Coordinate Geometry · Mathematics

Ellipse Reflection & Lithotripsy: 3D Simulation

How a Class 11 ellipse breaks kidney stones, in 3D: a ray from one focus of an elliptical mirror that always reflects through the other, a pulse whose every path is 2a so it all arrives at once, a source dragged off the focus until the focus blurs, a cut-open lithotripter whose spark shocks a stone you slide onto the focus until it shatters, designing the reflector, ellipse vs parabola vs circle mirrors, a whispering gallery you walk round, and the JEE results behind it (SG = e·SP, p₁p₂ = b²).

Interactive virtual lab · no download · works on phones and laptops

Chapter: Conic Sections
For: Class 9–12 · JEE Main · JEE Advanced
Time: ~30 min

How to use it

1Pick a module from the list (8, in three groups); set your level at the top right.
2Every scene runs by itself — drag the orange knobs (the wall point, the source, the patient's table, the bowl's rim, the listener) to drive it.
3Use Predict before you look, then Derive, Questions, Practise and Compare; tap a live value marked ƒ to see its formula.

What to notice

A ray from one focus reflects through the other, because the normal at $P$ bisects $∠ F_{1} P F_{2}$ .
Every path $F_{1} \to P \to F_{2}$ has length $P F_{1} + P F_{2} = 2 a$ , so a pulse from $F_{1}$ arrives at $F_{2}$ all at once, at $t = 2 a / v$ .
Only the foci work: off the focus the reflected rays cross over a blurred zone and arrive spread out in time.
A lithotripter puts the spark at $F_{1}$ of a half-ellipsoid and the stone, found by imaging, at $F_{2} = 2 c$ away.
The normal meets the major axis at $G = (e^{2} x_{1}, 0)$ with $S G = e \cdot S P$ ; the product of the perpendiculars from the foci to a tangent is $b^{2}$ .

For teachers

Teach it in 32 minutes

A lesson plan, questions to put to the class, a quick quiz you can project or print, and the formulas behind Ellipse Reflection 3D Lab.

1Hook
~2 min
- Open the simulation full screen on the projector. Before touching anything, ask the class what they expect to see, and write two or three predictions on the board.
2Demonstrate
~18 min
- Pick a module from the list (8, in three groups); set your level at the top right.
- Every scene runs by itself — drag the orange knobs (the wall point, the source, the patient's table, the bowl's rim, the listener) to drive it.
- Use Predict before you look, then Derive, Questions, Practise and Compare; tap a live value marked ƒ to see its formula.
3Pause and ask
~5 min
- Have a student say this in their own words: A ray from one focus reflects through the other, because the normal at P bisects angle F₁PF₂.
- Have a student say this in their own words: Every path F₁ → P → F₂ has length PF₁ + PF₂ = 2a, so a pulse from F₁ arrives at F₂ all at once, at t = 2a/v.
- Have a student say this in their own words: Only the foci work: off the focus the reflected rays cross over a blurred zone and arrive spread out in time.
- Have a student say this in their own words: A lithotripter puts the spark at F₁ of a half-ellipsoid and the stone, found by imaging, at F₂ = 2c away.
- Have a student say this in their own words: The normal meets the major axis at G = (e²x₁, 0) with SG = e· SP; the product of the perpendiculars from the foci to a tangent is b².
4Check
~7 min
- Run the 7-question quiz below: project it and take answers by show of fingers (1–4), or print it as a worksheet.

About the Ellipse Reflection 3D Lab simulation

The Ellipse Reflection Lab shows how a property you meet in Class 11 conic sections is used by doctors to break kidney stones. On a 3D elliptical mirror wall, a ray from one focus hits the wall and reflects through the other focus while the point it hits sweeps round by itself; drag the point anywhere and the two angles with the normal stay equal. Then a whole pulse leaves the first focus: its dots draw the wavefront, which reflects off the wall and shrinks onto the second focus, every part arriving together because every path is 2a long.

Drag the source off the focus and the meeting point smears out — the property belongs to the foci only. The lithotripter module cuts open a half-ellipsoid reflector under a water bath: a hand presses the fire button, a spark flashes at the first focus, the shock wave reflects and converges, and the kidney stone cracks and breaks into fragments only once you slide the patient so that the stone sits on the second focus. You can then design the reflector from the stone's depth (b = √(a² − c²), e = c/a, the share of the shock the bowl catches), compare elliptical, parabolic and circular mirrors (headlight, dish, concave mirror), walk a listener round a whispering gallery, and finish with the JEE results behind the property: the normal meets the major axis at G = (e²x₁, 0) with SG = e·SP, and the perpendiculars from the foci to a tangent multiply to b².

Also covers: reflection property of ellipse, ellipse reflection property, lithotripsy, lithotripsy ellipse, kidney stone ellipse, extracorporeal shock wave lithotripsy, eswl, whispering gallery, whispering gallery ellipse, elliptical mirror, ellipsoid reflector, focus of ellipse, foci of ellipse, applications of ellipse, real life application of ellipse, normal bisects focal radii, sg = e sp, product of perpendiculars from foci b^2, auxiliary circle feet of perpendiculars, parabolic mirror vs elliptical mirror, conic sections class 11 application.

Frequently asked questions

What is the reflection property of an ellipse?

A ray (of light, sound or a shock wave) that leaves one focus of an elliptical mirror is reflected through the other focus, wherever it hits the mirror. It works because the normal at every point of the ellipse bisects the angle between the lines to the two foci, so the law of reflection turns the line from one focus into the line to the other. Since PF₁ + PF₂ = 2a for every point, all the reflected parts of a pulse also arrive at the second focus at the same time.

How is the ellipse used to treat kidney stones?

In extracorporeal shock wave lithotripsy (ESWL), the original electrohydraulic machines use a reflector shaped like half of an ellipsoid. A spark at one focus makes a shock wave in water; the reflector focuses it onto the second focus. Using X-ray or ultrasound imaging, the patient is positioned so the kidney stone sits on that focus. Each shock arrives there concentrated and all at once, while the skin it passes through sees the energy spread thin; after many shocks the stone breaks into pieces small enough to pass in the urine.

Why must the stone be exactly at the second focus?

Only rays that start at one focus pass through the other. If the stone is off the focus, most of each shock passes beside it. The lab shows that the focus is much more forgiving along the axis than across it: the reflected rays all arrive at small angles to the axis, so the focal zone is long and thin.

What is a whispering gallery?

A room with elliptical walls or ceiling in which a whisper at one focus can be heard clearly at the other, even across the room, because the sound reflected off the whole wall converges there at the same moment. Someone standing elsewhere — even closer to the speaker — hears much less.