Cartesian geometry

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Example of a Cartesian coordinate plane from 1905
Example of a Cartesian coordinate plane from 1905

Cartesian geometry, also known as analytic geometry, unified the algebra and geometry by interpreting the variables of algebraic equations in terms of a coordinate system to create geometric objects. The methods were initially developed by rivals Rene Descarte and Pierre Fermat in the early 1600’s. The introduction of algebra and coordinates to geometry made the language of geometry much more powerful. Cartesian geometry immediately made it possible to describe many more curves without having to rely on special names and mechanical constructions, and lead within a generation to the development of differential calculus.

The Nautilus shell

A living chambered nautilus A chambered nautilus shell cross-section which we will measure

The chambered nautilus is one of the most charismatic “living fossils” on the planet. At first glance, they appear to be some unusual combination of snail, fish, and jellyfish. But the chambered nautili are actually cephelopods, like squid, cuttle fish, and octopi. Nautili have been around since the Cambrian explosion, 500 million years ago and look much the same today as they did then. The famous documentary series “The Undersea World of Jacque Cousteau” memorably filmed this undersea creature, and in 2015 the “fuzzy nautilus” Allonautilus scrobiculatus was finally also filmed.

One of the many mysterious things about a chambered nautilus is the shape of its shell. A nautilus shell coils around itself, and when cut in half, reveals a spiral with the regular partitions that gave rise to the moniker “chambered”. (Unfortunately, the nautilus is again threatened with extinction, in part because of our human attraction to this shell). The spiral itself is so attractive that one almost can not help but ask if it might be revealing some deeper law of nature. One way of getting at this question is asking what equation, if any, describes this spiral shape. And fortunately, the math of polar coordinates is just the thing for drawing spirals.

A spiral is a curve that winds a path around a center in a regular manner. One of the oldest definitions of a spiral comes from Sicily.

“If a straight line one extremity of which remains fixed be made to revolve at a uniform rate in a plane until it returns to the position from which it started and if, at the same time as the straight line is revolving, a point move at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.” – Archimedes, On spirals, circa 250 BC

This is what we now call an “Archimedean” or linear spiral. If we let the symbols \(t\) represent time, \(\theta\) be the angle of the line, and \(\rho\) be the distance of a point out from the center, then Archimedes’ definition is equivalent to the parametric polar system \[\theta(t) = a t, \quad \rho(t) = b t \] where \(a\) and \(b\) are the velocities of change of the angle and the radius respectively. By solving the first equation for \(t\) and then substituting into the second, we find that this means the Archimedean spiral has its radius proportional to its angle, specifically \[\rho = \frac{b}{a} \theta .\] Archimedes wrote a treatsie on this spiral and its uses.

A little more thought about polar equations reveals that spirals come in many forms. A spiral might grow quadratically (\(\rho = b \theta^2\)), exponentially (\(\rho = e^{b \theta}\), sometimes called “logarithmic spiral” when written \(\ln \rho = b \theta\).), or according to some other crazy function (e.g. \(\rho = 2 \theta + \sin(6 \theta)\)).

4 examples of spirals with different equations

We can guess at which of these matches the Nautilus shell the best, but we will postpone further consideration until after we’ve developed the method of least squares.

Kentucky Do-nothing’s

Another example of Cartesian modelling of a mechanical system is one you might find in a kid’s toy chest – a Kentucky Do-Nothing, sometimes also called Archimede’s Trammel, shown below. The handle end is free to be turned, but only to positions that the slider’s movement allows. But what curve, precisely is that?

Photo of a wooden Archimedes trammel Labelled diagram of trammel for calculation

Experimentation with the trammel reveals the handle’s motion traces an oval shape – roughly circular, but taller than it is wide. A natural guess, for those familiar with geometry, is that the motion traces an ellipse. But can we be certain of that?

To answer the question, we can use some geometric modelling and algebra. Let’s set the center of the trammel (where the slider grooves cross) to be the origin of our system, and let \((x,y)\) be the position of the handle. In Cartesian geometry, an ellipse centered at the origin and with principle axes aligned with the coordinate axes has an equation of the form \[ \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = c^2,\] where \(a\), \(b\), and \(c\) are constants determining the ellipse’s shape. Every point \((x,y)\) on the ellipse will be a solution of this equation. If we can show that there are some constants \(a\), \(b\), and \(c\), such that \(x\) and \(y\) always satisfy this equation, we will know the trammel traces an ellipse.

On close inspection, we see the trammel consists of two sliders trapped in perpendicular grooves. Each slider can move freely, but every time we move one slider, the other slider gets moved in response. The handle’s screw-attachments control this – the screws are always the same distance apart. So when one slider gets moved out, the other sider gets moved in to compensate for the distance change.

There are two constants in the trammel construction that may change from one toy to another, but are never changed once the trammel is completed – the distance between the handle and the first screw, and the distance between the first and second screws. Let \(r\) be the distance between the handle and the first screw and let \(s\) be the distance between the two screws.

The things that can change in the trammel are the positions of the slider screws and the position of the handle. Let \(p\) be the horizontal slider screw’s position and \(q\) be the vertical slider screw’s position. Now, let’s look for a curve that relates the handle position to the trammel’s built in constants, without regard to the slider positions – something like a function that can be written \(f(x,y; r,s) = 0\) without using \(p\) and \(q\).

