Jeśli jesteś właścicielem tej strony, możesz wyłączyć reklamę poniżej zmieniając pakiet na PRO lub VIP w panelu naszego hostingu już od 4zł!

Version of 2012-05-13

Wersja polskaBilanguage versionWersja dwujęzyczna

Grzegorz Jagodziński

Curves of Perseus

Curves of Perseus have been named after the Greek geometer who lived ca. 150 BC and who investigated them. They are curves obtained by intersecting a torus with a plane parallel to the rotational symmetry axis (the line through the centre of the hole of the torus, perpendicular to the torus midplane). The term “spiric sections” is also in use1.

General information on curves of Perseus

The basic definition of the curve of Perseus

Let’s start from three-dimensional Cartesian coordinates such that the `y`-axis be vertical. Given a circle in the `xy`-plane, with the centre in `O = (c,0,0)` (so, on the `x`-axis), and with the radius `r`. The equation of the circle (in two dimensions) is:

`(x - c)^2 + y^2 = r^2`

Now rotate the circle around the `y`-axis. Thus we have obtained a torus.

Let’s assume also `c >= 0`. In this case it is important to consider also the other circle `(x + c)^2 + y^2 = r^2` as rotating. When `c > r`, we have a “normal” torus with a hole, termed a ring torus. When `c = r`, there is no hole and the torus is self-tangent – we call it a horn torus. When `0 < c < r`, the torus is self-intersecting – we have a spindle torus. And finally when `c = 0`, we obtain a sphere rather than a torus2. We will describe this very special case separately, assuming everywhere `c > 0` for now.

ring torus horn torus spindle torus

Source of the images: Wikipedia

The torus is lying on the “table” `y = -r`. Let’s find its equation in Cartesian coordinates. Notice that when making a surface of revolution, each point `M = (x,y,z)` of the rotated line stays on the circle with the centre on the axis of revolution. Our axis is the `y`-axis, and the circle made with each rotating point lies in a plane parallel to `xz`. The distance from the axis to the rotating point is constant and equal to `sqrt (x^2 + z^2)`. Thus, in order to obtain the equation of the torus, let’s take the equations of both rotating circles `(x - c)^2 + y^2 = r^2` and `(x + c)^2 + y^2 = r^2`, and replace `x` with `sqrt (x^2 + z^2)`. Thus the equation of the torus is:

`(sqrt (x^2 + z^2) -+ c)^2 + y^2 = r^2`

or:

`y^2 = r^2 - (sqrt (x^2 + z^2) -+ c)^2`

Now let’s cut the torus by a plane parallel both to the axis of torus (the `y`-axis) and to the `x`-axis. All points of the plane have a constant `z` coordinate. We will mark this constant value with `f`. Thus the equation of the plane is `z = f`. Because of the symmetry of the torus, we may assume `f >= 0`. Notice also that `f <= c + r`, otherwise the plane and the torus would not intersect. When `f = c + r`, we obtain a single point. And when `0 <= f < c + r`, we obtain a curve of Perseus. Because of `z = f`, its equation, in the implicit form, is:

`y^2 = r^2 - (sqrt (x^2 + f^2) -+ c)^2`

or, when written in the explicit form,

`y = -+sqrt(r^2 - (sqrt (x^2 + f^2) -+ c)^2)`

We can write it in a different form, applying the following transformations:

`(sqrt (x^2 + f^2) -+ c)^2 = r^2 - y^2`,
`x^2 + f^2 -+ 2c sqrt (x^2 + f^2) + c^2 = r^2 - y^2`,
`x^2 + y^2 + c^2 + f^2 - r^2 = -+2c sqrt (x^2 + f^2)`,

which finally gives:

`(x^2 + y^2 + c^2 + f^2 - r^2)^2 = 4c^2(x^2 + f^2)`

where:

We may also write the equation in a little different way. Let’s transform it:

`(x^2 + y^2)^2 + 2(x^2 + y^2)(c^2 + f^2 - r^2) + (c^2 + f^2 - r^2)^2 = 4c^2x^2 + 4c^2f^2`,
`(x^2 + y^2)^2 + 2(c^2 + f^2 - r^2)x^2 - 4c^2x^2 + 2(c^2 + f^2 - r^2)y^2 + (c^2 + f^2 - r^2)^2 - 4c^2f^2 = 0`,
`(x^2 + y^2)^2 + 2(-c^2 + f^2 - r^2)x^2 + 2(c^2 + f^2 - r^2)y^2 + (c^2 + f^2 - r^2)^2 - 4c^2f^2 = 0`,
`(x^2 + y^2)^2 - 2(c^2 - f^2 + r^2)x^2 + 2(c^2 + f^2 - r^2)y^2 + (c^2 + f^2 - r^2)^2 - 4c^2f^2 = 0`.

