Distance from a point on a circle to an arbitrary axis

In summary, the equation for the shortest distance from a point on a circle perimeter to an arbitrary axis in a circle with angle theta is y=-\dfrac{1}{\tan\beta}x+b. The intercept b is found by using the information in items 3 and 4 to find the coordinates of the intersection of the two lines.
  • #1
Slipjoints
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Homework Statement
Formulate an equation for the shortest distance from a point to a line
Relevant Equations
Pythagorean Theorem / Thales?
Hi all! In this assignment I have to formulate an equation for the shortest distance from a point on a circle perimeter to an arbitrary axis in a circle with angle theta. I included an image with the sketch. Anyone that can help?
 

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  • #2
Rotate the whole points around the Origin with angle ##-\beta##
Then you see new y coordinate of thus rotated P is what you want.
 
  • #3
Thanks for the reply, I get what you mean but can't seem to get the new point P' through rotation. Do you know how to write it down? I added what you meant to my sketch for clarification.
 

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  • #4
Do you know about rotation matrices?
 
  • #5
Find in your sketch, say OP has angle ##\theta## to X axis, OP' has angle ##\theta-\beta## so you get y coordinate of P'.
 
  • #6
1. You know that the equation of the reference line is ##y=(\tan\beta)x.##
2. You know that a line perpendicular to the reference line has the general form ##y=-\dfrac{1}{\tan\beta}x+b.##
3. You also know the coordinates ##\{x_P,~y_P\}## of the given point P.
4. Use the information in item 3 to find the intercept ##b## in item 2.
5. Find the coordinates of the intersection of the two lines.
 
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  • #7
Slipjoints said:
Homework Statement:: Formulate an equation for the shortest distance from a point to a line
Relevant Equations:: Pythagorean Theorem / Thales?

Hi all! In this assignment I have to formulate an equation for the shortest distance from a point on a circle perimeter to an arbitrary axis in a circle with angle theta. I included an image with the sketch. Anyone that can help?
Hi @Slipjoints.

Let the circle’s centre be ‘O' and let ‘Q’ be the point on the line closest to P.

Have you given us the complete/accurate question? I’m guessing that you are told the circle’s radius is R and are told θ (which you haven’t marked) is the angle, measured anticlockwise, between the +x axis and OP.

Edit: And you want the distance (PQ) as a function of R, θ and β.
____________

Draw triangle OPQ. Note it is right-angled and that OP = R. Can you work out ∠POQ (or ∠QPO)?

If you can, the rest is (very) simple trigonometry.
 
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FAQ: Distance from a point on a circle to an arbitrary axis

1. What is the formula for finding the distance from a point on a circle to an arbitrary axis?

The formula for finding the distance from a point on a circle to an arbitrary axis is d = |r - a|, where d is the distance, r is the radius of the circle, and a is the distance from the point to the axis.

2. Can this formula be used for any point on the circle?

Yes, this formula can be used for any point on the circle as long as the radius and distance from the point to the axis are known.

3. How is the distance affected if the point is located inside or outside of the circle?

If the point is located inside the circle, the distance will be negative. If the point is located outside of the circle, the distance will be positive.

4. Is there a different formula for finding the distance from a point on a circle to a horizontal or vertical axis?

No, the same formula can be used for finding the distance from a point on a circle to a horizontal or vertical axis. The only difference is the value of a, which will be the distance from the point to the horizontal or vertical axis, respectively.

5. Can this formula be used for any type of circle, such as an ellipse or a parabola?

No, this formula is specifically for circles. Different formulas would need to be used for finding the distance from a point on an ellipse or a parabola to an arbitrary axis.

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