Geometry Review
A function is a mathematical algorithm that satisfies a very convenient requirement. For each input-value, there is precisely one output-value. You may remember this as the rule that a function must pass the 'vertical-line test'. The 'vertical-line test' means that, as I was taught, if you run a vertical line (ie. roll a pencil) over the graph, with the pencil pointing toward yourself, the pencil will never touch more than one point on the line at once. The graph below mostly passes, but I can't draw apparently, so it does fail. Woops.
This ensures that the graphs show what they're supposed to show; an object moving through physical space. An object cannot be in two places at the same time, nor can a car have two velocities at once. To describe motion such as this, you may use notation such as y = mx + b. This is a very useful notation because we have one input: x. Every unique x has its own, single value for y. This is great because computers like to follow a list of very specific instructions to get their output. If we want to tell our computer to draw a shape, we need a very simple list of instructions for it to follow. However, we do not want to draw lines; We are trying to predict the motions of the planets.
We will try to construct the necessary foundation for reducing the motions of the planets to a set of coordinates generated in time-steps.
We need to reduce our orbit to meaningful variables that operate independently of each other. Since we are simulating planetesimal orbits, we already know we are stepping through time. This is an iterative process; each point helps determine the next point. It may be useful to stop and ponder the division of number sets I'm implying here. There are some sets wherein the time dimension is useful and unique -- when a planet moves around a central object according to the laws of intertia and gravitational attraction, its current mass and acceleration determine what its future behavior will be. Alternatively, we have time-independent data-sets, wherein the order of events does not matter and data-points do not affect each other, such as with electrons in the double-slit experiment. We may observe that gravity is the missing element here; the double-slit experiment should adhere to a time-dimension if electrons were considered to be affected by gravity. We currently say the electrons are either not affected by gravity or to such a small degree as not to be measurable by current instrumentation. But let us turn our attention back to planetary orbits, which are affected by order-of-operations, and thus will be referred to me as iterative.
If I wanted to help my kiddo draw a circle on a piece of paper, I would get a pin, a pencil, and a length of yarn. Affix one side of the yarn to the pin and place in the center of the paper. The other side of the yarn should be tied about the pencil. Use the now constant length of yarn to draw around the pin while keeping the string taut. The line drawn by the pencil on the paper denotes the 'circle'. One affectionate to the classical language of formal mathematics might note that the circle is the set of all points equidistant from a singular (central) point. We may even define this distance as r, the radius of the circle. However, a circle does not adhere to our definition of a function because for every line we draw across a circle, it intersects exactly twice. A given input x needs to return at most two output y.
To try to create this 'fork', I introduce the sine and cosine functions. A circle, as we define it, is the locus of all points that satisfy:
x = r cos (t)
y = r sin (t)
where r is a constant#
where t is an iterated set, e.g. time
Recall that sine and cosine are two basic trigonometric functions; expressions of a waveform. It may be prudent at this point, if you are not in the habit of using trig in your every-day life, to find a suitable summary of basic trigonometry from an outside source. There is something both intuitive and arcane about trig, so any non-professional mathematician may be served by a moment of review. Go ahead, I'll wait for you~
a kilometer, but are symbols to represent a ratio. Computers tend to prefer whole numbers and basic iterations may start with zero or one and count upwards through the 'natural numbers'. Degrees divide the circle into sixty parts twice, and may hearken to an ancient affection for the number twelve, which is seen in a classical clock design. Radians, too, are a sort-of-a-clock, where a rotation of pi is equivalent to 180°, or the difference between 12 o'clock and 6 o'clock. An important distinction, however, is that radians conventionally place their 0-value (origin) at the right-horizontal axis. That is to say, radians are a clock that rotates starting at 3 o'clock and moves counter-clockwise through 12, then 9, then 6 o'clock, before finally terminating to start cycling again.
I am sure most of you are familiar with the Pythagorean formula, wherein Pythagoras showed us a special formula of graph theory -- In a rectangle, the lengths of two adjacent sides form a constant ratio with the distance between the non-adjacent vertices.
A*A + B*B = C*C
Pythagorean theorem
where ABC forms a triangle and C is its longest side
I will leave it as an exercise to the reader to confirm that this can be rearranged to create another formula referred to as the 'distance formula' and is commonly sold as a method of finding the distance between two points in the x,y plane.
that is constant for all values of t, we must allow for variation in the distance between a given point on the surface and the center of the ellipse.
At this point we must review the properties of ellipses as well as consider their relation to the first of Kepler's laws of planetary motion (1609-1619). Johannes Kepler (1571-1630) was an astronomer, mathematician, and astrologer. He was born and died in German lands, then the Holy Roman Empire. His laws of planetary motion expanded Copernicus' circle-based view to one considering ellipses. His first law of motion states that the orbit of a planet is an ellipse with the Sun at one of two foci.