If we apply new parametres:

`A = r^2 + c^2 - f^2`,
`B = r^2 - c^2 - f^2`,
`C = (r^2 - c^2 - f^2)^2 - 4c^2f^2`,

we will obtain a different form of the equation:

`(x^2 + y^2)^2 - 2Ax^2 - 2By^2 + C = 0`

Notice that the parametres are not unrestricted because `A >= B`. Besides,

`A + B = 2(r^2 - f^2)`,
`A - B = 2c^2`,
`B^2 - C = (2cf)^2`,

thus `B^2 >= C`, and naturally `A^2 >= C`.

These are dependencies between `A, B, C` and `r, c, f`:

  `A > 0` `A = 0` `A < 0`
`B > 0` `B = 0` `B < 0` `B < 0` `B < 0`
`C > 0` `f < r - c`, `c < r` `f < c - r`, `c > r`
`C = 0` `f = r - c`, `c < r` `f = 0`, `c = r` `f = c - r`, `c > r`
`C < 0` `r - c < f < sqrt (r^2 - c^2)`, `c < r` `f = sqrt (r^2 - c^2)`, `c < r` `c - r < f < sqrt (c^2 + r^2)`, `c > r` `f = sqrt (c^2 + r^2)` `sqrt (c^2 + r^2) < f < c + r`
`sqrt (r^2 - c^2) < f < sqrt (c^2 + r^2)`, `c <= r`

Note that `A < 0`, `B < 0`, `C > 0` would imply `f > c + r`, and `A = 0`, `B = 0`, `C = 0` would imply `c = 0`, which are excluded as stated above.

It is easy to find the inverse relations of the parametres:

These are dependencies between `r, c, f` and `A, B, C`:

  `f = 0` `0 < f < |c - r|` `f = |c - r|` `|c - r| < f < sqrt (r^2 - c^2)` `f = sqrt (r^2 - c^2)` `sqrt (r^2 - c^2) < f < sqrt (c^2 + r^2)` `f = sqrt (c^2 + r^2)` `sqrt (c^2 + r^2) < f < c + r`
`c < r` `A > 0`, `B = 0`, `C = 0` `A > 0`, `B > 0`, `C > 0` `A > 0`, `B > 0`, `C = 0` `A > 0`, `B > 0`, `C < 0` `A > 0`, `B = 0`, `C < 0` `A > 0`, `B < 0`, `C < 0` `A = 0`, `B < 0`, `C < 0` `A < 0`, `B < 0`, `C < 0`
`c = r` `A > 0`, `B = 0`, `C = 0` `A > 0`, `B = 0`, `C = 0` `A > 0`, `B = 0`, `C = 0` `A > 0`, `B < 0`, `C < 0` `A = 0`, `B < 0`, `C < 0` `A < 0`, `B < 0`, `C < 0`
`c > r` `A > 0`, `B = 0`, `C = 0` `A > 0`, `B < 0`, `C > 0` `A > 0`, `B < 0`, `C = 0` `A > 0`, `B < 0`, `C < 0` `A > 0`, `B < 0`, `C < 0` `A > 0`, `B < 0`, `C < 0` `A = 0`, `B < 0`, `C < 0` `A < 0`, `B < 0`, `C < 0`

The definition of the curve of Perseus as a locus

The curve of Perseus can also be defined in a totally different way, namely as the geometric locus of points `M = (x,y)` such that:

`F_1M^2*F_2M^2 = d*OM^2 + g^4`

where `F_1, F_2` are foci: `F_1 = (c,0), F_2 = (-c,0)`; `O` is the point `O = (0,0)`; `d, g` are real parametres (so they may also be negative)3. Note that `2c`, termed focal, is the distance between the foci.

