Custom-Normed Trigonometries

My means whereby I've defined the L₁ and L_∞ trigonometric functions, among some more cursèd ones.

It's rather early in higher math's education, usually during high school or even middle school years, that we are introduced to the trigonometric functions done unto an angle.

One may wonder, thereby, as to what would happen if we had different trigonometric functions. Well, that's a massive rabbit hole. But first, let's review the conventional six circular and six hyperbolic ones, before we go into alternate L_p norms and Jacobi elliptic functions and their generalisations to said other norms, and even norms outside of the L_p set.

Circular & Hyperbolic Trigonometry

For instance, take a clean angle such as 60°, or \frac{\pi}{3} radians. Going this far lets us draw half of a chord line intersecting the x-axis at the origin whence it begins, ending at the unit circle's very boundary. This line's radius is always 1, for it is literally just a line representing the radius of the unit circle, which is always 1.

But the interesting thing happens when you take the endpoint of that half-chord, which we'll call P, and draw a line segment down to the x-axis. The point on the x-axis where our segment intersects is the cosine of the angle, denoted \cos{\frac{\pi}{3}}, which is equal to \frac{1}{2}. You can similarly draw a line from P to the y-axis, and the y-coordinate of that is the sine of the angle, denoted \sin{\frac{\pi}{3}}, which in this case is equal to a perhaps-uglier \frac{\sqrt{3}}{2}.

If you're wondering why I didn't just say to take P's X and Y coordinates (\frac{1}{2},\frac{\sqrt{3}}{2}) to get the cosine and sine respectively, it's due to the Pythagorean Identity, wherein \sin^2{\theta}+\cos^2{\theta}=1 for any value \theta. Why is this the case? When we drew the lines from the intersection point, we really made two triangles. There's two right triangles, and one has \theta itself whereas the other has the angle complementary to \theta. Two angles are shared by the triangles since the non-right angles of a right triangle add up to \frac{pi}{2} radians, which is also a property had by complementary angles like the ones there. And since the triangles share a hypotenuse, via the Reflexive Property of Congruence and the angle-side-angle (ASA) theorem, we see that these triangles are congruent. We can use the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem or the hypotenuse-leg (HL) congruence theorem to now show that one of the legs of each triangle is sine and the other is cosine. Et voilá, we can conclude via the Pythagorean Theorem that \sin^2{\theta}+\cos^2{\theta}=1.

Remember for later that the Pythagorean theorem is what defines distance! In arbitrary dimensions, a point's magnitude is (\sum_{k=1}^{\text{no. of dimensions}}x_k)^{\frac{1}{2}}, which in 2D becomes a simple (|x|^2 + |y|^2)^{1/2}. If we set that latter equation equal to 1, we get every single point whose distance from the origin is 1 per L₂ norm.

Hyperbolic trigonometry is as easy as taking \sin i\phi for \operatorname{sinh}\phi and \cos i\phi for \operatorname{cosh}\phi.

That's enough rambling on basics now. Let's move on to breakin' (I mean, generalising) this system.

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Taxicab Trigonometry

Let's take a little look at distance itself, shall we? In L₂, I just gave the definition above. But why oughtn't we use exponents besides 2? Let me introduce or reintroduce you to the L_p norms, wherein we change just that.

Note that we aren't yet using Jacobi elliptic functions. It's a misfounded first instinct to think that they contain taxicab trigonometry somewhere within as they already do for the curves x^2 + y^2 = 1 for m=0, x^2 - y^2 = 1 for m=1, and x^4 + x^2 y^2 + y^4 = 1 for m=\frac{1}{2}.

Theorem 2.1 The L_p norm (generalised distance) between the origin and a point with x₁, x₂, ..., x_d as coordinates wherein d\in(\mathbb{N}\cap\lnot\{0\}), p\subset\mathbb{R}\ngeq 0, and \forall{\{x_k\}}\in\mathbb{R} is (\sum_{k=1}^{\text{d}}|x_k|)^{\frac{1}{p}}. I shall instead go the piecewise linear route and then map it to the standard function for a periodic triangular wave.

Proposition 2.1.1 Thereby, the L_p norm in 2 dimensions with coordinates (x,y) is (|x|^p + |y|^p)^{\frac{1}{p}}

Proposition 2.1.2 Thence we conclude that the L₁ norm in 2D with coordinates (x,y) is |x|+|y|.

