I am trying to find angle of yellow line from center of circle. I know circle radius, red and blue point coordinates and angle between red and yellow lines.
What kind of formula should I use?
3 Answers
Think the situation is as follows:
Points R and B are given, as well as angle θ. What is asked for is angle φ.
I place a coordinate system on the center of the circle and expressed R in polar coordinates
d = sqrt( (x_R-x_B)^2 + (y_R-y_B)^2 )
ψ = atan2( (y_R-y_B), (x_R-x_B) )
Then use the law of cosines to find l
l = sqrt(r^2 + d^2 -2*r*d*cos(θ))
Now to find φ and ψ we use the following two equations
d*cos(ψ) = r*cos(φ)-l*cos(θ-φ)
d*sin(ψ) = r*sin(φ)+l*sin(θ-φ)
This is where I am stuck now.
Assuming the following are known:
- side
BC = a - radius
CA = b - angle
∡BAC = α
It follows from the law of sines that sin ∡ABC = AC sin ∡BAC / BC = b sin α / a. The right-hand side is a known quantity, so the equation can be solved for ∡ABC then the third angle of △ABC is ∡BCA = π - α - ∡ABC. This gives the angle between the yellow line and the known segment BC.
Yes, it is possible.
Assuming the vector length is the same, you could define the first point as the origin (0,0).
Then describe the first vector in polar coordinates, (r, \displaystyle \thetaθ), where \displaystyle r = \sqrt{x_{1 }^{2} + y_{1 }^{2}}r=
x
1
2
+y
1
2
and \displaystyle \thetaθ = \displaystyle arctan(y_{1 }/x_{1 })arctan(y
1
/x
1
).
Then create the second vector by adding k degrees to \displaystyle \thetaθ: \displaystyle \theta_{2 }θ
2
= \displaystyle \thetaθ + k
Then convert from polar coordinates back to rectangular coordinates--if you need them that way:
\displaystyle x_{2 } = r cos(\theta_{2 })x
2
=rcos(θ
2
) and
\displaystyle y_{2 } = r sin(\theta_{2 })y
2
=rsin(θ
2
)
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