Math Is Fun Forum

onako

Member

Registered: 2010-03-28

Posts: 43

Circle to tangent mapping

Assume you're given a circle with the line AB containing its center O, such that A and B are on the circle (OA=OB=radius). A tangent t is drawn on the point A, and
I should calculate the mapping of certain points (a,b,c,d...) of the circle to the points on the tangent (at, bt, ct, dt, ...) such that the distance Aa (the distance along the circle) is the same as the distance Aat (the distance along the tangent) (and the same for the distances Ab, Ac, Ad). But, here, certain constraint should be considered: those points of the circle (among (a, b, c, d)) that are from one side of the circle from A to B should be placed on one side of the tangent (the nearer), and those from the other side of the circle form A to B should be placed on the other side. Basically, the circle should be split at B, and then mapped to the tangent. I hope this explanation is sufficient enough.

It should be noted that I have information about coordinates of A, B, O, a, b, c, d. I supposed to calculate (at, bt, ct, dt).
For solving this problem, I have two approaches, but I'm not sure how I could make sure they always work correctly.

1) I calculate the equation of the tangent at point A. Then for each point (a, b, c, d) I calculate the distance from A (along the circle), and use these distances for calculating (at, bt, ct, dt...) along the tangent. What I dont know here is how to calculate the distances
from A to (a, b, c, d). The problem is the 'proper side' determination, meaning how should I determine whether the point should be mapped on one side of the tangent or the other. What would be the way to determine this.

2) I calculate the equation of the tangent at point A. Then for each point (a, b, c, d) I calculate the distance from A (along the circle), and use these distances for calculating (at, bt, ct, dt...) along the tangent. To determine the 'proper side' of a given point, I might use the projection of that point to the tangent. But, even with this, how I know 'which side is which'? Perhaps there are much simpler ways to do this.

Any suggestion on how to do this is welcome. In case I was not precise enough, I'll elaborate.
Thanks.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Circle to tangent mapping

Hi onako;

It should be noted that I have information about coordinates of A, B, O, a, b, c, d.

Please provide all the info you know and even anything you think you know. Just do not provide the story of how the dinosaurs were wiped out, I know that one.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

onako

Member

Registered: 2010-03-28

Posts: 43

Re: Circle to tangent mapping

I dont understand your comment (how constructive can it be?).

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Circle to tangent mapping

Hi onako;

How about the coordinates you speak of. If you want distances between points? Or do you just want general expressions?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.