# Seeking an "Atlas of Mathematical Curves" **URL:** \ **Category:** Factual Questions\ **Created:** [December 28, 2015, 8:37pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592 "2015-12-28T20:37:37Z")\ **Posts on this page:** 14\ **Page:** 1
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [December 28, 2015, 8:37pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/1 "2015-12-28T20:37:37Z")
I want a picture book of what zillions of different mathematical curves look like. OK, OK, everybody get in line and take turns explaining to me why this is a dumb thing to want. Yes, of course it depends on too many considerations. Really, there’s a legitimate need. I often have to model data empirically, fitting smooth simple curves to groups of points. If there’s an a priori reason to expect a particular form of curve, that’s sweet, but if there isn’t, I start wondering if an exponential, or a rational polynomial, or some hyperbolic or trig function, or what else, would work. And I maintain that many of us have at least a little ability to distinguish between for example exponential looking curves versus polynomial looking curves. There are broad families. Sometimes I definitely want a function that increases monotonically from zero to one as its argument increases from zero to one, but what function? Or other times, I want something that in some transform is asymptotic to two different straight lines in the extremes, and yet is a nice fit in that ticklish transitional region between them. People who can recognize kinds of functions by their curves are at a great advantage in this. I’m looking for something that helps do this. Anybody know of such a thing? ---
**Author:** ![Dorjan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dorjan/32/12440_2.png) [@Dorjan](https://boards.straightdope.com/u/Dorjan)\ **Post date:** [December 28, 2015, 9:18pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/2 "2015-12-28T21:18:03Z")
Curves are pretty well defined, you can even find a handy list of them and the math on wikipedia: > **[List of curves](https://en.wikipedia.org/wiki/List_of_curves)** > > This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), physics, engineering, economics, medicine, biology, psychology, ecology, etc. > Rational curves are subdivided according to the degree of the polynomial. > Plane curves of degree 2 are known as conics or conic sections and include > Cubic plane curves include Not sure why you would need a picture book specifically when just about any math program can generate curves on request? ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [December 28, 2015, 9:24pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/3 "2015-12-28T21:24:12Z")
**Napier** , I do hope you already know what y=ln(x) looks like? ---
**Author:** ![Stranger\_On\_A\_Train](https://avatars.discourse-cdn.com/v4/letter/s/13edae/32.png) [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)\ **Post date:** [December 28, 2015, 9:54pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/4 "2015-12-28T21:54:16Z")
Here is the [_NIST Handbook of Mathematical functions_](http://www.amazon.com/Handbook-Mathematical-Functions-Paperback-CD-ROM/dp/0521140633) and [Springer’s _An Atlas of Functions_](http://www.amazon.com/An-Atlas-Functions-Function-Calculator/dp/0387488065). Abramowitz’ [_Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables_](http://www.amazon.com/Handbook-Mathematical-Functions-Formulas-Graphs/dp/161427617X) is the old standby although there hasn’t been a new version in decades; both this and the Dover edition are just printings of the 1964 editions. However, I’m not really sure that this is going to be of fundamental use to you. The shape of many curves, and especially anything more complex than the conic and hyperbolic curves are fairly arbitrary, depending on parameters. For instance, although a real polynomial will have as many reversals as it has exponents minus one. For any given set of n points there are an infinite number of polynomials of n-1 exponents which can be fit to them, notwithstanding higher order polynomials or other functions. A fit should represent some anticipated behavior in a system, e.g. a linear or logarithmic change, or orbital motion in a conic, or somesuch. Overfitting by assigning arbitrary functions often masks behavior and can lead to erroneous values in interpolation and (especially) extrapolation. There are times, such as with controls, that you’ll specifically pick a polynomial of a particular order to fit to and then use a least squares approach to find the “best” polynomial in the range that you intend to operate in (e.g. one that can be most easily made piecewise linear). If you are just fitting data mapped to a surface, piecewise interpolation to a spline is probably the best approach rather than trying to fit to a specific global function. If you do want some kind of arbitrary data fitting there are a wide variety of different packages and tools to do so for both commercial and open source applications and frameworks. _Mathematica_/Wolfram, the Matlab Curve Fitting and Optimization toolboxes, and scipy.optimize. The GNU project has a number of different regression and fitting tools. All of these will be of much more utility in trying to find a “best” fit than looking through a book of curves and truing to discern a suitable function. > [@Riemann](#): > > **Napier** , I do hope you already know what y=ln(x) looks like? _golf clap_ Stranger ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [December 29, 2015, 3:11am UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/5 "2015-12-29T03:11:35Z")