Observe that we have two similar triangles in action in the trammel, and by the proportionality of similar triangles, we should have \[\frac{q}{y} = \frac{p}{p-x} = \frac{s}{s+r}\] These proportions can be solved for \(p\) and \(q\). \[q = \frac{ys}{s+r}, \quad p = -\frac{sx}{r}\] By the Pythagorean theorem, we should have \(s^2 = p^2 + q^2\) always, so substituting for \(p\) and \(q\), we find \[s^2 = \left( \frac{sx}{r} \right)^2 + \left(\frac{sy}{s+r}\right)^2\] \[r^2 = x^2 + \left(\frac{1}{\frac{s}{r}+1}\right)^2 y^2\] Thus, the path traced by the trammel’s handle will be an ellipse. We can go a step further, and observe that as the ratio \(s/r\) gets smaller, the motion will become less elliptic and more circular.

The cycloid

image/svg+xml x 1 2 y P

At night in the summer, we sometimes see bicycles ride by with lights upon their wheels. The lights revolve in a circle around the wheel axles, but the path the lights trace appear as cheerful looping bounds down the road as the bicycles move on. These traces are called cycloids.

A cycloid is the curve traced by a bangle on the rim of a wheel. It may have been studied in ancient times, but this work, if it existed, is largely lost. It’s modern study began around 1500, and it became closely connected to many of the interesting mathematical arguments of the next two centuries, including the quadrature, the pendulum clock, optics, and the brachistochrone – it was nicknamed the “Helen of mathematics” in its time because of the wars it ignited.

To find an equation for the cycloid, we can make use of parametric coordinates. Let’s suppose we start with a wheel of radius 1 rolling on a level and flat surface, and we are tracking a bangle a distance \(r\) from the wheel axle. As a wheel rolls, its current state can be represented with a variable \(p\) tracking the position of the axle of the wheel, and a variable \(\theta\) tracking the angle of rotation of the wheel. As long as there is no slippage of the wheel, these two must be related – each time the wheels angle makes a full revolution of \(2 \pi\), the wheel’s position must move forward by a factor of the circumference of the wheel. Since we assumed our wheel has radius 1, one rotation of the wheel will move the axle forward \(2 \pi\). If, initially, \(p=0\) and \(\theta=0\), then \(p=\theta\) as the wheel rolls, whatever the specifics of the motion.

Now, let \((x,y)\) represent the position of a bangle on our wheel, a distance \(r\) from the axle. Since the axle’s position can be expressed in terms of the angle as \((p,1) = (\theta, 1)\), we can express the position of the bangle with trigonometry as \[x(\theta) = \theta - r \sin \theta,\quad y(\theta) = 1 - r \cos \theta,\] assuming the initial position is directly below the axle.

Exercises

  1. Show that the intersection between a cylinder and a plane is an ellipse

  2. Find the equation for a torus using a 3-dimensional Cartesian coordinate system.

  3. In a flat area of sand on a beach, a piling with circular cross-section of radius \(1\) foot is driven vertically into the ground. A string has been wrapped tightly around this piling at ground level, with one end free. Grab the free end and walk around the piling to unwind the rope, while keeping the string taut.
    1. Find an equation for the path the free end of the rope traces in the sand as you unwind it. (Hint: This curve is not an Archimedean spiral.)
    2. Show that the curve is well-approximated by an Archimedean spiral asymptotically as the length of unwound rope gets larger using a plot.
  4. The Conchoid of Nicomedes was a curve traced by a linkage mechanism where one pin is fixed to the vertical axis, while the second pin can move freely along the x axis. This conchoid was designed to help divide an angle into thirds, solving one of the classical Greek puzzles of geometry.
    1. What are the two physical length parameters that are determined by the linkage construction and control the curves shape. Call these parameters \(a\) and \(b\).
    2. Using \(x\) to represent the horizontal position of the movable pin, and \((p,q)\) representing the point of the linkage, find a pair of parametric curves for the Conchoid in the forms \((p(x,a,b), q(x,a,b))\).
  5. In class we found that Archimedes’ trammel traced out an ellipse, based on the separation between the pins and the handle. But it is not the only way to construct a perfect ellipse. Imagine two thumb tacks placed on cardboard a distance \(d\) apart. Take a string of length \(L\), tie it in a circle, loop it over the thumb tacks, and draw the curve of the farthest the string can reach. Show using Cartesian geometry that this curve is also an ellipse, as long as as \(L > 2 d\)

  6. A pair of sophomore students are moving a large mural painting on plaster board into a new apartment, but to get to the room, you have to turn the rectangular mural around a tight corner between two hallways. The mural is about as tall as the hallway, so tilting it only makes the fit harder.
    1. What 3 variables control whether or not you can get the couch
    2. Find an equation for the possible width of the first and second hallways for which you will be able to get the couch to your apartment.
  7. Plot the cycloid path in the following cases.
    1. Plot the path when R=1 and r = 0.6 for 4 revolutions.
    2. Imagine that in our cycloid problem, we could extend the position of the bangle beyond the rim of the wheel.
      Plot the path when R=1 and r = 1.5 for 3 revolutions.

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