That Proposition 2.1.2 right there is basically what we need to formulate taxicab trigonometry. Before I go further I'd like to point out that I'm definitely not the first, for Thompson and Dray have long formulated their interpretation hereof. However, I believe my interpretation is perhaps completer (due to my hyperbolic taxicab trigonometric functions soon hereafter).

Likewise to the unit circle x^2 + y^2 = 1, let's look at the taxicab unit diamond |x| + |y| = 1. It's important to call this a diamond, since in higher dimensional spaces, the arbitrarily-dimensioned implicit curve \sum_{k=1}^{no. of dimensions}|x_k|=1 doesn't magically become a rotated measure polytope (for which the 2D version is the square and the 3D version the cube); it becomes that dimension's measure polytope's dual polyhedron. Hence, the surface of |x| + |y| + |z| = 1 is not a rotated cube but actually an octahedron, the dual of the cube.

Let's begin with the basic thing. Taxicab angle. The ratio of a circle's radius to its circumference is 2π, and that's also how many radians make up a full revolution. Likewise, there are 4 times the radius units around the "circumference" of the L₁ diamond, so 4 "tabdians" around the diamond. If we wanted to, we could even define a degree equivalent such that 360^◊ go all the way around, so therein, 90^◊ equals 1 tabdian.

With these defined, we can now make a table of the trivial values, wherein x is in tegrees since I was too lazy to write both radian and tabdian inputs, and cost(x) and sin(t) are the taxicab sine and cosine, whence other functions like tangent, secant, cosecond, versine, and coversine are deriveable. One may even

Furthermore, since a unit diagonal in L₂ that's √2 distance becomes L₁ distance of exactly 2, we can take the midpoints of the diagonals of the diamond to get more values also shown below:

x	cost(x)	sint(x)
0◊	1	0
45◊	1	1
90◊	0	1
135◊	-1	1
180◊	-1	0
225◊	-1	-1
270◊	0	-1
305◊	1	-1
360◊	1	0

Were we to go to the midpoints of those midpoints, we could literally go on endlessly and land with a piecewise linear function for taxicab sine:

\operatorname{cost}(x)=\begin{cases}\operatorname{mod}(x,2)-1\;\;\text{if } \operatorname{mod}(x,4)\ge 2 \\1-\operatorname{mod}(x,2)\;\;\text{if } \operatorname{mod}(x,4)\ngeq 2\end{cases}

\operatorname{sint}(x)=\operatorname{cost}(1-x)

Heuristic 2.2 L_p trigonometric functions cos_ph(x) and sin_ph(x) MUST satisfy that |\cos_p\text{h}(x)|^p + |\sin_p\text{h}(x)|^p = 1

The definition of sint(x) as above is to satisfy the identity that |\operatorname{cost}(x)|+|\operatorname{sint}(x)|=1, as shown in Heuristic 2.2. And there we have it: piecewise linear definitions.

But I ain't gon' stop there, of course. Per Wikipedia, there exists a clean triangular wave function in terms of L₂ circular trigonometric functions:

y(x) = \frac{2a}{\pi} \arcsin\left(\sin\left(\frac{2\pi}{p}x\right)\right)

Let's modify this so that the period a thereof is equal to 4 like in our definitions. Et voilá, we get an equivalent alternate definition of the taxicab diamond trigonometric functions:

\operatorname{sint}(x)=\frac{2}{\pi}\arcsin(\sin(\frac{2\pi x}{4}))

\operatorname{cost}(x)=\text{sint}(1-x)

Thence, due to the sine in the definition we can also derive "hypobolic" functions; that is, they trace |x|-|y|=1 like an L₂ hyperbola is (partially) traced by x^2 - y^2 = 1. Hyperbolics' Pythagorean identity is \cosh^2 (x) - \sinh^2 (x) = 1, hence the hypobolics' is \operatorname{costh}^2 (x) - \operatorname{sinth}^2 (x) = 1.

Theorem 2.3 (heuristic) A hypobola is defined as the L₁ equivalent of the hyperbola.

Proposition 2.3.1 For a horizontal unit hypobola, the implicit function making a horizontal one is |x|-|y|=1.

Proposition 2.3.2 A unit hypobola CANNOT be parametrised in terms of X(t)=sect(t) (\operatorname{sect}(t)=\frac{1}{\operatorname{cost}(t)}) and Y(t)=tant (\operatorname{tant}(t)=\frac{\operatorname{sint}(t)}{\operatorname{cost}(x)}).

