polar coordinates

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Music this time is Perpetuate 303 from Tetrisphere XD

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Ron the Cartesian Bear explains the difference between the taxicab metric and the Euclidean metric. Creepy as hell.

“バラ曲線(バラきょくせん、英: Rose Curve)は極座標の方程式

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または
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によって表される曲線である。バラに似た形のため、このように名付けられた。" 

In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates.

- Wikipediaより

Polar Coordinates and Polar Curves

The polar coordinate system is a useful alternative to the rectangular (Cartesian) coordinate system for describing the location of points on a plane.In the polar coordinate system, there is an origin (or pole), O, and a polar axis, which is a ray extending from O horizontally to the right. The position of any point P in the plane is then determined by its polar coordinates (r,θ), where r is the distance from P to the origin and θ is the angle between the ray OP and the polar axis.

The rectangular axes x and y are often shown in the polar graph, where the polar axis coincides with the positive x-axis.

Unlike rectangular coordinates, the polar coordinates of a point are not unique. Increasing or decreasing the angle θ by an integer multiple of 2π does not change the distance r from the origin, so (r,θ ± 2π) denotes the same point as (r,θ), such as the following:

(1,π/4) = (1,9π/4) = (1,-7π/4)
(2,π) = (2,-π)
(2,5π/4) = (2,-3π/4)
(3,0) = (3,2π)

In addition, the origin O has the polar coordinates (0,θ) for any value of θ. There are also points (r,θ) where r < 0. The appropriate interpretation for this “negative distance” r is that it represents the position distance -r measured in the opposite direction, or the direction of θ + π. Therefore:

(-r,θ) = (r,θ + π)

For example, (-1,π/4) = (1,5π/4).

If both rectangular and polar coordinate systems are in the same plane, the positive x-axis is the polar axis, then the relationship between the rectangular coordinates of a point and its polar coordinates is shown in the following graph:

A single equation in x and y generally represents some curve in the plane with respect to the rectangular coordinate system. Similarly, a single equation in r and θ generally represents a curve with respect to the polar coordinate system.

The following relationships can be used to transform one representation of a curve to another:

Some Polar Curves
When the equation is of the form r = f(θ), this graph is called the polar graph of the function f. Some polar graphs can be recognized easily if the polar equation is transformed to rectangular form. However, for others, such a transformation does not help, because the rectangular equation may be too complicated to be recognizable. In these cases, one must resort to constructing a table of values and plotting points.

r = cosθ, r = sinθ

r = 2cosθ, r = 2sinθ

r = cos2θ, r = sin2θ (0 < θ < π)

r = cos2θ, r = sin2θ (0 < θ < 2π)

r = 1 + cosθ, r = 1 + sinθ

r = 1 + cos2θ, r = 1 + sin2θ

r = 2 + cosθ, r = 2 + sinθ

r = 2 + cos2θ, r = 2 + sin2θ

Conics
Suppose a point moves in the plane that its ratio of its distance from a fixed point F (the focus) to its distance from a fixed straight line L (the directrix) is a constant e (the eccentricity). The path of the point is a conic, where it is an ellipse if e < 1, a parabola if e = 1, or a hyperbola if e > 1.

If the focus F is at the origin of the polar coordinate system and its directrix L is vertical and is p units to the left of the focus, then the arbitrary point P = (r,θ) on the conic satisfies the following:

Ellipses
If e < 1, the above equation represents an ellipse with the major axis running from the following:

In addition, it has one focus at the origin and the second focus at the following point:

Parabolas
If e = 1, the above equation represents a parabola opening to the right with the polar axis at the axis, its focus at the origin, and a vertex at the point (p/2,π).

Hyperbolas
If e > 1, the above equation represents a hyperbola with vertexes at the following points:

There is one focus at the origin and a second at the following point:

The center is at the following point:

The asymptotes pass through the center in the following direction:

θ = Arccos(1/e) or when 1 - ecosθ = 0

The Slope of a Polar Curve
The polar graph r = f(θ) can be regarded as a parametric curve in rectangular coordinates, where the following parametric equations have the parameter θ:

x = rcosθ = f(θ)cosθ
y = rsinθ = f(θ)sinθ

The slope of this curve at the point P with polar coordinates (f(θ), θ) is the following:

If the curve r = f(θ) passes through the origin at θ = θ₀, then f(θ₀) = 0. If f’(θ₀) = 0, then the following is true:

Therefore, the tangent line at the origin is θ = θ₀. If f’(θ₀) = f(θ₀) = 0, then the curve may or may not be smooth at the origin.

