The Ellipse Machine

 

A mechanical device demonstrates that an ellipse is a tilted slice through a cone.

 The Ellipse in Physics

The ellipse is one of the most basic shapes in nature that describes a wealth of phenomena. Some examples include:

Orbit of two masses

A small mass ($m_2)$ travels around a much larger mass ($m_1)$  in an elliptical orbit.

Orbit of Spherical Pendulum

A spherical pendulum will generally follow an elliptical orbit about the vertical (gravitational) axis at low energies.

Focus of light and sound

The line perpendicular to the tangent of the ellipse (labeled s) always bisects the angle $\beta$, so the angles $\theta_i$ and $\theta_r$ are always equal. This is also the law of reflection in optics if the two dashed lines are light rays. This means that any light ray emanating from an ellipse foci will be reflected by an elliptical wall toward the opposing foci, so that the foci image one another, either for light or sound.

Conic Sections

Shapes like a circle, ellipse, parabola or hyperbola can all be thought of as conic sections (cross sections through a cone), as illustrated below (www.onlinemathlearning.com/image-files/conic-sections.png>).

A conical surface can be traced by sweeping a tilted rod about the vertical axis.  

A rod rotating about the vertical axis by an angle $\alpha$ should then be able to trace a cutout of each of these shapes in an appropriately tilted plane. The intent of the project is to demonstrate this for the case of an ellipse.

Mathematics of an Ellipse

Stretching a Circle

One way to describe an ellipse mathematically is to start with a circle of radius b represented in xy  coordinates:

$(\frac{x}{b})^2 + (\frac{y}{b})^2 = 1$

If we stretch a circle along x by a factor of $\frac{a}{b}$, the result is an ellipse with xy coordinates related by:

$(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$

Tilted Ellipse and Cylinder

Slicing a cylinder at an angle results in an elliptical cross-section. A side view of the long axis of an ellipse (red line) and a cylinder intersected by the ellipse (green line) is shown in the figures below with coordinates $(x’, z’)$.  The cylinder is symmetric about the z axis and has a radius r. The ellipse is tilted from the vertical by an angle $\gamma$.  The end points of the ellipse are at $(r, h)$ and $(-r, 0)$. The center of the ellipse is at $(0, z_c)$. 

These coordinates and angle are related as follows:

$a= \frac{r}{\sin\gamma}$

$h=2a\cos\gamma$

$z_c=\frac{h}{2}$

$b=r$

Which gives:

$h=2a\cos\gamma$

$z_c=a \cos\gamma$

$\frac{a}{b}=\frac{1}{\sin\gamma}$

These expressions show that the shape of the ellipse, described by the aspect ratio a/b, only depends on the ellipse tilt angle. The size of the ellipse is then determined by the cylinder radius r. 

Any ellipse can be thought of as a circle of radius r that has been rotated by an angle $\gamma$ about the y axis. The equation for a circle is $x’^2 + y’^2 = r^2$ in the coordinates $(x’, y’, z’)$ of the cylinder. The rotation into the (x,y,0) coordinates gives:

$x’ = \cos\gamma  –  z\sin\gamma$

$y’ = y$

$z’ = x\sin\gamma  +  z\cos\gamma$

Substituting into the circle equation gives $(\cos\gamma)^2 x^2 + y^2 = r^2$, which can be re-written as the equation for an ellipse $(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$, where $b = r$ and $\frac{a}{b} = \frac{1}{\cos\gamma}$.

Tilted Ellipse and Cone

A side view of the long axis of an ellipse (red line) and a cone intersected by the ellipse (green line) is shown in the figure below.  The cone is symmetric about the y axis and has a half angle $\alpha$ with its vertex at the origin. The ellipse is tilted from the vertical by an angle $\gamma$.  The end points of the ellipse are denoted $(x_0, h)$ and $(-x'_0, h’)$. The center of the ellipse is at $(x_c, z_c)$, and the intersection of the y axis and the ellipse is at $(0, z_{cc})$. 

These coordinates and angles are related as follows:

$x_0 = h\tan\alpha$

$x'_0 = h' \tan\alpha$

$h  –  h’ = 2a\cos\gamma$

$x_0 + x'_0 = 2a \sin\gamma$

$x_c = x_0 - a \sin\gamma$

$z_c = h - a \cos\gamma$

$\tan\gamma = \frac{x_0}{h-y_{cc}}$

$b^2 + {x_c}^2 = (z_c \tan\alpha)^2$

giving:

$h = a \frac{\sin\gamma}{\tan\alpha + \cos\gamma}$

$h’ = a \frac{\sin\gamma}{\tan\alpha - \cos\gamma}$

$x_0 = a (\sin\gamma + \cos\gamma \tan\alpha)$

$x'_0 = a (\sin\gamma - \cos\gamma \tan\alpha)$

$x_c = a \cos\gamma \tan\alpha$

$z_c = a \frac{\sin\gamma}{ \tan\alpha}$ 

$z_{cc} = h (1 - \frac{\tan\alpha}{\tan\gamma})$

$b = a \frac{(\sin^2\gamma – \sin^2\alpha)^{\frac{1}{2}}}{\cos\alpha}$

$\frac{a}{b} = \frac{\cos\alpha}{(\sin^2\gamma – sin^2\alpha)^{\frac{1}{2}}}$

The expressions show that the shape of the ellipse, described by the aspect ratio $a/b$, only depends on the cone angle and the ellipse tilt angle. The size of the ellipse is then determined by its distance along the cone axis, measured by $z_{cc}, z_c, h’$ or $h$. Both the cone and ellipse can be determined by the two angles $\gamma$ and $\alpha$ and any one of the characteristic lengths. 