The distance `F_1M` is equal to `sqrt ((x - c)^2 + y^2)` (from the distance formula of two points, or directly from the Pythagorean theorem). Similarly, the distance `F_2M` is equal to `sqrt ((x + c)^2 + y^2)`. It is also easy to determine the distance `OM` as equal to `sqrt (x^2 + y^2)`. When having used the above formulation and having applied the distances, it appears that the Cartesian equation of a curve of Perseus is:

`((x - c)^2 + y^2)((x + c)^2 + y^2) = d(x^2 + y^2) + g^4`,

which can be transformed in the following way:

`(x - c)^2(x + c)^2 + (x - c)^2y^2 + (x + c)^2y^2 + y^4 = dx^2 + dy^2 + g^4`,
`(x^2 - c^2)^2 + (x^2 - 2cx + c^2)y^2 + (x^2 + 2cx + c^2)y^2 + y^4 = dx^2 + dy^2 + g^4`,
`x^4 - 2c^2x^2 + c^4 + (x^2 - 2cx + c^2 + x^2 + 2cx + c^2)y^2 + y^4 = dx^2 + dy^2 + g^4`,
`x^4 - 2c^2x^2 + c^4 + 2x^2y^2 + 2c^2y^2 + y^4 - dx^2 - dy^2 - g^4 = 0`,
`x^4 + 2x^2y^2 + y^4 - 2c^2x^2 - dx^2 + 2c^2y^2 - dy^2 + c^4 - g^4 = 0`,

and finally:

`(x^2 + y^2)^2 - (2c^2 + d)x^2 + (2c^2 - d)y^2 + c^4 - g^4 = 0`

When having compared the equations, we can see that `-2A = - (2c^2 + d)`, `-2B = 2c^2 - d`, and then:

`A = 1/2 d + c^2`, `B = 1/2 d - c^2`, `C = c^4 - g^4`,
`A + B = d`, `A - B = 2c^2`.

Both definitions of curves of Perseus are (almost) equivalent. Because of the relations `C = (r^2 - c^2 - f^2)^2 - 4c^2f^2`, `A + B = 2(r^2 - f^2)`, `A - B = 2c^2`, we have also:

`c^4 - g^4 = (r^2 - c^2 - f^2)^2 - 4c^2f^2`,
`g^4 = c^4 - (r^2 - c^2 - f^2)^2 + 4c^2f^2`,
`g^4 = c^4 - (r^4 + c^4 + f^4 - 2c^2r^2 - 2f^2r^2 + 2c^2f^2) + 4c^2f^2`,
`g^4 = c^4 - r^4 - c^4 - f^4 + 2c^2r^2 + 2f^2r^2 - 2c^2f^2 + 4c^2f^2`,
`g^4 = 2(c^2f^2 + c^2r^2 + f^2r^2) - f^4 - r^4`,

and finally4:

`d = 2(r^2 - f^2)`,
`g = root (4) (2c^2(r^2 + f^2) - (r^2 - f^2)^2)`.

Note that `g` cannot be computed as a real number for `c = 0`. This is why the equivalency of both definitions of the curve of Perseus is not full.

The inverse relations are:

In order to make sense of the above expressions, we must assume `-2f^2 <= d < 2r^2` and `c != 0`. Moreover:

The parametre `c` exists in both definitions and it is the same parametre as `A - B = 2c^2` in either. It is equal to half of the focal but it is also equal to the midradius of the cut torus.

Special cases of curves of Perseus

Four special types of curves of Perseus will be considered below.

  1. The curve of Booth (also known as the hippopede of Proclus) is a type of the curve of Perseus when the intersecting plane is tangent to the interior (inner circle) of the torus: `f = |c - r|`.
  2. The curve of Cassini is a type of the curve of Perseus when the distance of the cutting plane from the centre of the torus is equal to the inner radius: `f = r`.
  3. The lemniscate of Bernoulli is both the curve of Booth and the curve of Cassini: `c = 2r`, `f = r`.
  4. The circle appears to be the curve of Perseus5 when the midradius of the torus is equal to zero: `c = 0`.