Heuristic 2.3.3 A unit hypobola MUST be parametriseable in terms of X(t)=sinth(t) and Y(t)=costh(t), wherein sinth(x) and costh(x) are the hypobolic L₁ trigonometric functions.

Heuristic 2.4 A hyperbolic version cos_ph(x), sin_ph(x) of any L_p trigonometric function MUST satisfy that |\cos_p\text{h}(x)|^p - |\sin_p\text{h}(x)|^p = 1

\operatorname{sinth}(x)=\frac{2}{\pi}\arcsinh(\sinh(\frac{2\pi x}{4}))

\operatorname{cost}(x)=\operatorname{sint}(1-x)

The parametric (a\operatorname{cost}(t),b\operatorname{sint}(t)) beautifully draws out the diamond "ellipse" with horizontal radius a and vertical radius b, and we just get the regular diamond when a=b=1. Similarly we have (a\operatorname{costh}(t),b\operatorname{sinth}(t)) tracing out the right half of a hypobola. All Pythagorean-esque identities have also been satisfied in both of these models. It's crazy.

But we oughtn't stop even here, oh we shall not. Let's move into some odder coordinate systems. Diamond polar coordinates work as you'd expect, just replacing the circle of circular polar with a diamond. Similar goes with hypobolic polar coordinates just replacing a hyperbola with a hypobola. These coordinate systems are found by just replacing the circular or hyperbolic trigonometric functions in the definitions thereof with diamond or hypobolic trigonometric functions.

Toroidal coordinates (the 3D extension of bipolar) become, well, interestingly shaped, with the conversion from 3D toroidal to 3D Cartesian coordinates being parametrised from L₂ (\frac{a\sinh\eta \cos\phi}{\cosh\eta - \cos\theta}, \frac{a\sinh\eta \sin\phi}{\cosh\eta - \cos\theta}, \frac{a\sin\theta}{\cosh\eta - \cos\theta}) into L₁ (\frac{a\operatorname{sinth}\eta_1 \operatorname{cost}\phi_1}{\operatorname{costh}\eta_1 - \operatorname{cost}\theta_1}, \frac{a\operatorname{sinth}\eta_1 \operatorname{sint}\phi_1}{\operatorname{costh}\eta_1 - \operatorname{cost}\theta_1}, \frac{a\sin\theta_1}{\cosh\eta_1 - \cos\theta_1}) wherein the Greek eta, theta, and phi in this latter one with a subscript of "1" represent diamond rather than circular angle.

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Uniform Trigonometry

Rather than giving the dual of the measure polytope, the L_∞ norm, when parametrised in the manner I'll later state, gives the actual measure polytope. This is by far my favourite norm, and it's also known as the Chebyshev norm. Therein, a diagonal distance we'd usually in L₂ call "√2 units" or in L₁ call "2 units" becomes known as equal to a horizontal or vertical shift, thereby becoming known in L_∞ as "1 unit". Y'all may sense one problem herewith, though: how in the actual heck can I calculate this norm without taking to the infinitieth power? But that's the thing: we don't. There's a reason why the uniform norm is also called the maximum norm, and it's since its unit circle is \operatorname{max}(|x|,|y|)=1. Yup, a square, unrotated, in its true beauty. In 3D, it becomes \operatorname{max}(|x|,|y|,|z|)=1, a cube. So these aren't duals of measure polytopes: these are the measure polytopes outright.

We can also confirm these to be the piecewise definitions of the uniform sine and cosine, which are trapezoidal waves with period 8:

\operatorname{cosu}(x)=\begin{cases}1-\operatorname{mod}(x-1,8)\;\;\text{if } \operatorname{mod}(x-1,8)\le 2 \\-1\;\;\text{if } 2\le\operatorname{mod}(x-1,8)\\\operatorname{mod}(x,8)-6\;\;\text{if } 4\le\operatorname{mod}(x-1,8)\le6\\1\;\;\text{if } 6\le\operatorname{mod}(x-1,8)\le8\end{cases}

\operatorname{sinu}(x)=\operatorname{cosu}(2-x)

Here, each 45°L₂ is 1 unidian (what I'll call the radian equivalent in L_∞). 8 unidians go around the entire square, each one corresponding to \frac{\pi}{4} radians. 1 "ungree" can be the uniform equivalent of the degree, wherein 360^□ go around the full square, and 45^□ = 1 unidian.