Distance from a point on a circle to an arbitrary axis

AI Thread Summary
The discussion centers on formulating an equation for the shortest distance from a point on a circle's perimeter to an arbitrary axis, incorporating an angle theta. Participants suggest using rotation matrices to find the new coordinates after rotating the point around the origin. The conversation emphasizes the importance of understanding the angles involved and using trigonometry to derive the necessary relationships. Key equations mentioned include the reference line's equation and the perpendicular line's general form. The overall goal is to express the distance as a function of the circle's radius, angle theta, and the angle of the axis.
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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Rotate the whole points around the Origin with angle ##-\beta##
Then you see new y coordinate of thus rotated P is what you want.
 
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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Do you know about rotation matrices?
 
Find in your sketch, say OP has angle ##\theta## to X axis, OP' has angle ##\theta-\beta## so you get y coordinate of P'.
 
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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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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