Curves of Booth

The curve of Booth is also known as the hippopede of Proclus. Basically, it is a type of the curve of Perseus, although it has different definitions as well.

Curves of Booth as curves of Perseus

The curve of Booth can be obtained by cutting a torus with a plane which is tangent to its inner circle (the circle formed by the inner radius), therefore `f = |c - r|`. Let’s compute the values of the parametres `A`, `B`, `C`:

`f^2 = (c - r)^2 = c^2 - 2cr + r^2`,
`r^2 + c^2 - (c^2 - 2cr + r^2) = r^2 + c^2 - c^2 + 2cr - r^2 = 2cr`,
`r^2 - c^2 - (c^2 - 2cr + r^2) = r^2 - c^2 - c^2 + 2cr - r^2 = 2cr - 2c^2 = 2c(r - c)`,
`(r^2 - c^2 - (c^2 - 2cr + r^2))^2 - 4c^2(c^2 - 2cr + r^2) = (2c(r - c))^2 - 4c^2(c^2 - 2cr + r^2) = 4c^2r^2 - 8c^3r + 4c^4 - 4c^4 + 8c^3r - 4c^2r^2 = 0`,
`2(r^2 - (c^2 - 2cr + r^2)) = 2(r^2 - c^2 + 2cr - r^2) = 2c(2r - c)`,
`2c^2(r^2 + c^2 - 2cr + r^2) - (r^2 - (c^2 - 2cr + r^2))^2 = 2c^2(2r^2 + c^2 - 2cr) - (r^2 - c^2 + 2cr - r^2)^2 =`
`= 2c^2(2r^2 + c^2 - 2cr) - c^2(2r - c)^2 = 4c^2r^2 + 2c^4 - 4c^3r - c^2(4r^2 - 4cr + c^2) = 4c^2r^2 + 2c^4 - 4c^3r - 4c^2r^2 + 4c^3r - c^4 = c^4`,

which finally gives:

`A = 2cr`, `B = 2c(r - c)`, `C = 0`,
`d = 2c(2r - c)`, `g = c`

(compare this result with the tables of dependencies between parametres).

As it can be seen, the curve of Booth is also the instance of the curve of Perseus when `g = c`. Therefore the equation of the curve can be presented as:

`(x^2 + y^2)^2 - (2c^2 + d)x^2 + (2c^2 - d)y^2 = 0`

As `c > 0`, `r > 0`, we have always `A > 0`. Depending on the sign of `B` we can distinguish 3 main types of curves of Booth:

  1. The oval of Booth, or the elliptic curve of Booth, formed by cutting a spindle torus (`c < r`, `B > 0`, `d > 2c^2`, `f = r - c`). The oval of Booth is an oval curve plus a point at (0,0) which is the point of tangence of the cutting plane and the spindle of the torus. The name “elliptic lemniscate of Booth” seems not to be suitable for the oval of Booth but it is also used.
  2. Two tangent circles, formed by cutting a horn torus (`c = r`, `B = 0`, `d = 2c^2`, `f = 0`).
  3. The lemniscate of Booth, or the hyperbolic curve of Booth, also known as “hyperbolic lemniscate of Booth”, formed by cutting a ring torus (`c > r`, `B < 0`, `d > 2c^2`, `f = c - r`).
Curves of Booth as pedals of conics

Besides the above, there exist two other definitions of the curve of Booth. The first of them says that the curve is a pedal of a conic curve6. This definition gives a good reason for terming curves of Booth “elliptic” and “hyperbolic”.

The tangent to the ellipse `x^2/a^2 + y^2/b^2 = 1` in the point `M_1 = (x_1,y_1)` is the line `(x x_1)/a^2 + (yy_1)/b^2 = 1`. The perpendicular to it which runs through (0,0) is `y = (a^2y_1)/(b^2x_1)x`. The point of intersection of both lines lies on the pedal curve. From the two equations we have `x_1 = (a^2x)/(x^2 + y^2), y_1 = (b^2x)/(x^2 + y^2)`. But `x_1, y_1` are points of an ellipse, hence `(a^4x^2)/(a^2(x^2 + y^2)^2) + (b^4y^2)/(b^2(x^2 + y^2)^2) = 1`. Finally, we have

`(x^2 + y^2)^2 - a^2x^2 - b^2y^2 = 0`

which is the equation of an elliptic curve of Booth.

