CODEWITHSUNDEEP
c#.netmath
anon
anon

Reputation:

How to find a point that is on the same circle given an angle?

I have the following setting:

alt text

I know P1, P2 and the angle alpha, now how do i calculate the coordinates of P3? (Note: P3 is on the same circle with origin P1 and radius P1P2)

The blue lines indicate the coordinate system

Upvotes: 2

Views: 609

Answers (5)

Josephine
Josephine

Reputation: 1449

Complex_To_Vector(Vector_To_Complex(P_2 - P_1) * ei*alpha) + P_1.

(Just for fun -- not a serious suggestion)

Upvotes: 2

ChrisF
ChrisF

Reputation: 137188

If you rotate the vector P1->P2 by alpha about P1 you'll get the vector P1->P3. Then knowing P1 you can get P3.

The basic equation for rotation about the origin is:

[ cos(α) -sin(α) ] [x]
[ sin(α)  cos(α) ] [y]

You might have to change the signs with your coordinate system, but I always have to it by trial and error as I can never remember!

Don't forget - as S.C. Madsen says sin and cos expect angles to be in radians not degrees.

Wikipeida's article on rotation has more information.

Upvotes: 4

Simon Fischer
Simon Fischer

Reputation: 3886

The formula stated above from Wikipedia is usable here to rotate the vector P1->P2 (V12). 

V12 = [0, -100]

When rotated (beware of α is -30 degrees in your drawing) the vector P1->P3 becomes

x' = V12(x)*cos(α) - V12(y)*sin(α) = 0*cos(-30) - (-100)*sin(-30) = -50
y' = V12(x)*sin(α) + V12(y)*cos(α) = 0*sin(-30) + (-100)*cos(-30) = -86.6

When translated with the point P1 the coordinates for P3 becomes

[x, y] = [-50+150, -86.6+210] = [100, 123.4]

Upvotes: 7

S.C. Madsen
S.C. Madsen

Reputation: 5256

The angle has to be in radians when invoking sin and cos, you state the angle alpha is 30, so it seems to be in degrees. Other than that I think 'gspr' and 'ChrisF' have given excellent advice on how to solve this.

Upvotes: 1

gspr
gspr

Reputation: 11227

Let r be the distance from P1 to P2. Then P3 lies r*sin(α) in the negative x-direction from P1, and r*cos(α) in the negative y-direction from P1. For more details, see Wikipedia on trigonometry. Hence P3 has coordinates P1 - (r*sin(α), r*cos(α)).

Sidenote: It's a shame SO doesn't have LaTeX support, like MO.

Upvotes: 1

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