The problem consists of a circle with center O, and a point P. The angle φ is defined as ∠HPI, and the distance d is defined between points P and I with and angle of φ. I'm trying to derive an expression for the distance d, when variying the angle φ in the interval [-10°:10°].
If you have an idea on how to solve this, i'm all ears.
Looking forward to your answers! Thank you in advance!
I have linked to an illustration of the problem as I cannot post images yet. Visualization of the problem
We need to assume that the radius $R$ of the circle is known. Shift the coordinate system such that the center of the circle is at $(x,y)=(0,0)$ for simplicity [i.e., subtract the coordinates of O from P...] Let P be at $(x_p,y_p)$ in that normalized Cartesian coordinate system and assume furthermore that $R=|x_p|$ as the figure suggests (otherwise rotate the coordinate system by calculating H first). Then H is at $(x_p,0)$. In the figure $x_p>0$ and $y_p<0$. I is at $(x_p-d\sin\varphi,y_p+d\cos\varphi)$ if we assume that in the figure $\varphi>0$. Because I is on the circle, the sum of the squares must be $R^2$ as Pythagoras argued, $$ (x_p-d\sin\varphi)^2+(y_p+d\cos\varphi)^2=R^2 $$ This is a simple quadratic equation for $d$ which can be solved by elementary methods: $$ d^2+2d(y_p\cos\varphi-x_p\sin\varphi)+x_p^2+y_p^2-R^2=0 . $$