On the Properties of Certain Conchoidal Families

L. I. GANDZHA

Let, in the axes \(x—y\), a certain central curve (Fig. 1) be given in such a way that the coordinates of the center of this curve are equal to:

\[ x_c=0,\quad y_c=c. \]

Let, further, a segment of a straight line \(AB=l\) move in the coordinate plane so that the point \(A\) always lies on the \(x\)-axis, while the straight line always touches the given central curve. Then the point \(B\) will describe in the coordinate plane a plane curve which, as will be shown below, belongs to a family of conchoids.

Let us find the equation of this curve in parametric form:

\[ y=f(\varphi), \]

\[ x=f'(\varphi), \]

where \(\varphi\) is the angle of inclination of the straight line \(AB\) to the \(x\)-axis, measured from the \(x\)-axis counterclockwise to the direction \(AB\). We shall restrict our conclusions to the particular case when the central curve is an ellipse.

Fig. 1

Denoting the current coordinates of the required curve by \(x\) and \(y\), we find (Fig. 1)

\[ y=l\sin\varphi. \tag{1} \]

Let us further denote by \(x_e'\) and \(y_e'\) the coordinates of the ellipse for the case when its center coincides with the origin of coordinates; then

\[ \frac{(x_e')^2}{a^2}+\frac{(y_e')^2}{b^2}=1. \tag{2} \]

For the transition to the coordinates of Fig. 1 we have the relations:

\[ x_e=x_e'\quad \text{and}\quad y_e=y_e'+c, \]

and after substituting \(x_e'\) and \(y_e'\) in equation (2) we obtain:

\[ \frac{x_e^2}{a^2}+\frac{(y_e-c)^2}{b^2}=1. \tag{3} \]

According to Fig. 1, the equation for \(x\) can be obtained in the following form:

\[ x=l\cdot \cos\varphi-y_{\mathrm e}\cdot \operatorname{tg}\left(\frac{\pi}{2}-\varphi\right)+(\mp x_{\mathrm e}), \tag{4} \]

where the minus sign must be taken in the case when the line \(AB\) is tangent to the ellipse in the region of negative \(x\)'s for acute angles \(\varphi\), and the plus sign in the region of positive \(x\)'s (the line \(A_1, B_1\)).

Let us substitute the values of \(x_{\mathrm e}\) and \(y_{\mathrm e}\) in (4).
From (3) we find:

\[ y_{\mathrm e}=c\pm \frac{b}{a}\sqrt{a^2-x_{\mathrm e}^{\,2}} \tag{5} \]

Taking the derivative of equation (5), we obtain:

\[ \operatorname{tg}\varphi=\frac{dy_{\mathrm e}}{dx_{\mathrm e}} =\mp \frac{b}{a}\frac{x_{\mathrm e}}{\sqrt{a^2-x_{\mathrm e}^{\,2}}}. \tag{6} \]

From (6) we have:

\[ x_{\mathrm e}=\pm \frac{a^2\operatorname{tg}\varphi}{\sqrt{b^2+a^2\operatorname{tg}^2\varphi}}. \tag{7} \]

Solving (4), (5), and (7) simultaneously, we obtain:

\[ x=l\cdot \cos\varphi- \left\{c\pm \frac{b^2}{\sqrt{b^2+a^2\operatorname{tg}^2\varphi}}\right\} \operatorname{tg}\left(\frac{\pi}{2}-\varphi\right) + \left\{\pm \frac{a^2\cdot \operatorname{tg}\varphi}{\sqrt{b^2+a^2\cdot \operatorname{tg}^2\varphi}}\right\}. \tag{8} \]

Bearing in mind that, by comparing (4) and (8), the equalities

\[ y_{\mathrm e}=c\pm \frac{b^2}{\sqrt{b^2+a^2\cdot \operatorname{tg}^2\varphi}} \]

and

\[ x_{\mathrm e}=\frac{a^2\cdot \operatorname{tg}\varphi}{\sqrt{b^2+a^2\cdot \operatorname{tg}^2\varphi}}, \]

follow, we obtain the following rules for choosing the signs in (8):

if the central curve is specified above the \(x\)-axis, then

1) the plus sign in the second term corresponds to the conditions

\[ 0<\varphi<\frac{\pi}{2}\quad \text{and}\quad \frac{3}{2}\pi<\varphi<2\pi \]

when the tangency is in the region of negative \(x\)'s for acute angles \(\varphi\), and

\[ \frac{\pi}{2}<\varphi<\frac{3}{2}\pi \]

when the tangency is in the region of positive \(x\)'s for acute angles \(\varphi\);

2) the minus sign in the second term corresponds to the conditions

\[ -\frac{\pi}{2}<\varphi<\frac{3}{2}\pi \]

when the tangency is in the region of negative \(x\)'s for acute angles \(\varphi\), and

\[ 0<\varphi<\frac{\pi}{2},\qquad \frac{3}{2}\pi<\varphi<2\pi \]

when the tangency is in the region of positive \(x\)'s for acute angles \(\varphi\);

