length ab. If we call the angle Oa and Ob make, then, by the cosine law, [itex]ab^2= 2r^2- 2r^2cos(\theta)= 2r^2(1- cos(\theta))[/itex]. From that, [itex]cos(\theta)= (2r^2- ab^2)/2r^2[/itex] so that the angle between Oa and ab is [itex]\theta= arccos(2r^2- ab^2)/2r^2[/itex]. Since the angle between a tangent to a circle and a radius is 90 degrees, to find the angle between ab and the tangent, add 90 degrees to that:
[itex]arccos(2r^2- ab^2)/2r^2)+ 90[/itex] degrees.


Thanks so much. Quite a bit more work than I thought but thanks for making the effort to help :)
 

What is the angle of a curve?

The angle of a curve refers to the amount of rotation needed to change the direction of a curve at a specific point. It is typically measured in degrees or radians.

How is the angle of a curve calculated?

The angle of a curve can be calculated using the slope of the curve at a given point. This can be found by taking the derivative of the curve's equation and plugging in the x-value of the point.

What is the difference between the angle of a curve and the curvature of a curve?

The angle of a curve refers to the direction of the curve at a specific point, while the curvature of a curve refers to its overall shape and how much it deviates from a straight line.

Can the angle of a curve be negative?

Yes, the angle of a curve can be negative if the curve is turning in a clockwise direction at a given point. A positive angle indicates a counterclockwise turn.

How is the angle of a curve used in real-world applications?

The angle of a curve is used in various fields such as engineering, physics, and navigation. It helps to determine the direction of a moving object, the design of roads and highways, and the trajectory of projectiles.

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