There is a simple formula that can be used to determine the direction of the tangent line to a polar curve r = f(θ) at a point P = (r,θ) other than the origin.

Let Q be a point on the curve nearby P, where Q corresponding to the polar angle θ + h. A line SP is drawn that connects from the line OQ to P. Then PS = f(θ)sinh and SQ = OQ - OS = f(θ + h) - f(θ)cosh. If the tangent line to r = f(θ) at P makes an angle ψ with the radical line OP, then ψ is the limit of the angle SQP as h approaches 0.

Therefore:

Therefore, the following equation can be used to determine the angle between the tangent line at P and the radical line OP for any point P on the curve, except the origin:

Mandalic geometry and polar coordinates - I

THE SPHERICAL MANDALA  

                             

The I Ching mandala demonstrated in this blog has been derived using Cartesian coordinates. However the system works with the approach of polar coordinates in three dimensions as well. There are, of course, methods for translating from one to the other. In some ways the polar coordinate approach is more revealing. For one, it looks more like the kind of mandala most of us are familiar with (although in three dimensions.) For another, the various symmetry relationships are much more obvious.

A short derivation/description of the polar coordinate approach follows:

For reference see 1, 2, 3, 4, 5, 6, 7, 8, 9. These are all 2-dimensional representations necessitating mental compositing. Visualization becomes much easier with a 3-cube labeled appropriately with the 64 hexagrams. Rotate the cube so the hexagram with all solid lines is at the south pole* and the hexagram with all broken lines is at the north pole. These become the pole stars, so to speak, of the system.

Now imagine that the 64 hexagrams shown describe a single sphere consisting of four distinct shells (or alternatively describe four related concentric spheres.) We will refer to these as Shell 1, 2, 3 and 4 heading from outermost to innermost. The outermost shell is composed of the 8 hexagrams with doubled trigrams (1 at each of 8 positions) which were the 8 vertices of the cube seen in the Cartesian description. The next shell heading toward the center consists of the 24 hexagrams (2 at each of 12 positions) which were the cube edge centers in the Cartesian description. The third shell heading inward consists of the 24 hexagrams (4 at each of 6 positions) which were the cube face centers in the Cartesian description. The fourth shell is the single center point of the sphere viewed as a 3-dimensional structure. It consists of the 8 hexagrams which were at the central point or origin in the Cartesian description.

 

*Recall that in ancient Chinese cartography South was drawn where we show North and vice versa. This is purely convention, but an important one to the system described and we shall follow it here.

                                                                         (Continued here.)

© 2013 Martin Hauser

The Shapes Of Scenes

Evidently, Kurt Vonnegut was once gunning for a master’s degree in anthropology. Who knew? Luckily for literature, he didn’t get it, but graphic designer Maya Eliam has taken the key ideas of Vonnegut’s rejected master’s thesis and turned them into an infographic. The shapes of a culture’s stories, Vonnegut thought, are at least as interesting as the shapes of their pots or spearheads:

It’s an interesting way of thinking about narrative, and a simple classification system for stories in general. Although in improvisation we tend to avoid narrative like the plague, what shape our scenes tend to take is still an interesting question to ask.

In Vonnegut’s analysis (or, at the very least, Eliam’s visualisation of Vonnegut’s analysis), the x-axis is defined by time, and the y-axis is a spectrum ranging from positive to negative emotion. For sake of determining the shape of an improv scene (and, for the record, things rarely get as speculative and wondering aloud for me as they are right now), I would think that leaving the x-axis as a measure of time would still be fitting, as was done in my poorly illustrated attempt to explain editing.