In a similar way we can show that the pedal of a hyperbola (with the pedal point at (0,0)) is a hyperbolic curve of Booth, and

`(x^2 + y^2)^2 - a^2x^2 + b^2y^2 = 0`

is the equation of a hyperbolic curve of Booth.

We can see from the equations that:

Further relations are:

Note that `c = 1/2 sqrt (a^2 - b^2)` for the elliptic curve and `c = 1/2 sqrt (a^2+b^2)` for the hyperbolic curve, i.e. the focal (`2c`) of the curve is equal to half of the focal of the related conic.

Curves of Booth as cissoids

And finally, there exists one more definition of the curve of Booth. Let’s take a circle `(x - c)^2 + y^2 = R^2` (we will call it a generating circle; `R` need not to be equal to `r`, which is the radius of the circle which builds our original torus) and a straight line through `O = (0,0)`, i.e. the line `y = kx` with variable `k`. Let `P, Q` be the points of transsection of the straight line and the circle. The locus of points `M` of the straight line `y = kx` such that `OM = PQ` is a curve of Booth. Thus the curve of Booth is a cissoid of two equal circles7.

Let’s analyse the first one. As the points `P, Q` lie on both the line and the circle, their coordinates must satisfy both equations `y = kx` and `(x - c)^2 + y^2 = R^2`. We can compute them, taking `kx` for `y` in the equation of circle:

`(x - c)^2 + k^2x^2 = R^2`,
`x^2 - 2cx + c^2 + k^2x^2 - R^2 = 0`,
`(k^2 + 1)x^2 - 2cx + c^2 - R^2 = 0`,
`Delta = 4c^2 - 4(k^2 + 1)(c^2 - R^2)`,
`Delta = 4c^2 - 4(c^2k^2 - k^2R^2 + c^2 - R^2)`,
`Delta = 4c^2 - 4c^2k^2 + 4k^2R^2 - 4c^2 + 4R^2`,
`Delta = - 4c^2k^2 + 4k^2R^2 + 4R^2`,
`Delta = 4(k^2(R^2 - c^2) + R^2)`,

so finally:

`x_(1,2) = (c -+ sqrt (k^2(R^2 - c^2) + R^2))/(k^2 + 1)`, `y_(1,2) = k(c -+ sqrt (k^2(R^2 - c^2) + R^2))/(k^2 + 1)`.

The square of the distance `PQ` can be computed as:

`PQ^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2`,
`PQ^2 = ((c + sqrt (k^2(R^2 - c^2) + R^2))/(k^2 + 1) - (c - sqrt (k^2(R^2 - c^2) + R^2))/(k^2 + 1))^2 + (k(c + sqrt (k^2(R^2 - c^2) + R^2))/(k^2 + 1) - k(c - sqrt (k^2(R^2 - c^2) + R^2))/(k^2 + 1))^2`,
`PQ^2 = ((2sqrt (k^2(R^2 - c^2) + R^2))/(k^2 + 1))^2 + (k(2sqrt (k^2(R^2 - c^2) + R^2))/(k^2 + 1))^2`,
`PQ^2 = (1 + k^2)((2sqrt (k^2(R^2 - c^2) + R^2))/(k^2 + 1))^2`,
`PQ^2 = (1 + k^2)(4(k^2(R^2 - c^2) + R^2))/(k^2 + 1)^2`,
`PQ^2 = 4(k^2(R^2 - c^2) + R^2)/(k^2 + 1)`,
`PQ^2 = 4(k^2R^2 - c^2k^2 + R^2)/(k^2 + 1)`,
`PQ^2 = 4(R^2(k^2 + 1) - c^2k^2)/(k^2 + 1)`,

and finally:

`PQ^2 = 4(R^2 - (c^2k^2)/(k^2 + 1))`.

Let’s consider the condition `OM = PQ`. As `OM^2 = x^2 + y^2`, we have:

`x^2 + y^2 = 4(R^2 - (c^2k^2)/(k^2 + 1))`.

