Web links to power-function definition & articles

[RogerOwens]

Lest anyone think that I made up the term "power-function" and its definition, let me post these links to web articles that define "power-function":

https://en.wikipedia.org/wiki/Power_function
http://study.com/academy/lesson/what-is ... raphs.html
http://wmueller.com/precalculus/families/1_41.html
http://www.biology.arizona.edu/biomath/ ... asics.html
https://www.brightstorm.com/math/precal ... functions/

Michael Ossipoff

[daan]

Power functions are normal and common. What’s nonstandard is your application to maps. The reason you’re having trouble with “quartic”, for example, is that your application of power function is unrelated to how the literature has evolved. The quartic authalic means the meridians consist of sections of a quartic curve. Craster parabolic means the meridians are sections of parabolas.

— daan

[RogerOwens]

Power functions are normal and common. What’s nonstandard is your application to maps.

Of course. If I'd heard of power-function maps other than the Craster Parabolic then I wouldn't be proposing that they be tried.

The reason you’re having trouble with “quartic”, for example, is that your application of power function is unrelated to how the literature has evolved.

At first I assumed that the Quartic map had x as a quartic power-function of y. Of course that isn't so. For one thing, such a map's outer meridian would be quite flat at the equator.

(Not that that's necessarily a bad thing--Not having seen that map, I don't know that it isn't something that people would like.)

So, alright, then that would mean that the Quartic must have x as a quartic polynomial function of y. ...or maybe vice-versa.

The quartic authalic means the meridians consist of sections of a quartic curve.

What else would it mean? That's what I'd assumed it to mean.

But, when, from Snyder's formulas for the Quartic Authalic, I wrote x as a function of y, what I got was a quadratic function divided by the square-root of another quadratic function. Not a quartic function.

Then I wrote y as a function of x, and, again, the answer I got wasn't a quartic function.

But, because so many people say that the Quartic Authalic has meridians that are quartic curves, with x and y related by a quartic polynomial, then I'd better re-check my results for x(y) and y(x) before I say otherwise.

Craster parabolic means the meridians are sections of parabolas.

Of course. ...And the Craster Parabolic formulas that were in the Album of Map Projections a few days ago were for a quadratic power-function.

Michael Ossipoff

[daan]

It’s not a problem of writing x in terms of y, or vice versa. A quartic curve is any combination of whole powers of x and y, up to and necessarily including 4th power. Meanwhile you can eliminate the roots you’re getting by judicious algebraic manipulations, resulting in an expression for a meridian that meets the criteria for a quartic curve.

— daan

[RogerOwens]

It’s not a problem of writing x in terms of y, or vice versa. A quartic curve is any combination of whole powers of x and y, up to and necessarily including 4th power. Meanwhile you can eliminate the roots you’re getting by judicious algebraic manipulations, resulting in an expression for a meridian that meets the criteria for a quartic curve.

— daan

Ok, thanks; that explains it. I'd thought they meant that x was a quartic function of y. (or maybe vice-versa). When they said "quartic equation", I misread it as "quartic function".

Michael Ossipoff