Rotate a line around origin to pass through a given point, how to find the rotate angle

Question

A given coefficients line : $$Ax + By + C = 0$$ I wish it rotate around the origin $O(0,0)$ and pass through a given point $P(p_x, p_y)$, how to find the rotate angle? Of course the distance of point P to $O$ is greater than the distance point O to the line $Ax+By+C=0$ or it will have no solution. The result can be have two angle.

Answer 1

Distance of line $L_1: Ax+By+C=0$ to origin is $R=\dfrac C{\sqrt{A^2+B^2}}$.

Hence $L_1$ is tangent to the circle $C_1: x^2+y^2=R^2$.

Rotating the line such that it passes through $P(h,k)$ (say) is equivalent to finding a tangent to the same circle that passes through $P$.

The equation of any line which is a tangent to the circle $C_1$ is given by $r=\dfrac R{\cos(\theta-\beta)}\tag{1}$.

The polar coordinates for $P$ are $(r,\theta)=(\sqrt{h^2+k^2}, \arctan\frac kh)\tag{2}$

Putting $(2)$ in $(1)$ gives $$\beta=\arctan\left(\frac kh\right)-\arccos\left(\frac R{\sqrt{h^2+k^2}}\right)\tag{3}$$

Substituting $(3)$ in $(1)$ gives the equation of the rotated line passing through $P$.

Answer 2

I'm not sure if this is going to work, but maybe it will be helpful. You need a matrix of rotation around origin for the angle $\phi $, which is $[a_{ij}]$, $a_{11}=cos\phi$, $a_{12}=-sin\phi$, $a_{21}=sin\phi$, $a_{22}=cos\phi$. Coordinates of the point on that line are $[x,\frac{C-Ax}{B}]^T$. When you multiply coordinates and matrix of rotation, for a certain x and $\phi$ you are supposed to get coordinates $[p_x, p_y]^T$.

Answer 3

Let $d$ be the distance from the line to the origin. This is constant when you rotate the line about the origin. Let $e_\theta$ be the unit vector pointing from the origin to the closest point on the line to the origin, after it has been rotated counter-clockwise through an angle of $\theta$. Note $e_\theta$ is just $e_0$ rotated counter-clockwise through an angle of $\theta$.

When the line has been rotated so that it contains the point $P$, the equation $\vec{OP}\cdot e_\theta = d$ is satisfied. That is, the component of $\vec{OP}$ in the direction of $e_\theta$ is $d$. This amounts to solving the equation $$\begin{pmatrix}p_x \\ p_y \end{pmatrix}\cdot \left(\begin{pmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}e_0\right) = d.$$ You are given the coordinates $p_x$ and $p_y$, and the constant $d$ and the vector $e_0$ can be found from the equation of the line. The rest is a trigonometry problem.

Answer 4

The rotation that you describe maps to the point $P$ to the point(s) $Q$ on the line that are equidistant from the origin to $P$. Finding these points is a matter of applying the Pythagorean theorem, after which you can use your favorite way of computing the angle between two points to determine the required rotation angle.

The vector $(A,B)$ is normal to the line, so $(B,-A)$ is parallel to it. The square of distance of the line from the origin is ${C^2\over A^2+B^2}$ and a straightforward computation gives the point on the line nearest the origin as $R\left(-{ac\over a^2+b^2},-{bc\over a^2+b^2}\right)$. Let $d=\sqrt{p_x^2+p_y^2}$, the distance of $P$ from the origin. Then $$Q=R\pm\sqrt{d^2-{C^2\over A^2+B^2}}{(B,-A)\over\sqrt{A^2+B^2}}=R\pm{\sqrt{d^2(A^2+B^2)-C^2}\over A^2+B^2}(B,-A).$$

Now, if what you ultimately want is a rotation matrix, you don’t need to compute any angles at all. The rotation matrix can be found directly from the coordinates of $P$ and $Q$ as $$\begin{bmatrix}p_x&-p_y\\p_y&p_x\end{bmatrix}\begin{bmatrix}q_x&-q_y\\q_y&q_x\end{bmatrix}^{-1}.$$ You can, of course, extract the rotation angle $\theta$ (up to sign) from the resulting matrix since the trace of a 2-D rotation matrix equals $2\cos\theta$.

Answer 5

From polar equation of straight line

$$ x \cos \beta + y \sin \beta =p \tag1$$

and corresponding form

$ \dfrac{Ax+By+C}{{\sqrt{A^2+B^2}}} =0 $ the pedal distance is $p=\dfrac {-C}{\sqrt{A^2+B^2}}\tag2$

Distance to origin $\sqrt{x_p^2+ y_p^2}, p$ are invariant in rotation. It acts just like a rigid $T$ joint while entire straight line rotates around origin on pedal line length $p$.

Using $ ( x_p,y_p) $ for $P$ coordinates in place of $ (p_x,p_y),$

$$ x_p \cos \beta + y_p \sin \beta =\dfrac {-C}{\sqrt{A^2+B^2}} ,\ \beta= \tan^{-1} \frac{B}{A} \tag3$$

Can you go from there?

EDIT1:

Again apply same trick.

$$\dfrac{ x_p \cos \beta + y_p \sin \beta}{{\sqrt{x_p^2+y_p^2}}} =\dfrac {-C}{\sqrt{A^2+B^2}} \cdot \dfrac{ 1}{{\sqrt{x_p^2+y_p^2}}} =RHS \tag4 $$

Letting

$$\dfrac{ [x_p , y_p] }{{\sqrt{x_p^2+y_p^2}}}= [\cos \gamma , \sin \gamma] ,\, \gamma= \tan^{-1} \frac{y_p}{x_p}\tag5$$

Left hand side of (4) is now

$$ \cos ( \beta - \gamma)= RHS \tag6$$

So rotate given line by $\cos^{-1}$ of RHS.

EDIT2:

If given line $ x+y = \sqrt2 $ is rotated to pass through given point $ P[2, 2( \sqrt 3 +1)], $ then angle between tangents should be $ 60^0-(-45)^0 = 105^0$, and it tallies with calculation by formula given in (6) within construction errors.

EDIT3:

Can you now guess how to arrive at the sketched second tangent drawn from P?