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## Re: Seven math curves

 Werner Exinger написал(а): Or use more than one focal point. What about a curve, where the sum of the distances from three points is constant. Regards WE

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## Re: Seven math curves

Nice find!

I vaguely remember having heard (a long time ago) from a Tschirnhaus-curve (but not sure - I may mix up things and be wrong about this).

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## Re: Seven math curves

 Werner Exinger написал(а): Why be content with seven when you can have eleven or more ? Regards WE

We have 7 operators in Mathcad Highlighted

## Re: Seven math curves

1. black+blue Highlighted

## Re: Seven math curves

2. |black-bleu| Highlighted

## Re: Seven math curves

3. black * blue = const (ovals Cassin) Highlighted

## Re: Seven math curves

Nice worksheets.

I see that you, too, determine the limits of the radius L by hand (trial and error, I guess).

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## Re: Seven math curves

4. black / blue = const (Apolinarian circle) Highlighted

## Re: Seven math curves

5. black^blue=const Highlighted

## Re: Seven math curves

6. log(black,blue) Highlighted

## Re: Seven math curves

7. blue-root of black Highlighted

## Re: Seven math curves

7. blue-root of black

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## Re: Seven math curves

7. blue-root of black - with new focuses and with two branches

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## Re: Seven math curves

6. log(black,blue)

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## Re: Seven math curves

With two branches: one log(Black, Blue) and others log(Blue, Black):

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## Re: Seven math curves

> 5. black^blue=const

Nice, but the unit "m^m" is still wrong 😉

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## Re: Seven math curves

 Werner Exinger написал(а): > 5. black^blue=const Nice, but the unit "m^m" is still wrong 😉 What about the "???"

Why wrong?

I'm going to do at my dacha for a bed of flowers in the form of this oval with a physical characteristic constant 7 (m^m), not 6 (ft^m) or 5 (m^ft)!

> ??? - I do not know what is this little close curve Highlighted

## Re: Seven math curves

> Why wrong?

We had that discussion already in the other thread, so I won't repeat it here. Just one thought: (2m)^(3m) = 2^(3m) * m^(3m) = 8^m * (m^3)^m. So its NOT 8 m^m . Now do the same game with different numbers!

> I'm going to do at my dacha for a bed of flowers in the form of this oval

Why not, if a simple ellipse is not challenging  enough 😉

> with a physical characteristic

Sorry, but its no physical characteristics and its a characteristic dealing with the numbers only, not the units. But it sure is a nice leisure activity anyway.

> > ??? - I do not know what is this little close curve

Obviously, if you did not make a mistake,  its part of the solution curve. Just give it a try and see, if black^blue still is 4.

EDIT: There is a mistake: If L<1, then L2 would be negative and no solution exists!! But your "Coord" yields solution because L2 (as radius) is squared in the circle equation. You have to take care of this and simply let L run from1 upwards (L=1 would provoke a divison by zero, of course which could be dealt with an on error statement). Highlighted

## Re: Seven math curves

 Werner Exinger написал(а): > Why wrong? We had that discussion already in the other thread, so I won't repeat it here. Just one thought: (2m)^(3m) = 2^(3m) * m^(3m) = 8^m * (m^3)^m. So its NOT 8 m^m . Now do the same game with different numbers! > I'm going to do at my dacha for a bed of flowers in the form of this oval Why not, if a simple ellipse is not challenging  enough 😉 > with a physical characteristic Sorry, but its no physical characteristics and its a characteristic dealing with the numbers only, not the units. But it sure is a nice leisure activity anyway.

My friend in USA would like to create same bed of flowers (клумб с цветами) in his garten. He has a rule not with meter and cm but with feet and inch. I send to him the coordinate of 2 points (focuses) with feet and inch and the constant a with.... What numeric value and UNIT must I send in America? Or I must send the rule with meter and cm scale?

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## Re: Seven math curves

Better send him the ruler in metric units as your flowerbed does not scale (the property a^b=const. does not scale). And don't forget to tell him that he should only use the numbers taken w/o units when he calculates the powers 😉

You haven't answered (2m)^(3m)=? and now (6,562 ft)^( 9,843 ft)=?

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## Re: Seven math curves

Here are some curves which should scale. Simply stay with basic commutative operations + and * and simply use more points:

Here the points whose sum of distance from five points is constant: Highlighted

## Re: Seven math curves

 Werner Exinger написал(а): Here are some curves which should scale. Simply stay with basic commutative operations + and * and simply use more points: Here the points whose sum of distance from five points is constant: Animation:

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## Re: Seven math curves

And here is the same with constant product of distances Guess I have finished playing around with those curves. Can't think of a way to derive a closed parametric equation for those beasts.

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## Re: Seven math curves

Please do one more with 7 points - 7 is one fine number!

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## Re: Seven math curves

 Werner Exinger написал(а): And here is the same with constant product of distances Animation

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## Re: Seven math curves

 Werner Exinger написал(а): And here is the same with constant product of distances Guess I have finished playing around with those curves. Can't think of a way to derive a closed parametric equation for those beasts.

It is a lemniscate with 5 focuses - see please Лемниската — Википедия

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## Re: Seven math curves

The discription is not available in the English or German version of Wikipedia.

I am not sure if all those curves should be called lemniscates.

In the two focal point case we speak of Cassini ovals and only if the constant product is the square of half of the distance of the two focals, we speak of a (Bernoulli) lemniscate. So the two focal lemniscate is just a special case of the Cassini curves.

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## Re: Seven math curves

5. black^blue=const

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## Re: Seven math curves

Video Link : 7034 4. black / blue = const (Apolinarian circle)

Animation

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## Re: Seven math curves

Bicycle Apollinaire - two branches: Black/Blue and Blue/Black: 