The Cardioid
The Cardioid - a film by Trevor Fletcher
<< Read about this in Mathematics Teaching
<< Full size for projection available here
<< Trevor Fletcher's notes for The Cardioid film
Some additional notes by Eric Love on using the Cardioid film
I’ve tried to think about why this film - and the cardioid itself - might be included in the mathematics curriculum. The most potent reasons for me are: the multiple constructions that are offered, considering the constraints that determine them, and how they connect. These are, in Dick Tahta’s distinction, the ‘inner meanings’ of the film, as distinct from the ‘outer meanings’ - the properties of the cardioid treated as ‘content’.
A guiding question might be: “Why do these different constructions for the cardioid all produce the same curve?”
I’ve been surprised at how easy the Cabri conctructions are. The rolling circles are a bit of a fake, but are totally convincing on the screen.
In the classroom
In the classroom, to familiarise with the cardiod, it might be useful to start constructing some cardioids by hand: - for example:
- the envelope of circles passing through fixed point on base circle;
- as a pedal curve using set square.
Perhaps look at length of diameters of cardioid - appear constant.
Constructions with Cabri
A • For 1 rolling circle construction
- Draw base circle (centre O)
- Mark fixed point on it (P)
- Mark another point on it (Q)
- Draw tangent at Q (draw ray OQ
- Draw perpendicular to this ray at Q)
- Reflect circle in tangent (point P goes to P')
- Drag Q round the base circle
- Locus of P' is a cardioid.
B • For two rolling circles construction
- Draw base circle (centre O)
- Mark fixed point on it (P)
- Mark another point on it (Q)
- Draw line OQ, intersecting the base circle again at Q'
- Draw perpendiculars to this line at Q, Q' (i.e. tangents)
- Reflect circle in tangents (point P goes to P' and P")
- Drag Q round the base circle
- Locus of P' and P" is cardioid
C • Diameter construction
- Draw a segment (this will be the radius of the base circle)
- Draw the base circle
- Mark the fixed point on it (P) and another point on it (Q)
- Draw a line through P and Q
- Create a segment equal to the diameter of the base circle
- From Q mark 2 new points R and R' distant the length of the diameter from Q
- Drag Q round the base circle Locus of R and R' is (the same) cardioid
D • Pedal curve
- Draw base circle (centre O)
- Mark fixed point on it (P)
- Mark another point on it (Q)
- Draw tangent at Q Draw ray OQ
- Draw perpendicular to this ray at Q
- Draw perpendicular from P to the tangent
- Intersection point is R Locus of R as Q moves round circle is cardioid
How are these different constructions connected?
D the Pedal Curve (above) produces a different cardioid - but how is it related to the other?
Some of the later stages in the film can be examined in a similar fashion.
In these constructions varying the size of objects will give indications that the cardiod is a special case of the limaçons.
In (A Rolling Circle construction) and (B Two Rolling Circles construction) vary the size of the rolling circles.
In (C Diameter construction) vary the distance QR (and QR').