3) the sign \(+\) in the last term corresponds to the conditions:

\[ \pi<\varphi<2\pi \]

for tangency in the region of negative \(x\)’s at acute angles \(\varphi\), and

\[ 0<\varphi<\pi \]

for tangency in the region of positive \(x\)’s at acute angles \(\varphi\);

4) the sign \(-\) in the last term corresponds to the conditions:

\[ 0<\varphi<\pi \]

for tangency in the region of negative \(x\)’s at acute angles \(\varphi\), and

\[ \pi<\varphi<2\pi \]

for tangency in the region of positive \(x\)’s at acute angles \(\varphi\).

Analogous rules for choosing the signs may also be derived for the case when the central curve is situated below the \(x\)-axis.

In Fig. 2 the desired curves described by equations (1) and (8) are presented. Here the solid curve corresponds to the case of tangency in the region of negative \(x\)’s at acute angles \(\varphi\), and the dashed curve to the case of tangency in the region of positive \(x\)’s at acute angles \(\varphi\).

We see that the curves of both cases are symmetric with respect to the \(y\)-axis. If we consider both of the mentioned cases for negative \(c\), then one can obtain two more curves, symmetric to those shown in Fig. 2 with respect to the \(x\)-axis (not shown in the drawing).

Fig. 2

Restricting ourselves only to the consideration of one case with positive \(c\) and tangency in the region of negative \(x\)’s at acute angles \(\varphi\) (the solid curve in Fig. 2), we derive the following consequences.

1. For \(0<\varphi<\pi\) the curve comes from \(-\infty\), describes, under certain conditions, a loop about the central curve, and goes to \(+\infty\), lying above the \(x\)-axis.

2. For \(\pi<\varphi<2\pi\) the curve comes from \(-\infty\) and goes to \(+\infty\), lying below the \(x\)-axis.

3. On the basis of (1) and (8), for \(\varphi=0,\pi,2\pi\),

\[ y=0, \]

\[ x=\infty, \]

that is, both branches have an asymptote, namely the \(x\)-axis.

1. Investigating equation (1) for a maximum and a minimum, we find that the maximum \(y=f(\varphi)\) occurs at

\[ \varphi=\frac{\pi}{2} \]

and the minimum at

\[ \varphi=\frac{3}{2}\pi . \]

To determine the values of \(x\) corresponding to the maximum and minimum of \(y\), we substitute the found values of \(\varphi\) into equation (8); in doing so expression (8) becomes indeterminate. To remove the indeterminacy, we write (8) in the form:

\[ x=l\cos\varphi-\left\{c+\frac{b^{2}}{\sqrt{\,b^{2}+a^{2}\operatorname{tg}^{2}\varphi\,}}\right\} \operatorname{tg}\left(\frac{\pi}{2}-\varphi\right) +\left\{\frac{a^{2}}{\sqrt{\dfrac{b^{2}}{\operatorname{tg}^{2}\varphi}+a^{2}}}\right\}. \tag{9} \]

Substitution of the values \(\varphi=\dfrac{\pi}{2}\) and \(\varphi=\dfrac{3}{2}\pi\) in (9) gives the following values of \(x\):

\[ \begin{aligned} &\text{for the maximum of } y:\ x_{\max}=-a,\\ &\text{for the minimum of } y:\ x_{\min}=+a. \end{aligned} \tag{10} \]

Here it is necessary to take into account the rules for choosing signs given above.

Substitution of the same values \(\varphi=\dfrac{\pi}{2}\) and \(\varphi=\dfrac{3}{2}\pi\) in (1) gives the values of the maximum and minimum of \(y\):

\[ \left. \begin{aligned} y_{\max}&=l,\\ y_{\min}&=-l \end{aligned} \right\} \tag{11} \]

1. From Fig. 2 one may conclude that, for \(l>c\), the curve has a nodal point at which its branch intersects itself. The coordinates of the nodal point are found on the basis of the following considerations.

At the nodal point the angles of inclination of the tangents to the curve are different and satisfy the conditions:

\[ 0<\varphi_{1}<\frac{\pi}{2} \]

and

\[ \frac{\pi}{2}<\varphi_{2}<\pi . \]

Substituting \(\varphi_{1}\) and \(\varphi_{2}\) in (9), with observance of the rules for choosing signs, we obtain the values \(x_{1}\) and \(x_{2}\). It is obvious that at the nodal point

\[ x_{1}=x_{2}. \]

