Circle as degenerate conic (e = 0): Visual walkthrough

Step 0 — The words we are allowed to use

Before any algebra, let us pin down the cast of characters so no symbol arrives unannounced.

Definition The vocabulary (each is a picture)

Point $P (x, y)$ — a location; $x$ is how far right, $y$ is how far up.
Focus $S$ — a fixed special point inside a conic. An ellipse has two of them; we name the one we are measuring to $S$ .
Directrix — a fixed straight line. An ellipse has two of them, one for each focus. Distance from $P$ to it is measured straight across (perpendicular).
Foot of the perpendicular $M$ — the exact spot on the directrix directly "across" from $P$ , where the perpendicular from $P$ meets the line. So $P M$ is the perpendicular distance from $P$ to the directrix.
Eccentricity $e$ — one number that says how "stretched" the shape is. See Eccentricity.
Semi-major axis $a$ — half of the long width of the oval.
Semi-minor axis $b$ — half of the short height of the oval.
$c$ — the distance from the centre out to each focus.

The one rule that ties them together, the conic rule, is:

Figure — Circle as degenerate conic (e = 0)

Step 1 — Start one door away: the ellipse

WHAT. We draw a genuine ellipse with its two foci at $(\pm c, 0)$ and its widths $a$ (long, horizontal) and $b$ (short, vertical).

WHY. We cannot just write $e = 0$ in $\frac{P S}{P M} = e$ . If $e = 0$ then $P S = 0 \cdot P M = 0$ , so $P$ would have to be the focus — a single dot, not a circle. So instead we approach the circle through the ellipse next door and take a limit. The ellipse is the perfect launch pad because it already contains the circle as its "un-squashed" extreme.

PICTURE. In the figure: two orange dots are the foci at $(\pm c, 0)$ ; the teal arrow marks the half-width $a$ and the plum arrow the half-height $b$ . The oval is visibly wider than it is tall.

Formula The two ellipse facts we will squeeze

$e = \frac{c}{a} b^{2} = a^{2} (1 - e^{2})$

$e = \frac{c}{a}$ — WHY this holds: $e$ is defined as the focus distance $c$ compared to the half-width $a$ , i.e. how far out the focus sits as a fraction of the axis. It is literally "focus-distance measured in units of $a$ ." Big $c$ (foci far apart) → big $e$ → very stretched.
$b^{2} = a^{2} (1 - e^{2})$ — WHY this holds: stand at the top of the ellipse, the point $(0, b)$ . By the ellipse's own "two-foci" property, its distances to the two foci add to $2 a$ ; by symmetry each equals $a$ . That slanted distance is the hypotenuse of a right triangle with legs $b$ (up) and $c$ (across), so Pythagoras gives $a^{2} = b^{2} + c^{2}$ . Rearrange: $b^{2} = a^{2} - c^{2} = a^{2} - (a e)^{2} = a^{2} (1 - e^{2})$ , using $c = a e$ from the first fact. The short half-height $b$ is pulled down from $a$ by the factor $(1 - e^{2})$ ; only when $e^{2} = 0$ does $b$ climb all the way back up to $a$ .

Step 2 — Turn the knob: send $e$ toward $0$

WHAT. We keep $a$ fixed (the long half-width stays at, say, $5$ ) and slide $e$ down: $e = 0.8, 0.6, 0.3, 0.1, \dots$

WHY. Holding $a$ still is the honest way to compare shapes — we shrink only the stretch, not the overall size. Watch what each fact does as $e$ falls.

PICTURE. In the figure: a stack of ovals, each rounder than the last, all sharing the same left–right extent. The outermost (orange, $e = 0.85$ ) is squashed; the innermost (plum, $e = 0.12$ ) is nearly a circle.

Step 3 — The two foci collide at the centre

WHAT. At the very end, $c = a e = a \cdot 0 = 0$ . Both foci, which were at $(+ c, 0)$ and $(- c, 0)$ , land on the same spot: $(0, 0)$ — the centre.

WHY. A circle has no "long way" and no "short way," so it cannot possibly point at two different foci. Symmetry demands they merge. That single merged point is what we call the centre $C$ , and it is the only special point a circle has.

