]> App02, Conic sections

Conic Sections

Dual conic surface

As already stated, a conic section is defined by the intersection of a dual conic surface with a plane (1.2.5.22). The present study describing how the sections are obtained is important, but not necessary for viewing the Fig. Conic sections.

The use of the spherical coordinates (3.1.4.20) reproduced here:

 $\begin{array}{l}x=R\mathrm{sin}\theta \mathrm{cos}\varphi \\ y=R\mathrm{sin}\theta \mathrm{sin}\varphi \\ z=R\mathrm{cos}\theta \\ \text{restrictions:}\\ R\ge 0\\ 0\le \theta \le \pi \\ 0\le \varphi <2\pi \end{array}\right\}$ (A02.1)
 $\begin{array}{l}x=R\mathrm{sin}\theta \mathrm{cos}\phi \\ y=R\mathrm{sin}\theta \mathrm{sin}\phi \\ z=R\mathrm{cos}\theta \\ \text{restrictions:}\\ R\ge 0\\ 0\le \theta \le \pi \\ 0\le \phi <2\pi \end{array}\right\}$ (A02.1)

facilitates the definition of a dual conic surface of rotation:

 $\mathrm{sin}\theta =\mathrm{sin}{\theta }_{0}=\text{constant}$ (A02.2)

where the lower value of  θ0  represents the cone's angle of formation.

From (A02.1) one obtains:

 $\begin{array}{l}{x}^{2}+{y}^{2}={R}^{2}{\text{sin}}^{\text{2}}{\theta }_{0}\\ {z}^{2}={R}^{2}{\text{cos}}^{\text{2}}{\theta }_{0}\end{array}\right\}\text{\hspace{0.17em}}⇒\text{\hspace{0.17em}}{x}^{2}+{y}^{2}={z}^{2}{\mathrm{tan}}^{2}{\theta }_{0}$ (A02.3)

meaning that the intersecion of this conic surface with the plane  $z={z}_{0}$  yields a circle centred at the  $\left(x,y\right)$  origin with radius  $r=|{z}_{0}|\mathrm{tan}{\theta }_{0}$ . In the case of  ${z}_{0}=0$   the circle degenerates to a point.

Rotated conic surface

In order to study other conic sections, we'll rotate the dual cone about the y axis, so that the direction the cone's axis will form an angle  δ  with  z . We'll continue to use the plane  $z={z}_{0}$  for obtaining the conic sections. This transformation is a two dimensional rotation in the  $\left(z,x\right)$  plane, conserving the values of  y  and of  ${\rho }^{2}={z}^{2}+{x}^{2}$. By following similar steps to those carried out by (1.2.5.32-35) one obtains the relations:

 $\begin{array}{l}x\text{'}=x\mathrm{cos}\delta +z\mathrm{sin}\delta \\ z\text{'}=-x\mathrm{sin}\delta +z\mathrm{cos}\delta \end{array}\right\}$ (A02.4)

and their inverse

 $\begin{array}{l}x=x\text{'}\mathrm{cos}\delta -z\text{'}\mathrm{sin}\delta \\ z=x\text{'}\mathrm{sin}\delta +z\text{'}\mathrm{cos}\delta \end{array}\right\}$ (A02.5)

where x' and z' are the coordinates of the rotated conic surface.

Before obtaining the general expression of this rotated conic surface, for sake of simplicity we'll choose the angle:

 ${\theta }_{0}=\frac{\pi }{4}$ (A02.6)

The substitution of  x  and of  z  from (A02.5) into the conic surface (A02.3) and by taking care of (A02.6) yields:

 $\left({\text{cos}}^{2}\delta -{\text{sin}}^{2}\delta \right)\left(x{\text{'}}^{2}-z{\text{'}}^{2}\right)-4x\text{'}z\text{'}\mathrm{sin}\delta \mathrm{cos}\delta +{y}^{2}=0$ (A02.7)

The prime signs are removed:  $\left(x\text{'},z\text{'}\right)⇒\left(x,z\right)$  , and the trigonometric expressions - simplified, in order to obtain finally the required conic surface:

 $\left({x}^{2}-{z}^{2}\right)\mathrm{cos}\left(2\delta \right)-2xz\mathrm{sin}\left(2\delta \right)+{y}^{2}=0$ (A02.8)

Conic sections

The choice of  $z={z}_{0}$  represents the conic section corresponding to this particular plane:

 $\left({x}^{2}-{z}_{0}^{2}\right)\mathrm{cos}\left(2\delta \right)-2x{z}_{0}\mathrm{sin}\left(2\delta \right)+{y}^{2}=0$ (A02.9)

First the simplest cases will be studied:

1.  $2\delta =0\text{ }⇒\text{ }{x}^{2}+{y}^{2}={z}_{0}^{2}$ (A02.10)
which we already know, represents a circle with radius  $r=|{z}_{0}|$  centred at the  $\left(x,y\right)$  origin, except for  ${z}_{0}=0$  where the circle degenerates in a point.

2.  $2\delta =\frac{\pi }{2}\text{ }⇒\text{ }{y}^{2}=2x{z}_{0}=4\left(\frac{{z}_{0}}{2}\right)x$ (A02.11)
which is the canonical form of a parabola (1.2.5.23), except for the case of  ${z}_{0}=0$  corresponding to the  x  axis  $\left(y=0\right)$ .

3.  $2\delta =\pi \text{ }⇒\text{ }{x}^{2}-{y}^{2}={z}_{0}^{2}$ (A02.12)
which is the canonical form of a hyperbola (1.2.5.40), except for the case of  ${z}_{0}=0$  corresponding to two intersecting straight lines:
 $y=±x$ (A02.13)
which are actually the asymptotes of the hyperbolas (A02.12).

Before going to the more general case you may wish to observe these three simple cases at Fig. Conic sections

The therms of (A02.9) can be rearanged in a more suitable manner:

 ${\left[x-{z}_{0}\mathrm{tan}\left(2\delta \right)\right]}^{2}{\text{cos}}^{2}\left(2\delta \right)+{y}^{2}\mathrm{cos}\left(2\delta \right)={z}_{0}^{2}$ (A02.14)

The inquisitive user can easily prove the equivalence between the two expressions, just by opening up the parantheses of (A02.14). The therm into the square brackets of (A02.14) just defines a translation along x axis. The infinity of the  $\mathrm{tan}\left(2\delta \right)$  does not cause any problem as shown at (A02.11). If the sign of  $\mathrm{cos}\left(2\delta \right)$  is positive, (A02.14) describes an ellipse, on the other hand the negative sign indicates a hyperbola. In both cases the canonic form of the appropriate conic section is translated along the x axis, except for  ${z}_{0}=0$ . We are now ready to fill the gap of the  2δ values:

1.  (A02.15)

for  ${z}_{0}=0$  the ellipse degenerates in a point at the  $\left(x,y\right)$  origin. Example of such an ellipse is:
 $2\delta =\frac{\pi }{3};\text{ }\mathrm{cos}\left(2\delta \right)=\frac{1}{2}\text{ }⇒\text{ }\frac{{\left(x-{z}_{0}\sqrt{3}\right)}^{2}}{4{z}_{0}^{2}}+\frac{{y}^{2}}{2{z}_{0}^{2}}=1$ (A02.16)

2.  (A02.17)

for  ${z}_{0}=0$  the hyrebola degenerates into the asymptotes  $y=±\frac{b}{a}x\text{\hspace{0.17em}}$  intersecting at the  $\left(x,y\right)$  origin. An example of such a hyperbola is:
 $2\delta =\frac{2\pi }{3};\text{ }\mathrm{cos}\left(2\delta \right)=-\frac{1}{2}\text{ }⇒\text{ }\frac{{\left(x+{z}_{0}\sqrt{3}\right)}^{2}}{4{z}_{0}^{2}}-\frac{{y}^{2}}{2{z}_{0}^{2}}=1$ (A02.18)

All these five examples are shown in Fig. Conic sections