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