Teaching objectives
This is one of the most visually striking labs because it generates beautiful, colourful images. Designed for secondary-school students, it starts from the simplest possible question — "What happens if I connect points on two segments?" — and leads students, room by room, to discover parabolas, hyperbolas, cardioids, astroids, and more.
What you'll learn
First and foremost, that mathematics can be beautiful and full of colour! Students also develop geometric intuition and the ability to conjecture patterns: they manipulate interactive constructions and decide which curve they are looking at before receiving confirmation. The journey moves from the plane to space and from simple curves to curves generated by speeds or pursuit paths.
- In rooms 1–2, points on two segments sharing a vertex are connected (Apollonian construction), and students discover that the envelope is always a parabola, regardless of the angle between the segments.
- In room 3, the spacing changes to inverse heights (1, 1/2, 1/3…) and the hyperbola appears as the curve xy = c with its asymptotes.
- Room 4 (focus and perpendicular bisectors) shows that the auxiliary curve Γ determines the conic: a line → parabola, a circle with F inside → ellipse, a circle with F outside → hyperbola.
- Room 5 (saddle surface) introduces the third dimension and the property that two distinct families of lines pass through every point.
- Rooms 6–7 construct the cardioid and the nephroid by connecting points on a circle at double and multiple speeds; students discover that speed ×M produces M−1 cusps.
- Room 8 (Leonardo's ellipsograph) proves that the midpoint of a segment sliding between two axes traces a circle, not an ellipse.
- Room 9 closes with the pursuit curve on regular polygons and a calculation of the distance travelled.
- Room 13 (free workshop) lets students draw any curve by hand and see its tangent lines in real time, closing the loop with the definition of envelope.
Key mathematical ideas
- Envelope of a family of lines: the curve that is tangent to every line in the family at some point.
- Conics (parabola, ellipse, hyperbola) appear as envelopes of the perpendicular bisectors of FP depending on the position of focus F relative to the auxiliary curve Γ.
- The hyperbola xy = c arises from the inverse spacing 1/k; its coordinate axes are asymptotes.
- A ruled quadric (saddle surface) is a surface that contains two families of lines; through every point pass exactly two lines, one from each family.
- The cardioid is the envelope of chords on a circle with speed ratio 2:1; the nephroid with ratio M:1 has M−1 cusps.
- Leonardo's ellipsograph traces a circle because the distance from the midpoint to the origin is constant (half the length of the segment).
- The pursuit curve on a regular polygon with N vertices and side L converges to the centre; the distance travelled is L / (1 − cos(2π/N)).
Room-by-room contents
Room 1 · What is an envelope?
Animated opening: 36 tangent lines are drawn one by one, revealing that their shared "edge" is a hidden smooth curve. The concept of the envelope of a family of lines is introduced.
Student tasks
- Watch how the tangent lines appear and form the curve.
- Advance after reading the definition of envelope.
Room 2 · Choose a curve and see its tangents
The student selects from several classic curves (parabola, cardioid, cycloid, deltoid, rose, lemniscate) and sees in real time how the tangents reconstruct it. Final challenge: identify which curves have exactly one axis of symmetry.
Student tasks
- Click on at least three different curves and observe their family of tangents.
- Mark the curves with exactly one axis of symmetry for the final challenge.
Room 3 · Strings between two segments
Two segments share a vertex; connecting point k of one with point k of the other (from opposite ends), the strings graze a parabola. Classic Apollonius construction.
Student tasks
- Observe the construction and the curve that emerges.
- Answer which curve forms the envelope (multiple choice: parabola, hyperbola, cycloid, catenary).
Room 4 · Change the angle — what happens to the curve?
The same construction as the previous room but with 30 fixed points and the angle between the segments freely adjustable. Free investigation: does the curve remain a parabola for any angle, even 180° or very small?
Student tasks
- Drag the angle and observe how the shape of the curve changes.
- Answer whether the envelope is still a parabola or becomes another curve when the angle changes.
Room 5 · Points at inverse heights
The vertical axis has points at heights 1, 1/2, 1/3, …, 1/N (reciprocals). Connecting the k-th endpoint of the horizontal segment with the k-th point of the vertical axis reveals a hyperbola as the envelope.
Student tasks
- See how the inverse spacing generates a different curve.
- Identify the resulting curve from: circle, hyperbola, parabola, and ellipse.
Room 6 · Conics from a focus and perpendicular bisectors
A draggable focus F and an auxiliary curve Γ (line, circle, or parabola). For each point P on Γ the perpendicular bisector of FP is drawn; the envelope of all those bisectors is a conic that changes depending on the position of F relative to Γ.
Student tasks
- Drag F and change Γ (line → parabola, circle with F inside, circle with F outside).
- Answer in which configuration the envelope is an ellipse (or circle).
Room 7 · Saddle surface
3D visualisation of two crossed rods. Connecting their equidistant points, the strings form a ruled surface (one-sheeted hyperboloid), shaped like a saddle. The student can rotate the angle between the rods.
Student tasks
- Rotate the angle and observe how the saddle deforms.
- Answer how many lines pass through each point of the surface (one, two, three, or infinitely many).
Room 8 · All against all
N points on a circle connected by every possible segment. The strings cross to form nested rings. Combinatorics challenge: how many segments are there with N points?
Student tasks
- Vary N and observe the string patterns and inner rings.
- Calculate how many segments can be drawn with a given number of points.
Room 9 · Cardioid at double speed
The circle is divided into 36 parts; point k is connected to point 2k (the "fast" end moves at twice the speed). The strings trace a cardioid as the envelope.
Student tasks
- Observe how the cardioid emerges from the string construction.
- Answer how many full turns the fast end completes while the slow end completes one.
Room 10 · More speed, more foci
The number of points is increased and different speeds (×2, ×3, …, ×10) are tested. At speed ×M the nephroid and other curves with M−1 visible foci appear.
Student tasks
- Try increasing speeds and count the foci that form.
- Answer how many foci there would be at speed ×10.
Room 11 · Leonardo's ellipsograph
A fixed-length segment slides with each end along perpendicular axes (connecting-rod mechanism). The midpoint of the segment traces a precise curve — which one is it?
Student tasks
- Observe the movement of the segment and the trace of its midpoint.
- Identify the curve drawn by the midpoint (circle, rhombus, square, or oval).
Room 12 · Pursuit on a polygon
On a regular polygon, each point chases its neighbour at the same speed. All spiral towards the centre simultaneously. The view of the changing polygon can be enabled.
Student tasks
- Start the animation and observe the pursuit spirals.
- Calculate how far each point travels in a unit square of side 1 m before they all meet at the centre.
Room 13 · Draw your own curve
Creative closing room: the student freely draws any curve with the mouse and the system draws its tangents in real time, showing that every smooth curve is the envelope of its own tangents.
Student tasks
- Draw at least one curve and see its tangents in real time.
- Erase and experiment with different shapes.
Rooms to project
The most striking ones to show and discuss in class.