Leaving the y-axis as some measure of emotion is appropriate, I think, but in order to truly get a sense of what a scene does we would need to expand upon it. Plutchik’s diagram showing the relation of various emotions to one another comes to mind here:

What I’m picturing- and again, I can’t stress enough how much this is just caffeine fueled speculation here- is an x-axis for time coming straight out of the middle of this emotional compass rose. Doing so would form some kind of cylindrical coordinate system, which Wikipedia informs me means that we need to redefine some of these parameters a bit. So, let’s look at what we’ve got and see if we can make some sense of this needlessly rigorous and mathematical approach to a style of comedy which is neither rigorous nor mathematical.

Fundamentally, we’re still using Plutchik’s flattened emotional cone and time, we’re just orienting them differently. Let’s start by laying Plutchik’s diagram flat on a table, something I can’t picture here because I don’t know how to do any kind of 3D rendering.

Defining a location on this circular space now requires two variables: theta (θ), which determines the angle at which a line strikes out from the origin (the center of the wheel), and r, the length of the radius of said line striking out at angle θ. For reference sake, here’s a bit of polar graph paper to help you visualize what any given emotion’s value for θ would be:

For sake of the present discussion, we’ll limit r to having a value of 1 (the innermost ring of the circle/most intense emotions), 2 (the middle ring, where emotions aren’t particularly intense) or 3 (the outer ring of the circle/least intense emotions). Values of r greater than 3 indicate a kind of apathy, and negative values of r simply indicate a shift in emotion which would be better expressed using a new value for θ.

Our third variable is t, a measure of the length of the scene, extending along what the math nerds call a longitudinal axis perpendicular to the plane of emotions we started from. This is what turns our graph from being a set of comparatively simple polar coordinates on a flat plane into a set of three dimensional coordinates identifying a point in a cylindrical space. Any cross-section of this space which remains perpendicular to the longitudinal axis gives us the same array of emotions we’d experience at t=0

So now, looking at our absolutely unnecessary cylindrical graph of an improv scene, we can define the active emotion present at any point t in the scene by using and θ. If I’m doing this right (and that’s a BIG if), we could define any moment in the scene with three numbers in the format of (θr, t), where θ lets us know what kind of emotion is being expressed, r lets us know its intensity, and t lets us know how long the scene has been going on for.

In this system, then, what shapes are produced? Perhaps a scene where somebody starts out feeling apprehensive (0, 3, 0) and winds up feeling terrified after, say, a minute (0, 1, 60s) and then quickly moves on to disgust (210, 2, 90s) would look like a gentle counter-clockwise curve. A monoscene, where characters tend to be more fleshed out and textured, and where the scene last for long enough that several emotional cycles may go through would probably look more like an elongated spirograph. Perhaps it would be more accurate to plot curves for each individual character in a scene. Who knows? This is all speculative.

I should hope it goes without saying that no coach is ever going to sit down after a show, draw a couple of scenes out on graph paper, and point to some segment of a curve and say “that’s where we lost the game.” It isn’t necessary, and, much though it pains me to admit it, probably not possible.

However, in the same way that Vonnegut believed that the shapes of a culture’s stories was at least as important of a tool for understanding that culture as the shapes of their spearheads, perhaps it is also true that by undertaking a large scale attempt to identify and codify some common shapes of scenes we might be able to learn something about a show, a player, a team, or an entire community that we might not find out otherwise.

Perhaps there are shapes conspicuously missing from that survey, who’s construction requires a long form structure or a style of play not yet known.

Science historian and kooky British guy James Burke once said in an interview with non-historian but fan of history Dan Carlin that the most important thing you can do as a thinker is to make yourself aware of the box you’re in, so that you can think outside of it. A comprehensive survey of the shapes of scenes might very well tell us what our current box looks like.

What I’ll leave you with, then, is this: What assumptions has the current paradigm of long-form improv hard wired into us so completely that we don’t even realize we’re making them? And, having gone beyond the current paradigm, what does the new improv look like?

Don’t bother writing me and saying you’ve figured it out, because I don’t think there’s a concrete answer. Put up a show instead. Follow the lead of LA indie giants Room 101 and do monoscenes in cars for an audience of three, or of Shapeshift, who convinced a tattoo parlor to let an improviser do a set for a full audience while getting a tattoo. Don’t just try to retrace the footsteps of those that have come before you, because that path is already worn. Strike out into new territory and see if you find something that’s been hiding under our noses all along.