PICTURE. In the figure: faint ovals for $e = 0.85, 0.5, 0.2$ with their orange focus-pairs; the black arrows show the foci travelling inward, and the big plum dot marks where both fuse — the centre.

Step 4 — Both directrices flee to infinity

WHAT. An ellipse owns two directrices, one per focus, at $x = + \frac{a}{e}$ and $x = - \frac{a}{e}$ (mirror images across the centre). Feed the knob down: as $e \to 0$ , the fraction $\frac{a}{e}$ blows up, so both lines run off — one to the right, one to the left.

WHY. Dividing a fixed positive $a$ by a tinier and tinier $e$ gives a bigger and bigger answer — each line marches off the edge of the page and never comes back. This is exactly why a finished circle has no directrix at all: both retreated to infinity.

PICTURE. In the figure: the teal directrix line at $x = a / e$ (and its mirror at $x = - a / e$ ) marches outward, further with every smaller $e$ , until it is gone.

Step 5 — Write down the round equation

WHAT. Put $b = a$ into the standard ellipse equation and simplify.

WHY. The algebra must agree with the pictures. If our story is right, the ellipse formula should melt into the circle formula the instant $b = a$ .

PICTURE. In the figure: the dashed orange ellipse ( $b < a$ ) sits inside the solid plum circle ( $b = a$ ); the teal arrow is the radius $a$ , and it is the same length in every direction.

Step 6 — Build it a second way, from the definition itself

WHAT. Forget the ellipse for a moment. Use the circle's own definition: every point $P (x, y)$ is a fixed distance $r$ from the centre $C (h, k)$ .

WHY. This confirms the same equation appears from the ground up, using only the Distance Formula — no limits needed, once we already know "circle = constant distance."

PICTURE. In the figure: the orange circle with centre $C (h, k)$ (plum) and a rim point $P (x, y)$ (teal); the dashed legs are the horizontal gap $x - h$ and vertical gap $y - k$ , with the solid segment $r$ as their hypotenuse.

Step 7 — The degenerate edges (don't skip these)

WHAT. Expanding $(x - h)^{2} + (y - k)^{2} = r^{2}$ and renaming the constants gives the general form $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ (see Completing the Square).

WHY. A complete map must show every landing spot, including the broken ones. The sign of $g^{2} + f^{2} - c$ decides everything.

PICTURE. In the figure: three side-by-side panels — a full orange circle ( $r^{2} > 0$ ), a single plum dot ( $r^{2} = 0$ ), and an empty frame ( $r^{2} < 0$ ) — the shape's life story as the number under the root passes through zero.

The one-picture summary

Everything above is a single event viewed from four windows: foci merge, axes equalise, directrices flee, equation rounds.

PICTURE. In the summary figure: faint orange ovals collapse into the bold plum circle; the plum dot is the fused centre, the teal arrow shows the directrices departing, and the final equation $x^{2} + y^{2} = a^{2}$ is written beneath.

Recall Feynman retelling — the whole walk in plain words

Picture an oval balloon. Two pins inside it (the foci) hold its shape, and two walls, one on each side (the directrices), keep it honest. Now start letting the "stretch" out of the balloon — that stretch is the number $e$ . As you do, three things happen together, like one motion filmed from three angles. The two pins glide toward each other and click into a single pin right in the middle — that pin is now just the centre. Both walls slide away, further and further, until they're over the horizon and gone. And the balloon, no longer pulled long, puffs out until its width and height are the very same length. When the stretch is completely gone ( $e = 0$ ), what's left is the roundest thing possible: a circle, where the only special point is its middle and the only rule is "stay a fixed distance $r$ from it." Written down, the stretched formula $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ has its two bottoms become equal and collapses to $x^{2} + y^{2} = a^{2}$ . And if you squeeze even the radius down to nothing, the circle becomes a lone dot — the last, most degenerate ghost of a conic.

Connections

Parent topic — full note
Ellipse — the shape we squeezed; $a = b$ is its round limit.
Eccentricity — the single knob $e$ we turned to $0$ .
Conic Sections — the family this circle belongs to.
Parabola and Hyperbola — the other endpoints of the $e$ dial.
Distance Formula — the tool of Step 6.
Completing the Square — general form ↔ standard form (Step 7).
Degenerate Conics — point circle and imaginary circle live here.