But because of `y = kx` we can compute `k = y/x` and use it in the preceding formula:

`x^2 + y^2 = 4(R^2 - (c^2y^2/x^2)/(y^2/x^2 + 1))`,
`x^2 + y^2 = 4(R^2 - (c^2y^2/x^2)/((x^2 + y^2)/x^2))`,
`x^2 + y^2 = 4(R^2(x^2 + y^2)/(x^2 + y^2) - (c^2y^2)/(x^2 + y^2))`,
`(x^2 + y^2)^2 = 4(R^2x^2 + R^2y^2 - c^2y^2)`,
`(x^2 + y^2)^2 = 4R^2x^2 + 4(R^2 - c^2)y^2`.

The equation of the elliptic curve of Booth as a cissoid is then:

`(x^2 + y^2)^2 - 4R^2x^2 - 4(R^2 - c^2)y^2 = 0`

We can find the equation of the hyperbolic curve of Booth as a cissoid in a quite similar way and with the same result. However, because of `c > R`, we will prefer the following form:

`(x^2 + y^2)^2 - 4R^2x^2 + 4(c^2 - R^2)y^2 = 0`

Comparing these equations with the ones for curves of Booth as pedals, we can see that `a^2 = 4R^2`, therefore `a = 2R`, and

We can also notice that here the `c` parametre means the same that the `c` parametre used previously. In other words, the centres of the circle generating the torus and the circle generating the cissoid are in the same place. Moreover `c` is equal to half of the focal of the curve of Perseus, so it has three meanings altogether.

As we have stated previously, `a^2 = d + 2c^2` and `b^2 = d - 2c^2` for the elliptic curve of Booth, and `b^2 = 2c^2 - d` for the hyperbolic one. Thus we have:

therefore:

`d = 2(2R^2 - c^2)`

for both types.

We will also notice that because of `a^2 = 2(r^2 - f^2 + c^2)` and `a^2 = 4R^2` we have `4R^2 = 2(r^2 - f^2 + c^2)`, so:

`R = sqrt ((r^2 - f^2 + c^2)/2) = 1/2 sqrt (d + 2c^2)`

also for both types of the curves.

Detailed classification of curves of Booth

Up to 11 types of curves of Booth can be distinguished in detailed classification:

1 a typical convex elliptic curve of Booth `0 < c < 1/2 r` `1/2 r < f < r` `c < f` `0 < c < sqrt 2/2 R` `0 < R < sqrt 2/2 r` `d > 6c^2`
2 a flat elliptic curve of Booth `c = 1/2 r` `f = 1/2 r` `c = f` `c = sqrt 2/2 R` `R = sqrt 2/2 r` `d = 6c^2 = 3R^2`
3 a concave elliptic curve of Booth `1/2 r < c < r` `0 < f < 1/2 r` `c > f` `sqrt 2/2 R < c < R` `sqrt 2/2 r < R < r` `2c^2 < d < 6c^2`
4 two tangent circles: `a = 2c`, `b = 0` `c = r` `f = 0` `f = 0` `c = R` `R = r` `d = 2c^2 = 2R^2`
5 the hyperbolic curve of Booth with `a > b`, `d > 0` `r < c < 2r` `0 < f < r` `c > 2f` `R < c < sqrt 2 R` `r < R < sqrt 2 r` `0 < d < 2c^2`
6 the lemniscate of Bernoulli: `a = b = 2R`, `d = 0`, see below `c = 2r` `f = r` `c = 2f` `c = sqrt 2 R` `R = sqrt 2 r` `d = 0`
7 the hyperbolic curve of Booth with `a < b`, `d < 0`,
and the focus inside the curve loop
`2r < c < 4r` `r < f < 3r` `4/3 f < c < 2f` `sqrt 2 R < c < 2R` `sqrt 2 r < R < 2r` `-c^2 < d < 0`
8 the centres of the generating circle (being the foci)
lie on the curve
`c = 4r` `f = 3r` `c = 4/3 f` `c = 2R` `R = 2r` `d = -c^2 = -4R^2`
9 the foci lie outside the loop of the curve,
the generating circle intersects the curve
`4r < c < 9r` `3r < f < 8r` `9/8 f < c < 4/3 f` `2R < c < 3R` `2r < R < 3r` `-14/9 c^2 < d < -c^2`
10 the generating circle is tangent to the curve `c = 9r` `f = 8r` `c = 9/8 f` `c = 3R` `R = 3r` `d = -14/9 c^2 = -14R^2`
11 the curve does not intersect the generating circle `c > 9r` `f > 8r` `c < 9/8 f` `c > 3R` `R > 3r` `d < -14/9 c^2`

As we can see, foci can be sometimes located outside the curve loops (which is rather “unnatural”). This is so in certain instances of a hyperbolic lemniscate of Booth, when the midradius of the torus is sufficiently long.