> [@Riemann](#): > > **Napier** , I do hope you already know what y=ln(x) looks like? Sure, kind of like a zeta function without all those zeros. ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [December 29, 2015, 3:20am UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/6 "2015-12-29T03:20:14Z")
> [@Dorjan](#): > > Curves are pretty well defined, you can even find a handy list of them and the math on wikipedia: > > [List of curves - Wikipedia](https://en.wikipedia.org/wiki/List_of_curves) > Not sure why you would need a picture book specifically when just about any math program can generate curves on request? Hey, this looks useful! Especially the gallery of curves linked near the top! I had what looked like one branch of the swastica curve to model today and now I have a form to apply. Math programs can generate curves on request, but the need is to know what to ask for. When I’m looking at a scatter plot and wondering what form of function could imitate it well, the math programs aren’t helpful. ---
**Author:** ![Typo\_Knig](https://avatars.discourse-cdn.com/v4/letter/t/cdc98d/32.png) [@Typo\_Knig](https://boards.straightdope.com/u/Typo_Knig)\ **Post date:** [December 29, 2015, 3:33am UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/7 "2015-12-29T03:33:44Z")
If you want to fit points to a function, and you don’t care what the underlying function is, try spline interpolation [Spline interpolation - Wikipedia](https://en.wikipedia.org/wiki/Spline_interpolation?wprov=sfti1) ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [December 29, 2015, 3:45am UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/8 "2015-12-29T03:45:10Z")
> [@Napier](#): > > Sure, kind of like a zeta function without all those zeros. I think I set myself up for a fall with this username. Maybe I called have pulled off Pythagoras, or Unknown\_Babylonian or something. ---
**Author:** ![Richox](https://avatars.discourse-cdn.com/v4/letter/r/8797f3/32.png) [@Richox](https://boards.straightdope.com/u/Richox)\ **Post date:** [December 29, 2015, 4:14am UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/9 "2015-12-29T04:14:32Z")
> [@Napier](#): > > Math programs can generate curves on request, but the need is to know what to ask for. When I’m looking at a scatter plot and wondering what form of function could imitate it well, the math programs aren’t helpful. I haven’t been at university (college) for awhile, but as i recall the mathematical programs and/or scripts Stranger mentioned (and others generally) will both plot the data, then identify the most appropriate curve or curves based on a variety of regression analysis tools. So they should actually perform the step you hope to do by a visual comparison. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [December 29, 2015, 2:26pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/10 "2015-12-29T14:26:45Z")
Geez, the stuff people are citing here put my old _CRC Handbook of Mathematics_ section on curves to shame. ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [December 29, 2015, 3:53pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/11 "2015-12-29T15:53:20Z")
Abramowitz and Stegun’s _Handbook of Mathematical Functions_ and the _CRC Handbook of Chemistry and Physics_ have already been mentioned. There’s also J. Dennis Lawrence’s \* A Handbook of Special Plane Curves\*, published by Dover, and now in print for over 40 years: > **[A Catalog of Special Plane Curves](https://store.doverpublications.com/0486602885.html)** > > One of the largest and finest available collections, this catalog covers general properties of curves and types of derived curves. Illustrated by nearly 90 images from a CalComp digital incremental plotter. 1972 edition. There’s a lot in this book, but there’s also a lot to confuse the non-mathematical reader, and it’s not all adequately explained. ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [December 29, 2015, 9:52pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/12 "2015-12-29T21:52:26Z")
> [@Riemann](#): > > I think I set myself up for a fall with this username. Maybe I called have pulled off Pythagoras, or Unknown\_Babylonian or something. “Unknown\_Babylonian”? Seriously? Isn’t that kinda putting Descartes before the Horus? ---
**Author:** ![bonzer](https://avatars.discourse-cdn.com/v4/letter/b/45deac/32.png) [@bonzer](https://boards.straightdope.com/u/bonzer)\ **Post date:** [December 29, 2015, 10:37pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/13 "2015-12-29T22:37:55Z")
Not quite sure why, but I always preferred [Jahnke and Emde](http://blogs.mathworks.comA/cleve/2014/12/15/jahnke-and-emde-revisited/)’s graphs to those in Abramowitz and Stegun. Judging by the opening credits of _Pi_, I suspect Darren Aronofsky agrees. ---
**Author:** ![Hermitian](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hermitian/32/470_2.png) [@Hermitian](https://boards.straightdope.com/u/Hermitian)\ **Post date:** [December 30, 2015, 4:14pm UTC](https://boards.straightdope.com/t/seeking-an-atlas-of-mathematical-curves/741592/14 "2015-12-30T16:14:51Z")
> [@Dorjan](#): > > Curves are pretty well defined, you can even find a handy list of them and the math on wikipedia: > > [List of curves - Wikipedia](https://en.wikipedia.org/wiki/List_of_curves) I didn’t see on that list the [downward facing dog yoga pose curve](https://www.wolframalpha.com/input/?i=downward-facing+dog+yoga+pose+curve&lk=3) That is a major oversight.