What is the Derivation for a Point on a Circle?

In summary, the conversation is about a person who is working on a problem involving a bead on wire and is stuck on a derivation detail. They followed the same approach as Andrew Witkin and are now having trouble with the third line of the equation. They shared their derivation process and asked for help, but later on, they were able to solve the problem on their own.
  • #1
ivjohn
2
0
Hi All,

I am working on the problem of bead on wire and got stuck on some basic derivation detail.
I took the same approach as Andrew Witkin used in his slide (page 13):
http://www.cs.cmu.edu/~baraff/sigcourse/slidesf.pdf

Here is the screenshot of the slide page 13:
http://desmond.imageshack.us/Himg831/scaled.php?server=831&filename=pocs.png&res=medium

As we can see, C is the function of x and r is constant. x is a function of t as well.

I got stuck on deriving the third line of the equation (the derivative of N with respect to t). My derivation result seems different from what the author proposed.
Here is my step by step derivation:
http://img269.imageshack.us/img269/8623/derivative.png

Could somebody point out my mistake? Any help will be greatly appreciated.

Regards,
John.
 
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  • #2
Nevermind. I have solved this problem.
 

1. What is the formula for finding the coordinates of a point on a circle?

The formula for finding the coordinates of a point on a circle is (x,y) = (h + r cosθ, k + r sinθ), where (h,k) is the center of the circle, r is the radius, and θ is the angle from the positive x-axis to the point on the circle.

2. How do you derive the formula for finding the coordinates of a point on a circle?

The formula can be derived using basic trigonometric identities and the Pythagorean theorem. By drawing a right triangle with one side on the x-axis and another side on the circle, the coordinates of the point can be expressed in terms of the angle and the radius.

3. Can the formula be used for any type of circle?

Yes, the formula can be used for any type of circle, including circles that are not centered at the origin. However, the coordinates of the center and the radius must be known in order to use the formula.

4. How accurate is the formula for finding the coordinates of a point on a circle?

The formula is accurate as long as the center and radius of the circle are known and the point is on the circle. However, rounding errors may occur when using decimal values for the coordinates.

5. Are there any other methods for finding the coordinates of a point on a circle?

Yes, there are other methods such as using the distance formula or using the slope of the tangent line to find the coordinates of a point on a circle. However, the formula for finding the coordinates using trigonometry is the most commonly used and efficient method.

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