Curves of Cassini

Curves of Cassini as curves of Perseus

Looking at the torus, it is hard to guess why curves of Cassini are so special. They are generated when the cutting plane is as distant from the axis of the torus as long its inner radius is: `f = r`. However, the curve of Cassini can also be defined in another way. Namely, it is a geometric locus of points `M` such that the product of its distances from two foci `F_1 = (c,0), F_2 = (-c,0)` is constant and equal to `g^2` (`g` is a real parametre):

`F_1M*F_2M = g^2`

Using the distance formula, we can write this as:

`sqrt ((x - c)^2 + y^2) sqrt ((x + c)^2 + y^2) = g^2`,
`((x - c)^2 + y^2)((x + c)^2 + y^2) = g^4`,
`(x - c)^2(x + c)^2 + (x - c)^2y^2 + (x + c)^2y^2 + y^4 = g^4`,
`(x^2 - c^2)^2 + (x^2 - 2cx + c^2)y^2 + (x^2 + 2cx + c^2)y^2 + y^4 = g^4`,
`x^4 - 2c^2x^2 + c^4 + x^2y^2 - 2cxy^2 + c^2y^2 + x^2y^2 + 2cxy^2 + c^2y^2 + y^4 = g^4`,
`x^4 + 2x^2y^2 + y^4 - 2c^2x^2 + 2c^2y^2 + c^4 = g^4`,

and finally:

`(x^2 + y^2)^2 - 2c^2(x^2 - y^2) + c^4 - g^4 = 0`

Let’s compute the values of the parametres `A = r^2 + c^2 - f^2`, `B = r^2 - c^2 - f^2`, `C = (r^2 - c^2 - f^2)^2 - 4c^2f^2`, `d = 2(r^2 - f^2)`, `g = root (4) (2c^2(r^2 + f^2) - (r^2 - f^2)^2)` for `f = r`:

`A = c^2`,
`B = - c^2`,
`C = c^2(c^2 - 4r^2)`,
`d = 0`,
`g = sqrt (2cr)`.

It appears that `A = -B` for the curve of Cassini.

Detailed classification of curves of Cassini

We can distinguish several types of curves of Cassini:

1 the bioval of Cassini (a curve with two ovals) `c > 2r` `f < c/2` `g < c`
2 the lemniscate of Bernoulli, see below `c = 2r` `f = c/2` `g = c`
3 the concave oval of Cassini `r < c < 2r` `c/2 < f < c` `c < g < sqrt 2 c`
4 the flat oval of Cassini `c = r` `f = c` `g = sqrt 2 c`
5 the convex oval of Cassini `0 < c < r` `f > c` `g > sqrt 2 c`

Lemniscate of Bernoulli

The lemniscate of Bernoulli is both a curve of Booth and a curve of Cassini: `f = |c - r|`, `f = r`, `c = g = 2r`. It is also defined as a geometric locus of points `M` such that the product of its distances from two fixed points named foci: `F_1 = (c,0), F_2 = (-c,0)` is a constant and equal to `c^2`.

