Linear eccentricity is a fundamental geometric property of conic sections, specifically ellipses and hyperbolas. It is defined as the distance of the foci to the center of the conic section, and it is also commonly referred to as the focal distance.
This concept is crucial for understanding the precise shape and orientation of these curves.
Understanding Linear Eccentricity
As per the provided reference, "The distance of the foci to the center is called the focal distance or linear eccentricity." This value helps in precisely locating the foci relative to the center of the conic section.
- Symbol: Linear eccentricity is typically denoted by the letter c.
- Context in Ellipses: For an ellipse, the line passing through the two foci and the center is known as the major axis. This axis intersects the ellipse at its two vertices, which are at a distance a (the semi-major axis length) from the center. The linear eccentricity c measures how far each focus is displaced from the center along this major axis.
- Context in Hyperbolas: Similarly, for a hyperbola, the foci are located along the transverse axis, and c represents the distance from the center to each focus.
Linear Eccentricity vs. Eccentricity
It is vital to distinguish linear eccentricity (c) from the general eccentricity (e), though they are closely related.
- Linear Eccentricity (c): This is an absolute distance, measured in units of length (e.g., meters, centimeters, units). It tells you how far the foci are from the center.
- Eccentricity (e): This is a dimensionless ratio, calculated as e = c/a, where a is the semi-major axis (for ellipses and hyperbolas). Eccentricity e describes the "roundness" or "openness" of the conic section.
- For a circle, e = 0 (since c = 0, meaning the foci coincide with the center).
- For an ellipse, 0 < e < 1. The closer e is to 0, the more circular the ellipse.
- For a parabola, e = 1.
- For a hyperbola, e > 1.
The relationship between linear eccentricity (c), the semi-major axis (a), and the semi-minor axis (b) for an ellipse is given by the formula:
c² = a² - b²
And for a hyperbola:
c² = a² + b²
These formulas highlight how linear eccentricity is intrinsically linked to the dimensions defining the shape of these conic sections.
