# Why is the center of a circle often given as (h, k)?April 17, 2014 10:21 AM   Subscribe

Why are the letters 'h' and 'k' often used to represent the center of a circle in the standard equation of a circle: (x - h)2 + (y - k)2 = r2 are 'h' and 'k' short for something? I might guess 'h' is short for 'horizontal', but I can't imagine what 'k' is for.

So why do we use (h, k) rather than (a, b) or some other pair of letters? My Googleing shows I'm not the first person to wonder this, but no one seems to have received a good answer.
posted by Reverend John to Science & Nature (16 answers total) 2 users marked this as a favorite

I don't think this is universal -- when I was a kid we used (a, b). The only convention I learned is that early letters (a, b, c, ...) are constants, and late letters (..., w, x, y, z) are variables.
posted by phliar at 10:44 AM on April 17, 2014 [1 favorite]

I think the choice of k has to do with the alphabetic sequence h,i,j,k from which I and j are excluded due to their use as index subscripts.
posted by SemiSalt at 10:57 AM on April 17, 2014 [4 favorites]

a, b, c, and d are too often used as corners of a quadrilateral, then you have e, f and g are too physics-y, i is i, and j looks too much like i.

So in the first part of the alphabet, you're left with h and k. This is all complete conjecture, but makes sense to me for the most part, though the f and g rationalizations are a bit of a stretch.
posted by papayaninja at 11:00 AM on April 17, 2014 [1 favorite]

f and g are also used as generic function names.
posted by mbrubeck at 11:13 AM on April 17, 2014

I don't think it's just a convenient alphabetic progression... I you look elsewhere, for instance at formulas for locating the vertex of a parabola, h and k are used there as well to express a horizontal and vertical offset, respectively. I did some Google searching to determine why this is the case, and the pages I could find that even addressed the "why" question literally had answers like "why? I don't know." So you are not the only one to notice a pattern! Perhaps you could get in touch with an undergrad math professor somewhere?
posted by Joey Buttafoucault at 11:16 AM on April 17, 2014 [1 favorite]

Circle equations can written in the form ax^2 + by^2 + cx + dy + e = 0. See "completing the square" for converting this form into the one you gave. You wouldn't be able to keep a,b in doing so. It doesn't explain why h,k were chosen in particular but the vertex (h, k) are also used for any given quadratic y = ax^2 + bx + c.
posted by dukes909 at 11:17 AM on April 17, 2014 [1 favorite]

k is often used for a constant ("konstant").
posted by grouse at 11:22 AM on April 17, 2014

It might also be that someone influential in math, especially conic sections, wrote the equation that way. Then all it takes is one influential textbook author to do the same.

From my limited recall of days of yore (aka college), I think Descartes was the man when it came to conic sections and algebra, so he'd be where I'd start, but it'd probably take a bit of effort to find the first occurrence.
posted by Fortran at 11:55 AM on April 17, 2014 [2 favorites]

xy fg hijk

No mathematical reason. Just the way things be, bro. Just the way things be.
posted by jjmoney at 1:21 PM on April 17, 2014

Fortran's suggestion of tracing it back to the mathematical discovers of the equations in question is probably on point. Descartes' La Géométrie, in which we can see analytic geometry's invention, may hold the answer. I suspect that "h" and "k" emerged through a complex process of algebraic elimination of all the other letters that stood for the various lines (i.e., a, b, c, d, e, f, g, i and j were eliminated). Euclidean proofs of any complexity require a dizzying amount of named lines and points, but analytic geometry can be much more terse.
posted by dis_integration at 2:07 PM on April 17, 2014 [1 favorite]

I agree with dis_integration. I doubt H and K were chosen or were convenient. This is geometry not algebra. Almost the full alphabet is used up in constructing geometric proofs.

Take a look at the diagram of Heaths 1898 translation of Proposition 2 by Archimedes and its not hard to see how H and K could emerge as the right variables. As to which treatise they first appeared in, that remains to be discovered.
posted by vacapinta at 2:25 PM on April 17, 2014

From what I can remember, here are some common uses for the Latin letters in math.

a, b, c - constants
d - used for differentials and derrivatives
e - 2.7181.....
f, g - functions
h - can be a function, but not always, can fill other rolls
i - sqrt{-1}, also used as an index in series and sequences
j - another index
k - whatever you need it for
l - god that looks too much like a 1. Burn it! I took to writing a script l when I was writing the natural log function.
m - used like n
n - some sort of natural number, generally the term in the sequence worth looking at or the last term of a summation. Frequently shows up in Number Theory.
o - used to show how fast something is growing. Otherwise avoided because of its resemblance to 0
p- prime number
q- prime number
r - radius, also used as an exponent in some sequences and series
s - I honestly can't remember. Capital S gets used for spheres. My s always looks too much like a 5
t - ditto to s, except I taught myself to put a tail on my t so that I could differentiate them from my +. Capital T gets used for Topological spaces. I guess you can use it for time, if you're into that sort of thing.
u - used as a placeholder for a larger expression (on the page) I seem to remember it cropping up a lot in Complex Analysis and when doing integration by parts
v- ditto u
w - variable for the fourth dimension (whoooo....)
x, y, z - variables for the common 3 dimensions, also used as general variables

Like everything on this list, the h and k are arbitrary, however, I suspect they are used because a great deal of other letters have preexisting usages.

Disclaimer- these are what were used at the college I went too, which I am 10 years removed from. My memory could be faulty/my school could have been idiosyncratic in this. Also these are valid only for mathematics. Physicists probably butcher everything sensible about using letters.

And don't get me started on when we start borrowing from other alphabets.

On Preview: Wow, this list looks ridiculous.
posted by Hactar at 2:54 PM on April 17, 2014 [12 favorites]

So, this is a really nice question. I have taught algebra, and I agree that (h,k) is often used for the center of a circle, and I can't think of any particular reasons why.

• x, y, z are continuous variables, except when they're not. But don't be trying to do something like sum as x goes from 1 to n. That's just wrong.
• In graph theory, G, H, J are graphs; e,f are edges, u,v,w are vertices; D is a digraph, P is a path, T is a tree. And graphs have n vertices.
• i, j, k are iterators
• m,n are integers, mostly positive integers; a,b,c,d are probably integers too, unless they're coefficients of some polynomial
• theta and phi are angles, except when phi is the golden ratio (also sometimes tau)
• sigma and tau, and sometimes pi, are permutations, sometimes

posted by leahwrenn at 3:14 PM on April 17, 2014 [4 favorites]

s - often used as the variable to integrate when x or y appears in the limits of integration
t - often used much the same way as s; also frequently used in parametric equations and vector functions
posted by The Great Big Mulp at 6:29 AM on April 18, 2014

Hactar: "From what I can remember, here are some common uses for the Latin letters in math.

<snip>...

e - 2.7181.....

</snip>

Disclaimer- these are what were used at the college I went too, which I am 10 years removed from. My memory could be faulty/my school could have been idiosyncratic in this. Also these are valid only for mathematics. Physicists probably butcher everything sensible about using letters.

And don't get me started on when we start borrowing from other alphabets.

On Preview: Wow, this list looks ridiculous.
"

I'm surprised you didn't define e = lim (x->∞) (1 + 1/x)x, and that 5th digit is another attempt to get out of building that hyperspace junction box again, isn't it?
posted by Reverend John at 6:21 PM on April 18, 2014 [1 favorite]

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