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What is the locus of points equidistant from two circles?

$$ x^2+y^2+ 2 h x + 2 g y + c =0 ;\; C =0 ;$$

Locus Equidistant from 2 Circles

Construction of circles with $(a,b,2h)= (3,2,3.6)$

Please help finding equation of the locus equidistant from two circles $ C_1=0, C_2=0 ,$ if possible in terms of $ C_1, C_2. $

.. Like we have radical axis $ C_1 = C_2$ for equal tangents condition.

EDIT1:

It now appears that $ e=\pm 1$ refer to hyperbola and ellipse cases respectively , but we should express eccentricities in terms of $ a,b,h$ and parametrizations as the next step.

Narasimham
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    The locus equidistant from two circles is going to be a conic section (or part of a conic section, and there may be two loci) depending on the two circles – Henry Jul 13 '20 at 08:29
  • What does $C_1=0,C_2=0$ mean? – Henry Jul 13 '20 at 08:29
  • Thanks,edited... Different constants $(h,g,c)$ in a two circle set.. – Narasimham Jul 13 '20 at 08:35
  • How do you define the distance between a point and a circle? – Allawonder Jul 13 '20 at 11:20
  • Distance along the normal ( i.e., along the radius passing through center of circle) dropped from the point onto the circle. – Narasimham Jul 13 '20 at 11:26
  • @Narasimham what signifies "THE three circles"? – user376343 Jul 13 '20 at 15:44
  • I am sorry, This applies only if the solution is a circle passing through concurrent points. Shall edit it. – Narasimham Jul 13 '20 at 16:23
  • What are $a, b$ and $h$? Do the occurrences of the letter $h$ on the second and third lines mean the same thing? How does the second line relate to the symbols $C_1, C_2$ used later? Also, what is $e$? (I'm guessing that it's related to the other question. Regardless of that, the use of the symbol $e$ here could be a little confusing in view of its frequent use to denote the eccentricity of a conic section.) – Calum Gilhooley Jul 15 '20 at 15:32
  • $(a,b,2h) $ are radii, and distance between centers. $h$ in conic equation is different from semi-center distance as we can see from context. $C$ are symbols for a circle. $C_1$ is circle at left, $C_2$ at right. Symbol $e$ has no meaning as eccentricity as these are not conics.Wecan have $-infty<e<+\infty$. – Narasimham Jul 15 '20 at 16:57

2 Answers2

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Let the center of $C1$ be $(x1,y1)$ and radius $r1$.

The center of $C2$ is $(x2,y2)$ and radius $r2$.

Z is the locus $(x,y)$.

The distance from $Z$ to the center of $C1$ is:

$$D1 = \sqrt{(x - x1)^2 + (y-y1)^2} \tag{1}$$

The closest point from $C1$ to $Z$ is extended from the radius.

The point on the perimeter is $P1$.

The distance from the perimeter of $C1$ to $Z$ is $L1$, from $P1$ to $Z$:

$$L1 = \left\lvert \sqrt{(x - x1)^2 + (y-y1)^2} - r1 \right\rvert \tag{2}$$

Similarly for $C2$:

$$L2 = \left\lvert \sqrt{(x - x2)^2 + (y-y2)^2} - r2 \right\rvert \tag{3}$$

$Z$ is at an equal distance form both circles : $L1 = L2$

$$\left\lvert \sqrt{(x - x1)^2 + (y-y1)^2} - r1 \right\rvert = \left\lvert \sqrt{(x - x2)^2 + (y-y2)^2} - r2 \right\rvert \tag{4}$$

The sign changes if $Z$ is inside the circle.

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The distance of a point $P$ from the circle centre $A,$ radius $r$ is $\|AP\| - r.$

The distance of the same point $P$ from the circle centre $A',$ radius $r'$ is $\|A'P\| - r'.$

(I am assuming here that "negative distances" are allowed, as is implied by the diagram in the question.)

The two distances are equal if and only if $\|AP\| - \|A'P\| = r - r'.$

If $r = r',$ then the locus of $P$ is the perpendicular bisector of $AA'.$

(That is, unless also $A = A',$ in which case any point $P$ satisfies the condition.)

If $r \ne r'$ and $\|AA'\| = |r - r'|,$ i.e., if the circles touch internally, then the condition on $P$ is $\|AP\| - \|A'P\| = \|AA'\|$ (if $r > r'$) or $\|A'P\| - \|AP\| = \|AA'\|$ (if $r < r'$). In either of these cases, the locus of $P$ is the ray from the centre of the smaller circle that passes through the circles' point of contact.

If $r \ne r'$ and $\|AA'\| < |r - r'|,$ i.e., if one circle is contained in the other, then by the Triangle Inequality no point $P$ satisfies the condtion.

If $r \ne r'$ and $\|AA'\| > |r - r'|,$ then the locus of $P$ is one branch of a hyperbola whose foci are $A$ and $A'$ and whose eccentricity is $\|AA'\|/|r - r'|.$

Calum Gilhooley
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  • [Also ellipse][2]

    [2]: https://math.stackexchange.com/questions/3755371/locus-of-a-point-with-constant-distance-ratio-e-to-two-circles?noredirect=1#comment7727483_3755371 Ellipse case is also admissible it seems now. Can you now indicate parametrization for e=+1 and e=-1 cases? That could be a more comprehensive answer perhaps.

     – Narasimham Jul 14 '20 at 22:17
  • Although I wrote my answer rather hastily, and I'll go through it again to check more thoroughly for errors, I'm almost sure that it doesn't overlook anything as big as the possibility of an elliptical locus. The other question you refer to seems to concern a more general problem, in which an elliptical locus might well be a possibility, although a hasty glance suggests that the locus in general won't be a conic section, and might be a curve of degree $4$ or something like that. I'm sorry I don't have time to look at it properly right now. (Already late for bed!) I'll delete this comment soon. – Calum Gilhooley Jul 14 '20 at 23:04
  • I see that the other question been rewritten since I glanced at it (yesterday?), and others have commented at length, in a similar vein, but in vastly more detail. I'm very unlikely to have anything useful to add, but I will try to read it tomorrow. – Calum Gilhooley Jul 14 '20 at 23:10
  • Thanks. (The other case link is only for information. Your answer here may relate to both conics). You have concluded that eccentricity $ \epsilon =\frac{2h}{a-b}$ which should result in hyperbola and ellipse depending on whether the circles intersect or stand separate, Is this correct? What then are their equations asked for? I could not readily arrive at these. – Narasimham Jul 15 '20 at 06:06