cardioid : exploration

CARDIOID : Exploration

Properties

The equation of the cardioid obviously depends on its positioning in the plane. It is customary to take the position shown below right, in which the cusp A of the cardioid is placed at the origin. The easiest way to represent the cardioid is by its polar equation:

r = 1 + cos

In this equation, r gives the distance from the origin (cusp) A to a general point P of the cardioid, and gives the (directed) angle between AP and the positive x-axis. The polar equation of the cardioid is often given as r = 2a(1 – cos) which allows for some scaling.

Notice that the cardioid cuts the positive x-axis at the point (2, 0), and the y-axis at the points (0, 1).

The Cartesian equation of the cardioid is less pleasant. If P has Cartesian coordinates (x, y), then r = (x² + y²), and cos = x/r = x/(x² + y²). Substituting in the polar equation and some algebraic manipulation gives the Cartesian equation:

(x² + y² – x)² = x² + y².

We observe that the above cardioid intercepts the positive x-axis at x = 2, and the y-axis at y = 1.

The generation property

Initially we generated the cardioid as an envelope of circles. The construction involved a given base circle of diameter ¹/₂.

However, we did observe that the cardioid also occurs as the locus of points which are reflections of A in the tangent lines to the base circle. Here is an applet which demonstrates this property. Click on the adjacent figure, and run the applet using the ‘Animate’ button and manually by dragging point Q around the circle.

What is the relationship between segments AQ and PQ? How do they relate to the blue tangent line? State a result about the generation of the cardioid.

Take any circle through point A. Then the cardioid is the locus of the reflections of point A in the tangents to that circle.

The cusp chord properties

The cusp is obviously a very special point of the cardioid. Suppose that PR is a chord which passes through the cusp. Does this chord have any special properties? Run the adjacent applet using the ‘Animate’ button and manually.

Think about the length of this chord. Remembering where the cardioid cuts the axes, what is the length of this chord in the horizontal position? the vertical position? Can you make a conjecture? Can you prove it?

Now let M be the midpoint of this chord. What is the locus of M? Can you prove it? [Hint: think about the perpendicular at M to the chord.]

The cusp chord always has length 2. We can calculate this using the r-values of P and R:

Length = (1 + cos ) + (1 + cos (– )) = (1 + cos ) + (1 – cos) = 2.

The locus of the midpoint M of the cusp chord is a circle through the cusp having centre (1/2, 0). We can prove this easily. In the diagram, let the blue circle meet the positive x-axis in point C. Now since M is the midpoint of the chord, MR = 1, and so OM = cos. If angle OMC = 90° everything fits, since OC = 1, and the chord makes angle with the x-axis. Otherwise, the perpendicular to the chord at M meets the x-axis in some other point C' with OC' 1, and we get a contradiction. Hence OMC is a right angle and the locus of M is the described circle.

Cusp chord tangent property

Suppose we now take a cusp chord PR of the cardioid, and construct a tangent at each end point. What can we say about these tangents?

Play with the applet at right, clicking the ‘Animate’ button, and dragging the point Q.

What appears to be true about the tangents? What can you conjecture about the angle at T?

Now, what do you think the locus of the point T might be? When you have an answer, click the ‘Show locus’ button. Describe this locus carefully.

The tangents to the end points of a cusp chord of a cardioid are perpendicular. The locus of the point T of intersection of the tangents at the endpoints of a cusp chord is a circle centre (0.5, 0) and radius 1.5 (assuming the above simple equation for the cardioid).

Cusp chord tangent and normal property

We saw above how the tangents at the end points of a cusp chord meet at right angles on a circle. In fact, the normals to the tangents at the end points have the same property, although the circle is different. Look carefully at the applet illustrated at right. You will probably find it most helpful to manually move the point Q.

Concentrate on the endpoints P, R of the cusp chord. The tangents give two edges of the yellow rectangle meeting on the large circle as before. Now look at the other two adjacent sides, PN, RN. How are these related to the cardioid? The point N appears to move on a circle. Describe this circle. What is its centre? radius?

We observe that edges PN, RN are perpendicular. In fact we know already that this must be the case (why is this?). The locus of point N is a circle which has center (0.5, 0) (the same centre as the large circle) and radius 0.5. The tangents and normals define a rectangle of changing shape which has two opposite vertices on the cardioid, and the remaining two opposite vertices lying on two concentric circles. Very pretty!

Actually, there is some even prettier structure here. Because of the right-angle at N, the edges PN, RN must intersect the small circle at the ends X, Y of a diameter. Also,we have seen above that the midpoint M of PR lies on this same circle. Now angles MXN, MYN are right angles, and MXNY is a rectangle. We can deduce from this that X, Y are midpoints of PN, RN , and MN passes through the centre of the small circle.

Parallel tangent property

In the diagram at right, we have three points P, R, S on the cardioid. Play with the applet to see what you can discover about this situation.

What appears to be true about the three tangents? Would you expect there to be four tangents with this property?

Clearly the three tangents do not behave like this for a random choice of points P, R, S. How are these points related? Check out the segments OP, OR, OS. What do you notice?

If P, R, S are three points on the cardioid with OP, OR, OS making equal angles of 2/3 at the cusp, then the tangents at these points are parallel.

[Programming note: I have a confession to make here. This applet constructs the tangent to the cardioid at P, and then draws the remaining two tangents at R and S as translates, rather than constructing them. Could you tell the difference? Should I feel guilty?! An applet can never be used as proof of a mathematical theorem. It can demonstrate a result, and exploring with applets can lead us to conjecture hitherto unknown results. Hence the choice here is between drawing accurate tangents which appear to be parallel, or drawing parallel lines which appear to be tangents. Take your pick.]

Bibliography

A book of curves, Lockwood, E. H. (Cambrdige University Press, 1967)

http://en.wikipedia.org/wiki/Cardioid

http://mathworld.wolfram.com/Cardioid.html

http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cardioid.html