Elipse

Elipse

An illustration of an ellipse.

The ellipse is a geometric shape that is defined as a closed curve in which the sum of the distances from any point on the curve to two fixed points, called foci, is constant. It is a conic section, along with the circle, parabola, and hyperbola. The term "ellipse" is derived from the Greek word elleipsis, meaning "deficiency" or "falling short," referring to the property that the sum of the distances is always less than the major axis.

Properties

An ellipse has several important properties:

1. Major Axis: The longest diameter of an ellipse, which passes through both foci and is perpendicular to the minor axis. 2. Minor Axis: The shortest diameter of an ellipse, which is perpendicular to the major axis and passes through the center of the ellipse. 3. Foci: Two fixed points inside the ellipse, which are equidistant from the center. The sum of the distances from any point on the ellipse to the two foci is constant. 4. Eccentricity: A measure of how elongated an ellipse is. It is defined as the ratio of the distance between the foci to the length of the major axis. The eccentricity of an ellipse is always between 0 and 1, with a value of 0 indicating a circle and a value of 1 indicating a line segment.

Equations

The equation of an ellipse in the Cartesian coordinate system is given by:

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where (h, k) represents the coordinates of the center of the ellipse, and a and b represent the lengths of the semi-major and semi-minor axes, respectively.

Applications

Ellipses have various applications in different fields:

1. Astronomy: The orbits of planets and other celestial bodies around the sun are often elliptical. 2. Optics: Elliptical mirrors and lenses are used in optical systems to focus or shape light. 3. Engineering: Elliptical shapes are commonly used in the design of gears, bearings, and other mechanical components. 4. Architecture: Elliptical arches and domes are frequently used in the construction of buildings for their aesthetic appeal and structural stability.

See Also

• Circle: A special case of an ellipse with equal major and minor axes.
• Parabola: Another conic section that has a single focus and a directrix.
• Hyperbola: A conic section that has two foci and two separate branches.