Determining whether a point lies within a circle

Question

I have several points $(a,b)$ and a circle with center point at $(x,y)$ and radius $r$. If point $(a,b)$ lies on the circle, then $(x-a)^2+(y-b)^2=r^2$. Given $a=12, b=288$ and $x=18.912, y= 290.912, r=7.5$.

So using that values:

$(x-a)^2+(y-b)^2=56.255$

and

$r^2=56.25$.

If you want to see the sketch, click this

according to the picture, it looks as if point $A$ lies on the circle but there is a slight difference between $56.255$ and $56.25$. My question is what is the maximum error between $(x-a)^2+(y-b)^2$ and $r^2$ for the point $(a,b)$ to be considered on the circle?

Thanks

From a purely mathematical point of view, A is outside the circle. — Andrei, Sep 01 '21 at 05:28
I think there is a typo in the line "... point $(a,b)$ to be considered inside the circle?" — Infinity_hunter, Sep 01 '21 at 05:29
@Infinity_hunter so if i have another point $(c,d)$ and the value of $(x-c)^2+(y-d)^2$ is $56.26$, I'm confused whether that point is inside /outside the circle — irvan98, Sep 01 '21 at 05:53

score 1 · Answer 1 · answered Sep 01 '21 at 06:46

If the point in question is at a distance exactly equal to the radius, then the point lies on the circle, if greater than the radius, outside the circle and if less than the radius, then inside the circle. These inequalities are strict.

Let the difference between the radius $r$ of a circle with center $(a,b)$ and the distance between a point $(p,q)$ and $(a,b)$ be $d$. Then

$$ d=\sqrt{(p-a)^2+(q-b)^2}-r $$

$(p,q)$ outside the circle if $d>0$
$(p,q)$ inside the circle if $d<0$
$(p,q)$ on the circle if $d=0$

The requirements are precise without any ifs and buts.

score 0 · Answer 2 · answered Sep 01 '21 at 05:48

The point is inside the circle iff (if and only if) the inequality is strict, on the circle iff the relation is an equality, and outside the circle otherwise.

In engineering though, where measurement imprecision is an issue, the error bounds (which indicate the acceptable maximum calculated distance for the point to be considered within the circle) depend on the measurement precision (the “resolution” of the measuring instrument or choice of “resolution” at which the data was recorded).

Determining whether a point lies within a circle

2 Answers2