Distance from a point on a circle to an arbitrary axis

[Slipjoints]

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?

 

[anuttarasammyak]

Rotate the whole points around the Origin with angle $-\beta$
Then you see new y coordinate of thus rotated P is what you want.

 

[Slipjoints]

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.

 

[PeroK]

Do you know about rotation matrices?

 

[anuttarasammyak]

Find in your sketch, say OP has angle $\theta$ to X axis, OP' has angle $\theta-\beta$ so you get y coordinate of P'.

 

[kuruman]

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.

 

[Steve4Physics]

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.