Find Center of Circle given Radius, Circumference Point, and that Point's Rotation

Question

I need to find the center point of a circle $(x,y)$ given:

• The radius $\mathbf r$ of the circle

• A point on the circumference of the circle $\mathbf (a,b)$

• The clockwise degrees of rotation $\mathbf t$ of the point $(a,b)$ about the center point $(x,y)$

Here's an illustration :

I've tried

$\begin{cases} x=a+r\;\cos(t) \\ y=b+r\;\sin(t) \end{cases}$

and it seemed to get me close but it's possible I'm missing an additional piece of the puzzle.

Answer 1

Consider that you know how to write $(a,b)$ using $x,y,\mathbf{r},\mathbf{t}$: $$\left\{ \begin{array}{l} a=x+\mathbf{r}\cos(2\pi-\mathbf{t}) \\ b=y+\mathbf{r}\sin(2\pi-\mathbf{t}) \end{array} \right.$$

Answer 2

Was a bit confused with the above answer, so here's how I got x and y:

(assuming: x and y axis increase in the north east direction)

1. draw a perpendicular line from (a;b) upwards (see attached drawing)
2. Simply solve the following

• x = a - cos(t-90) * r

• y = b + sin(t-90) * r

Edited drawing enter image description here

Answer 3

Draw it with small angles to get

$$\left.\begin{aligned}a & =x+r\sin t\\ b & =y+r\cos t \end{aligned} \right\} \;\left.\begin{aligned}x & =a-r\sin t\\ y & =b-r\cos t \end{aligned} \right\} $$