Caclulate X,Y coordinates of point after rotation around another point of given degrees

Question

There are Two Points A and B. The linear distance between the points is R. I have the coordinates of both points given as X and Y values, and these points can be anywhare in on the 2D plane.

I need to rotate Point A around point B for a given angle (for example 12.56 deg) and get the X and Y coordinates of the new location. The rotation is clockwise.

How can these new X and Y coordinates be calculated?

Thanks!

i have tried on paper to make some triangles up based on the horizonal zero, but came nowhere. — sharkyenergy, Apr 29 '15 at 10:45
@sharkyenergy the two are pretty closely linked, you should not be afraid! And if this is the kind of thing you want to do in programming you'll have to get used to the maths anyway ;) — danimal, Dec 14 '20 at 10:46

gammatester · Accepted Answer · 2015-10-07T06:59:09.273

6

Hint: If you know how to rotate about the origin, you have to calculate (using pseudo code notation) $$(X,Y) = (B_X, B_Y) + \mathrm{rotate\_origin}(A_X-B_X, A_Y-B_Y)$$

Edit: For 2D rotations you use http://en.wikipedia.org/wiki/Rotation_matrix#In_two_dimensions and remember to convert your angle to radians before using $\cos, \sin.$ $$X = B_X + (A_X-B_X)\cos\phi - (A_Y-B_Y)\sin \phi $$ $$Y = B_Y + (A_X-B_X)\sin\phi + (A_Y-B_Y)\cos \phi $$

edited Oct 07 '15 at 06:59

answered Apr 29 '15 at 10:55

gammatester

18,827

Thanks, this was one of my options, problem is i dont know at all how to rotate.. trying to figure it out via web searches but its difficult to find a non-matematician-friendly explanation. – sharkyenergy Apr 29 '15 at 10:57
I don't think there's any way to get away from the mathematics! – danimal Apr 29 '15 at 11:00
not trying to get away, just trying to understand them at a "user friendly" level of rappresentation.. – sharkyenergy Apr 29 '15 at 11:01
yay! this is what i needed! thanks! – sharkyenergy Apr 29 '15 at 11:02
Working like a charm! thanks! – sharkyenergy Apr 29 '15 at 11:15
I figured it out myself via the link you provided, but for completness it would be great to add also the formula for the clockwise rotation. – sharkyenergy Apr 29 '15 at 11:15
for clockwise rotation, replace $\phi$ with $-\phi$ – danimal Apr 29 '15 at 11:24
There is IMO no need for separate formula. A clockwise rotation with angle $\phi$ is the same as using a counter-clockwise rotation with $-\phi$. – gammatester Apr 29 '15 at 11:26
Should this not be $$X = B_X + (A_X-B_X)\cos\phi - (A_Y-B_Y)\sin \phi $$ $$Y = B_Y + (A_X-B_X)\sin\phi + (A_Y-B_Y)\cos \phi $$? You're subtracting off B_Y in all four places.... – Fake Name Oct 06 '15 at 20:17
@Fake Name:Yes of course, thank you for detecting the typo. – gammatester Oct 07 '15 at 07:00

score 3 · Answer 2 · answered Apr 29 '15 at 10:59

Hint: For rotating a position vector $\vec r$ to $\vec r'$ about the origin by an angle $\theta$, the transformation is given by the matrix $R$:

$$\vec r' =R\vec r=\left(\begin{matrix} \sin\theta & \cos\theta \\ \cos\theta & -\sin\theta \end{matrix}\right)\vec r$$

In your application, the vector $\vec r = \vec a- \vec b$

score 0 · Answer 3 · answered Jun 19 '23 at 12:05

The rotated coordinates are $x'=x \cos(\theta)-y \sin(\theta)$ and $y'=x \sin(\theta)+y \cos(\theta)$
although you'll need to convert you degrees angles to radians first by multiplying by the "degrees to radians" constant first: $\theta=a \frac \pi{360}$
Incase your programming language (you said you were a programmer, i am one too for your notice) doesn't support $\cos$ and $\sin$ by default you can use a different solution.

No one else used this formula, so it's useful, plus no need for matrices! — , Jun 22 '23 at 11:29

Caclulate X,Y coordinates of point after rotation around another point of given degrees

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