`F_1M*F_2M = c^2`

Using the distance formula, we can write this as:

`sqrt ((x - c)^2 + y^2) sqrt ((x + c)^2 + y^2) = c^2`,
`((x - c)^2 + y^2)((x + c)^2 + y^2) = c^4`,
`(x - c)^2(x + c)^2 + (x - c)^2y^2 + (x + c)^2y^2 + y^4 = c^4`,
`(x^2 - c^2)^2 + (x^2 - 2cx + c^2)y^2 + (x^2 + 2cx + c^2)y^2 + y^4 = c^4`,
`x^4 - 2c^2x^2 + c^4 + x^2y^2 - 2cxy^2 + c^2y^2 + x^2y^2 + 2cxy^2 + c^2y^2 + y^4 = c^4`,
`x^4 + 2x^2y^2 + y^4 - 2c^2x^2 + 2c^2y^2 + c^4 = c^4`,

and finally:

`(x^2 + y^2)^2 - 2c^2(x^2 - y^2) = 0`

Let’s compute the values of the parametres `A = r^2 + c^2 - f^2`, `B = r^2 - c^2 - f^2`, `C = (r^2 - c^2 - f^2)^2 - 4c^2f^2`, `d = 2(r^2 - f^2)`, `g = root (4) (2c^2(r^2 + f^2) - (r^2 - f^2)^2)` for `f = r`, `c = 2r`:

`A = 4r^2`,
`B = - 4r^2`,
`C = 0`,
`d = 0`,
`g = 2r`.

It appears that `A = -B`, `C = 0` for the lemniscate of Bernoulli.

Circle

The circle is a very special type of the curve of Perseus with `c = 0`. Because of this condition (meaning that the centre of the rotating circle is placed on the axis) we get a sphere instead of a torus when rotating the initial circle. The circle being a curve of Perseus originates when intersecting this sphere with a plane with the distance `f` such that `0 <= f < r`.

Let’s apply this condition to the original equation of the curve of Perseus `(x^2 + y^2 + c^2 + f^2 - r^2)^2 = 4c^2(x^2 + f^2)`:

`(x^2 + y^2 + f^2 - r^2)^2 = 0`,
`x^2 + y^2 + f^2 - r^2 = 0`,

and finally:

`x^2 + y^2 = r^2 - f^2`

Notice that `r` is the radius of the initial circle, not the generated one (which is equal to `sqrt (r^2 - f^2)`).

Let’s compute the values of the parametres `A = r^2 + c^2 - f^2`, `B = r^2 - c^2 - f^2`, `C = (r^2 - c^2 - f^2)^2 - 4c^2f^2` for `c = 0`:

`A = r^2 - f^2`,
`B = r^2 - f^2`,
`C = (r^2 - f^2)^2`.

It appears that `A = B` for the circle. Therefore the circle is not a curve of Cassini as for this type the condition is `A = -B`. The circle is not a curve of Booth, either, as it does not satisfy the condition `f = |c - r|` because it would mean `f = r` for circle which is not true.

The circle is so special because it cannot be presented using the definition of the curve of Perseus as a locus, and with the formula `(x^2 + y^2)^2 - (2c^2 + d)x^2 + (2c^2 - d)y^2 + c^4 - g^4 = 0`. It is so because for `c = 0` we could compute `d = 2(r^2 - f^2)` but not `g^4 = 2c^2(r^2 + f^2) - (r^2 - f^2)^2 = - (r^2 - f^2)^2` which would be negative, therefore `g` would have to be irreal.

If we accepted `a^2 = d + 2c^2`, `b^2 = |d - 2c^2|`, `a = 2R` for the circle as for the curve of Booth, we would get `a = b = sqrt d = sqrt (2(r^2 - f^2))`, `d = a^2 = b^2`, `R = 1/2 a`, `d = 4R^2`.

Footnotes

1. In the Polish literature the term “curves of Perseus” is used exclusively.

2. For more details about tori, consult Wikipedia or Wolfram Mathworld.

3. In the literature (in Polish: Niczyporowicz E., Krzywe płaskie [Plane Curves], PWN, Warszawa 1991) you can find a similar formula but with d4; this is a strange error of the author however (copied to other books without checking) because d (the factor in the d · OM2) can be negative; so if we took d4 instead of it, we should also consider irreal values of d.

4. It is clear from the formulas below that d can be negative – namely when f > r.

5. Another Niczyporowicz’s mistake is treating the circle as a type of the curve of Cassini. In fact they are two different and excluding cases. Namely, A = -B for the curve of Cassini while A = B for the circle.

6. For more information on pedal curves see Wikipedia, Wolfram Mathworld, 2d Curves or Famous Curves.

7. For more information on cissoids see Wikipedia, Wolfram Mathworld, 2d Curves or Famous Curves.