Then

\[ \begin{aligned} &l\cos\varphi_{1} -\left\{c+\frac{b^{2}}{\sqrt{\,b^{2}+a^{2}\operatorname{tg}^{2}\varphi_{1}\,}}\right\} \operatorname{tg}\left(\frac{\pi}{2}-\varphi_{1}\right) +\left\{\frac{a^{2}}{\sqrt{\dfrac{b^{2}}{\operatorname{tg}^{2}\varphi_{1}}+a^{2}}}\right\} \\ &= l\cos\varphi_{2} -\left\{c-\frac{b^{2}}{\sqrt{\,b^{2}+a^{2}\operatorname{tg}^{2}\varphi_{2}\,}}\right\} \operatorname{tg}\left(\frac{\pi}{2}-\varphi_{2}\right) +\left\{-\frac{a^{2}}{\sqrt{\dfrac{b^{2}}{\operatorname{tg}^{2}\varphi_{2}}+a^{2}}}\right\}. \end{aligned} \tag{12} \]

On the other hand, substituting $\varphi_1$ and $\varphi_2$ in (1) gives the values $y_1$ and $y_2$, which for the nodal point are also equal:

$y_1=y_2$, or $l\sin\varphi_1=l\sin\varphi_2$, or $\sin\varphi_1=\sin\varphi_2$, whence

\[ \varphi_2=\pi-\varphi_1. \tag{13} \]

Substituting $\varphi_2$ from (13) into (12), we obtain:

\[ \sin\varphi_1=\frac{c}{l}. \tag{14} \]

Substitution of (14) into (1) gives the value $y_1=y_2$ for the nodal point:

\[ y_1=l\sin\varphi_1=c. \tag{15} \]

Substitution of (14) into (9) gives the value $x_1=x_2$:

\[ x_1=\sqrt{l^2-c^2} -\left\{ c+\frac{b^2}{\sqrt{\,b^2+\dfrac{a^2c^2}{l^2-c^2}\,}} \right\} \frac{\sqrt{l^2-c^2}}{c^2} -\frac{a^2}{\sqrt{\,\dfrac{b^2(l^2-c^2)}{c^2}+a^2\,}}. \tag{16} \]

1. On the basis of (11), (14), and (15) we conclude that when $l=c$ the maximum point of the curve coincides with its singular point. In this case the nodal point turns into a point of return.

Fig. 3

If in (8) we put $a=b=r$, then we obtain the central curve specified by a circle; in this case (8) is simplified and takes the form:

\[ x=l\cos\varphi-\{c\pm r\cos\varphi\}\tg\left(\frac{\pi}{2}-\varphi\right)+\{\pm r\sin\varphi\} \tag{17} \]

and, as before, $y=l\sin\varphi$.

In Fig. 3 a family of conchoids is presented for various values of the parameter $l$, described by the system of equations (17). The method of investigating these curves remains the same.

Fig. 3 shows that as $l$ decreases, the branches of the curve approach their asymptote and, when $l=0$, coincide with it.

In Fig. 4 another family of circular conchoids is presented for various values of the parameter $c$. The curves for $c=r$ and for $c=0$ are especially characteristic.

In the first case the curve, coming from \(-\infty\) in the region of positive values of \(y\), forms a loop, reaches the \(x\)-axis and, turning sharply, goes off to \(+\infty\), merging with the \(x\)-axis. The lower branch, merging with the \(x\)-axis on the interval \(-\infty,\, l\), at the point \((x=l,\ y=0)\) turns sharply downward and, after passing through

Fig. 4

a minimum value, goes off to \(+\infty\), asymptotically approaching the \(x\)-axis.

In the second case the upper branch comes from \(-\infty\) and goes off to \(-\infty\); the lower branch comes from \(+\infty\) and, intersecting the \(y\)-axis twice, goes off to \(+\infty\).

Fig. 5

Fig. 5 represents a family of circular conchoids for different values of the parameter \(r\).

In the particular case when \(r=0\), the central curve turns into a point lying on the \(y\)-axis at a distance \(c\) from the origin.

In this case the circular conchoid becomes the conchoid of Nicomedes, which is shown in Fig. 5 for three values of the parameter \(l\):

\[ l=6, \]

\[ l=3, \]

\[ l=2. \]

Substituting \(r=0\) into equation (17), we obtain the equations of the conchoid of Nicomedes in parametric form:

\[ \left. \begin{aligned} y &= l\cdot \sin \varphi,\\ x &= l\cdot \cos \varphi - c\cdot \operatorname{tg}\left(\frac{\pi}{2}-\varphi\right) \end{aligned} \right\}. \tag{18} \]

Conclusions

1. By varying the parameters, families of conchoids can be formed around various central curves.

2. Families of conchoids may be designated by the name of the central curve around which they are formed. Thus, one may distinguish “elliptic conchoids,” “circular conchoids,” etc.

3. “Circular conchoids” constitute a special case of “elliptic conchoids,” when \(a=b=r\).

The conchoid of Nicomedes constitutes a special case of “circular conchoids,” when \(r=0\).