As with the cylinder example (a special case where $\alpha = 0)$, we can generate an ellipse starting with a horizontal circle in the cone and stretch it while tilting it to remain in contact with the cone surface. Unlike the cylinder case, the ellipse center shifts right as the tilt increases, leading to the horizontal offset $x_c$. As with the cylinder case $r = b = (x_0 + x'_0)/2 = a \sin\gamma$.

Ellipse Foci

A length $\ell = 2a$ of string is laid across the long axis of the ellipse along x. If the center of the string is then raised along y a distance b, then the ends of the string will slide in some distance to $x = \pm f$. From this we can conclude that $f^2 = a^2 – b^2$. 

If we move the raised point to some point $(x, y)$, keeping the string taut, then the string length $\ell = \ell1 + \ell2$, where $\ell1 = \sqrt{(f-x)^2 + y^2}$ and $\ell2 = \sqrt{(f+x)^2 + y^2}$. The fixed length of $\ell = 2a$ then provides a relationship between x and y. After some algebraic manipulation, we find that these equations reduce to that of an ellipse: $(y/b)^2 + (x/a)^2 = 1$, when $\ell = 2a$ and $f = \sqrt{a^2-b^2}$. An ellipse can therefore be defined as a set of points whose distance from two focal points is a constant. In the case of a circle $a = b, f = 0, \ell1 = \ell2 = r$ and the string of length $2r$ goes from the center to the perimeter and back to the center.

This result provides a convenient way to draw any ellipse using a string of length $2a$, tacking down the ends at $+f$ and $–f$ from center, and a pencil is moved against the string from the inside to trace the ellipse.

The Ellipse Machine

Design Method for the Ellipse

The apparatus creates a virtual cone by rotating two rods about a vertical axis that are tilted from the vertical by an angle $\alpha$. The ellipse with a major axis a and minor axis b was cut from sheet metal, tilted by an angle gamma from the vertical and shifted by $x_c$ from the cone axis to fit into the cone shape. To arrive at values for all these parameters, an aspect ratio of $a/b \approx 1.5$ with $a = 2.5”$ was chosen, for aesthetic and practical reasons. The aspect ratio determines a set of values for $\alpha$ and $\gamma$ as shown in the plot at the left. The red arrow indicates a reasonable choice, and the previous equations lead to the final design:

$\alpha = 30^\circ, \gamma = 50^\circ, a = 2.5”, a/b = 1.492, b = 1.675”, x_c = 0.928”, z_c = 3.317”, z_{cc} = 2.539”$

System Design

Piece by Piece

Note the elliptical cross section of the cylinder.

Video of Partial Assembly

Cutting the Ellipse

Trammel (Ellipsograph) of Archimedes

(HMK suggestion and lots of guidance)

Point $A = (x_1,0)$ travels only along x

Point $B = (0, -y_1)$ travels only along y

Rotating the red line about the origin traces the dashed circle ${x_1}^2 + {y_1}^2 = L1^2$, where L1 is the distance between A and B.

extending the red line a length L2 from A to $(x, y)$ means:

$(x-x_1)/L2 = x_1/L1$

$y/L2 = y_1/L1$

$x = x_1 (L2 + L1)/L1$

$y = y_1 L2/L1$

$(x, y)$ traces an ellipse if: 

$(x/a)^2 + (y/b)^2 = 1$

The circle and ellipse equations then require $(1/a)(L1 + L2)/L1 = 1/L1$ and $(1/b)(L2/L1) = 1/L1$, resulting in 

$L1 = a-b$

$L2 = b$

Trammel of Archimedes Details

For T slots, use ¼” mill for channels, then Woodruff keyway cutter for T slots. Attach a wood plate to the base, then put the Trammel bar in the mill vise with the wood plate up. Adhere the metal plate to be cut to the wood plate. The brass sliders are made from ¾” brass rod and machined to fit snugly in the channel and T slots. The slider should be slightly proud of the base to raise the bar just above the trammel surface. The bar attaches to each slider with a bolt through a threaded bushing.

Video: Trammel with bar + pencil to trace an ellipse

Complete assembly video, showing that a cone traced by the rotating rods just fits the tilted ellipse.

Acknowledgements

Hobby Machinist Keith (HMK) provided crucial design details and many of the methods required to create this demonstration.