Hyperbola transformed to a canonical form

[ValeryOchkov]

I bring the curve of the second order (hyperbola - red curve) to the canonical form. Why don't my starting blue points end up on a hyperbola transformed to a canonical form. Help me please!

Mathcad 15 in attach.

[Werner_E]

1) You used X instead of X1 in yor second plot

2) The center of the original hyperbola is not in the origin, so you have to translate X and Y by x.o and y0 before you rotate them

3) You angle phi is wrong. Here you see the original hyperbola (matrix H) translated by (x.0; y.0) and rotated by your angle phi

The correct angle to rotate is not 42.659° but rather 90° - 42.659° = 47.341°.

Using all these corrections we get

[ValeryOchkov]

Thanks, Werner!

Send please the sheet!

[Werner_E]

---

@ValeryOchkov wrote:

Thanks, Werner!

Send please the sheet!

---

Sorry, I did not save and keep the modified sheet.

But only very few changes are necessary and they can easily be taken from the pictures, I think.

• Simply add the angle to 90°,
• add the displacement at the calculation of X1 and Y1 as shown
• and use X1 instead of X in the plot.

That is all that is necessary.

[Werner_E · Accepted solution]

Instead of changing the value of the angle or using (pi/2-phi) instead of phi you may also just exchange sin and cos in your two rotation equations.

I also wondered why you used the "root" function to calculate phi instead of a direct calculation

This time the sheet is attached 😉

[Werner_E]

Some additional remarks:

You could get the coefficients for your implicit equation more efficient by using the symbolic "coeff" command so you can avoid copy&paste:

And while I think that implicitplot2D() is a most valuable tool, you don't need it to plot hyperbolas as you could use a parameter equation

[ValeryOchkov]

Dummy is x^3.

[Werner_E]

---

@ValeryOchkov wrote:

Dummy is x^3.

---

No, "dummy" is the coefficient of x^5 which would correspond to (non-existent) x^2*y in the original expression.

[ValeryOchkov]

Yes! With dynamic: sh(t), ch(t)...

[ValeryOchkov]

[ValeryOchkov]

[ValeryOchkov]

Origin seven (five) points (the Asteroid position in time), our blue planet, two branches of hyperbola, two focus, two asymptotes, two directrices and... new method  of hyperbola transforming to a canonical form. Or it is one old method?  

[Werner_E]

Sorry, I don't understand your question.

[ValeryOchkov]

Is it one new method of transforming one curve of 2-d order in canonical form. With using one Math package?

[Werner_E]

I guess its not really new but rather a bit different from the usual way of using matrices, eigenvalues, etc. But the basic math is quite similar, I suppose.

[ValeryOchkov]

Many students are horrified by words eigenvalues, etc. And so - "not aesthetically pleasing, but cheap, reliable and practical" using the numerical solution of systems of equations.

SMath

(67) Лёлик, но это же не эстетично! Зато дёшего, надёжно и практично! - YouTube

[Werner_E]

---

@ValeryOchkov wrote:

Many students are horrified by words eigenvalues, etc. And so - "not aesthetically pleasing, but cheap, reliable and practical" using the numerical solution of systems of equations.

---

In other words .. KISS 😄

[ValeryOchkov]

One article about it (MathCAD)

Методика приведения уравнений кривых и поверхностей второго порядка к каноническому виду с применением среды MathCAD (e-koncept.ru)