# Elliptic Trig: Circles, Hyperbolas & Ellipses

• JyN
In summary, the conversation discusses the lack of trigonometric functions for elliptic geometry and the possibility of defining them for non-square hyperbolas and ellipses. It is suggested that these functions could be useful in an elliptical coordinate system. There are existing papers on this topic, with one example being Jacobi elliptic functions.

#### JyN

Why aren't there trigonometric functions for elliptic geometry? There is trigonometry for circles and hyperbolas, but why not ellipses?

Hi JyN!
JyN said:
There is trigonometry for circles and hyperbolas, but why not ellipses?

ah, there is trigonometry only for the square hyperbola (ie with perpendicular asymptotes), just as there is only for the "square" ellipse (ie the circle).

Why can't trigonometry-like relationships exist for non-square hyperbolas and ellipses?

i expect they can be, but why would anyone bother with them, when the "square" functions can easily be adapted for the purpose?

An angle in radians is defined as the length of the arc of a circle over its radius.

A slight problem occurs if you want to extend that to an ellipse because an ellipse is defined by two variables, major axis and minor axis. It would be interesting how to define an elliptic angle, length of arc of an ellipse over major axis or minor axis or their algebraic combination? But in either case, I don't know how this could be useful other than perhaps in an elliptical coordinate system.

JyN said:
Why aren't there trigonometric functions for elliptic geometry? There is trigonometry for circles and hyperbolas, but why not ellipses?

Is this what you had in mind? They're called Jacobi elliptic functions.

EDIT: Cross post!

## 1. What is the difference between a circle and an ellipse?

Both circles and ellipses are closed curves, but the main difference is that a circle has a constant distance from its center to any point on its circumference, while an ellipse has two distinct distances from its center to any point on its circumference.

## 2. How do you find the equation of an ellipse?

The general equation of an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse and a and b are the semi-major and semi-minor axes, respectively. To find the specific equation of an ellipse, you need to know its center, its major and minor axes lengths, and its orientation (horizontal or vertical).

## 3. What is the focus-directrix property of an ellipse?

The focus-directrix property of an ellipse states that for any point P on the ellipse, the distance from P to one focus is equal to its distance from the corresponding directrix. This property is useful in constructing an ellipse using a string and two pins.

## 4. How do you find the eccentricity of an ellipse?

The eccentricity of an ellipse is a measure of how "flat" or "round" the ellipse is. It is calculated by taking the ratio of the distance between the center and one focus to the length of the semi-major axis. The closer the eccentricity is to 0, the more circular the ellipse is. The closer it is to 1, the more elongated the ellipse is.

## 5. What are some real-life applications of ellipses?

Ellipses can be found in various natural and man-made objects, such as the orbits of planets around the sun, the shape of egg yolks, and the design of satellite dishes. They are also used in engineering and architecture, such as in the design of arches and bridges, to distribute weight evenly and reduce stress on the structure.